Properties of Elementary Functions · CAS Activity 2-5a: Dilations of Logarithmic Functions Chapter...

63

If a mother mouse is twice as long as her o� spring, then the mother’s weight is about eight times the baby’s weight. But the mother mouse’s skin area is only about four times the baby’s skin area. So the baby mouse must eat more than the mother mouse in proportion to its body weight to make up for the heat loss through its skin. In this chapter you’ll learn how to use functions to model and explain situations like this.

Properties of Elementary FunctionsProperties of Elementary

222222

y

x

63

CHAP TE R O B J EC TIV ES

• Discover patterns in the graphs of linear, quadratic, power, and exponential functions.

• Given the graph of a function, know whether the function is exponential, power, quadratic, or linear and fi nd the particular equation algebraically.

• Given a set of regularly spaced x-values and the corresponding y-values, identify which type of function they fi t (linear, quadratic, power, or exponential).

• Find other function values without necessarily fi nding the particular equation.

• Learn the properties of base-10 logarithms.

• Use logarithms with base 10 or other bases to solve exponential or logarithmic equations.

• Show that logarithmic functions have the multiply–add property, and fi nd particular equations algebraically.

• Fit a logistic function to data for restrained growth.

63A Chapter 2 Interleaf: Properties of Elementary Functions

OverviewIn Chapter 2, students learn to tell which kind of function might fi t a given set of data by recognizing fi rst the geometric pattern of the graph and then the numerical pattern revealed by regularly spaced points. Th e numerical patterns include the add-multiply property for exponential functions and the multiply-multiply property for power functions. As they study logarithms, students learn that logarithmic functions have the multiply-add property. Students learn a verbal way to remember the defi nition of logarithm, namely that a logarithm is an exponent. Natural logarithms and common logarithms are presented. Th e chapter concludes with the modeling of restrained population growth with the logistic function.

Using This ChapterAt this point you may prefer to postpone Chapters 2–4 and skip to the study of periodic functions in Chapters 5–9. In this chapter students learn about graphical and numerical patterns associated with linear, exponential, and power functions. Th e logarithmic and logistic functions, important topics for calculus, are also studied. Chapter 2 sets the stage for Chapter 3, where functions are examined using regression to fi nd models for data. Section 2-3, Identifying Functions from Numerical Patterns, can be omitted without creating challenges later on for you and your students. If students will not continue on to calculus, then Section 2-7, Logistic Functions for Restrained Growth, may also be omitted.

Teaching ResourcesExplorationsExploration 2-2: Graphical Patterns in FunctionsExploration 2-3: Patterns for Quadratic Functions Exploration 2-3a: Numerical Patterns in Function ValuesExploration 2-3b: Equations from Given Values PracticeExploration 2-4: Introduction to LogarithmsExploration 2-4a: Introduction to Logarithmic FunctionsExploration 2-7: Th e Logistic Function for Population GrowthExploration 2-8a: Rehearsal for Chapter 2 Test

Blackline MastersSections 2-7 to 2-8

Supplementary ProblemsSections 2-2, 2-3, and 2-5 to 2-7

Assessment Resources Test 4, Sections 2-1 to 2-3, Forms A and BTest 5, Sections 2-4 to 2-7, Forms A and BTest 6, Chapter 2, Forms A and B

Technology ResourcesSketchpad Presentation SketchesLogistic Present.gsp

ActivitiesFathom: Moore’s LawFathom: Population GrowthCAS Activity 2-4a: Dilations of Exponential FunctionsCAS Activity 2-5a: Dilations of Logarithmic Functions

Properties of Elementary FunctionsProperties of Elementary FunctionsC h a p t e r 2

Chapter 2 Interleaf 63B

Standard Schedule Pacing Guide

Block Schedule Pacing Guide

Day Section Suggested Assignment

1 2-1 Shapes of Function Graphs 1–4

22-2 Identifying Functions from Graphical

Patterns

RA, Q1–Q10, 1–25 odd

3Optional 2–10 even, 14–24 even

42-3 Identifying Functions from Numerical

PatternsRA, Q1–Q10, 1–23 odd

5 25–27, 29–32, 35

6 2-4 Properties of Logarithms RA, Q1–Q10, 1–47 odd

7 2-5 Logarithms: Equations and Other Bases RA, Q1–Q10, 1, 2, 3–49 odd

8 2-6 Logarithmic Functions RA, Q1–Q10, 1–13 odd, 14

9 2-7 Logistic Functions for Restrained Growth RA, Q1–Q10, 1, 3–5, 7

102-8 Chapter Review and Test

R0–R7, T1–T28

11 Problem Set 3-1

Day Section Suggested Assignment

1 2-2 Identifying Functions from Graphical Patterns RA, Q1–Q10, 1–15

2 2-3 Identifying Functions from Numerical Patterns RA, Q1–Q10, 1–24 multiples of 3, 25, 30, 33

3 2-4 Properties of Logarithms RA, Q1–Q10, 1-47 odd

4 2-5 Logarithms: Equations and Other Bases RA, Q1–Q10, 1–7 odd, 15–37 odd, 41

52-6 Logarithmic Functions RA, Q1–Q10, 1–9 odd

2-7 Logistic Functions for Restrained Growth RA, 1, 3

62-7 Logistic Functions for Restrained Growth Q1–Q10, 5, 7

2-8 Chapter Review and Test R0–R7, T1–T28

72-8 Chapter Review and Test

3-1 Introduction to Regression for Linear Data 1–6

65Section 2-1: Shapes of Function Graphs

Shapes of Function GraphsIn this chapter you’ll learn ways to � nd a function to � t a real-world situation when the type of function has not been given. You will start by refreshing your memory about graphs of functions you studied in Chapter 1.

Discover patterns in the graphs of linear, quadratic, power, and exponential functions.

Shapes of Function GraphsIn this chapter you’ll learn ways to � nd a function to � t a real-world situation

2-1


Objective

1. Exponential Function Problem: In the exponential function f (x) 0.2 2 x , f (x) could be the number of thousands of bacteria in a culture as a function of time, x, in hours. Find f (x) for

and graph the function as in Figure 2-1b. � e number of bacteria is increasing as time goes on. How does the concavity of the graph tell you that the rate of growth is also increasing?

2. Power Function Problem: In the power function g (x) 0.1 x 3 , g (x) could be the weight in pounds of a snake that is xfor each foot from 0 through 6, and graph function g. Because the graphs of f 1 and gconcave up, what graphical evidence could you use to distinguish between the two types of functions? Is the following statement true or false? “� e snake’s weight increases by the same amount for each foot it increases in length.” Give evidence to support your answer.

3. Quadratic Function Problem: In the quadratic function q(x) 0.3 x 2 8x 7, q(x) could measure the approximate sales of a new product in the xth week since the product was

from 0 through 30, and graph function q. Which way is the concave side of the graph, up or down? What feature does the quadratic function graph have that neither the exponential function graph

4. Linear Function Problem: In the linear function h(x) 5x 27, h(x) could equal the number of cents you pay for a telephone call of length x, in from 0 through 18, and graph function h. What does the fact that the graph is neither concave up nor concave down tell you about the cents per minute you pay for the call?

Exploratory Problem Set 2-1

Figure 2-1b

Figure 2-1a shows the plot of points that are values of the exponential function f (x) 0.2 2 x . You can make such a plot by storing the x-values in one list and the f (x)-values in another and then using the statistics plot feature on your grapher. Figure 2-1b shows that the graph of f contains all the points in the plot. The concave side of the graph is up.

5

1 2 3 4 5

f (x)

x

5f (x)-intercept

Concaveup

1 2 3 4 5

f (x)

x

Figure 2-1a

64 Chapter 2: Properties of Elementary Functions

In this chapter you will extend what you have already learned about some of the more familiar functions in algebra, as well as some you may not yet have encountered. � ese functions are

You will study these functions in four ways.

You can de� ne each of these functions algebraically; for example, the logarithmic function is de� ned

y lo g b x if and only if b y x

You can � nd interesting numerical relationships between the values of variables x and yproperty: As a result of adding a constant to x, the corresponding y-value is multiplied by a constant.

y

Popu

latio

n

Time

Logistic function

x

Exponential functions can describe unrestrained population grow th,

such as that of rabbits i f they have no natural enemies. Logistic

functions start out like exponential functions but then level of f.

Logistic functions can model restrained population grow th where

there is a maximum sustainable population in a certain region.

ALGEBRAICALLY

NUMERICALLY

GRAPHICALLY

VERBALLY

In this chapter you will extend what you have already learned about

Mathematical Overview


both involve an exponent. In Problem 1, emphasize that in an exponential function the variable is the exponent, whereas in a power function the variable is the base. In addition, you may want to discuss these questions with your class.a. Th e f (x)-intercept is f (0), the value of

f (x) when x 5 0. What is the graphical signifi cance of the f (x)-intercept? Why does the f (x)-intercept equal 0.2, not 0, for this function?

b. Th e word concave means “hollowed out.” Th e words cave and cavity have the same origin. Why do you suppose the graph of f is said to be concave up?

c. By calculation, f (21) 5 0.1. Why is f (21) not meaningful in this real-world situation?

d. Why do you suppose the function in this problem is called an exponential function?

S e c t i o n 2-1S e c t i o n 2-1S e c t i o n 2-1S e c t i o n 2-1S e c t i o n 2-1S e c t i o n 2-1PL AN N I N G

Class Time 1 __ 2 day

Homework AssignmentProblems 1–4

TE ACH I N G

Important Terms and ConceptsConcavity (concave up, concave down)Exponential functionPower functionQuadratic functionLinear function

Section Notes

In this section, students investigate the shapes and features of the graphs of linear, quadratic, power, and exponential functions. You can assign this section for homework the day of the Chapter 1 test or as a group activity to be completed in class. No classroom discussion is needed before students begin the activity.

Diff erentiating Instruction• Pass out the list of Chapter 2

vocabulary, available at www.keypress.com/keyonline, for ELL students to look up and translate in their bilingual dictionaries.

• ELL students should do the Exploratory Problem Set in pairs. Th ey may need more time than other students. Let them write in their primary language, English, or a combination of the two.

PRO B LE M N OTES

Problems 1 and 2 present an exponential function and a power function, respectively. Students sometimes confuse these two types of functions because


Shapes of Function GraphsIn this chapter you’ll learn ways to � nd a function to � t a real-world situation when the type of function has not been given. You will start by refreshing your memory about graphs of functions you studied in Chapter 1.


Shapes of Function GraphsIn this chapter you’ll learn ways to � nd a function to � t a real-world situation

2-1


Objective

1. Exponential Function Problem: In the exponential function f (x) 0.2 2 x , f (x) could be the number of thousands of bacteria in a culture as a function of time, x, in hours. Find f (x) for

and graph the function as in Figure 2-1b. � e number of bacteria is increasing as time goes on. How does the concavity of the graph tell you that the rate of growth is also increasing?

2. Power Function Problem: In the power function g (x) 0.1 x 3 , g (x) could be the weight in pounds of a snake that is xfor each foot from 0 through 6, and graph function g. Because the graphs of f 1 and gconcave up, what graphical evidence could you use to distinguish between the two types of functions? Is the following statement true or false? “� e snake’s weight increases by the same amount for each foot it increases in length.” Give evidence to support your answer.

3. Quadratic Function Problem: In the quadratic function q(x) 0.3 x 2 8x 7, q(x) could measure the approximate sales of a new product in the xth week since the product was

from 0 through 30, and graph function q. Which way is the concave side of the graph, up or down? What feature does the quadratic function graph have that neither the exponential function graph

4. Linear Function Problem: In the linear function h(x) 5x 27, h(x) could equal the number of cents you pay for a telephone call of length x, in from 0 through 18, and graph function h. What does the fact that the graph is neither concave up nor concave down tell you about the cents per minute you pay for the call?

Exploratory Problem Set 2-1

Figure 2-1b

Figure 2-1a shows the plot of points that are values of the exponential function f (x) 0.2 2 x . You can make such a plot by storing the x-values in one list and the f (x)-values in another and then using the statistics plot feature on your grapher. Figure 2-1b shows that the graph of f contains all the points in the plot. The concave side of the graph is up.

5

1 2 3 4 5

f (x)

x

5f (x)-intercept

Concaveup

1 2 3 4 5

f (x)

x

Figure 2-1a


In this chapter you will extend what you have already learned about some of the more familiar functions in algebra, as well as some you may not yet have encountered. � ese functions are

You will study these functions in four ways.

You can de� ne each of these functions algebraically; for example, the logarithmic function is de� ned

y lo g b x if and only if b y x

You can � nd interesting numerical relationships between the values of variables x and yproperty: As a result of adding a constant to x, the corresponding y-value is multiplied by a constant.

y

Popu

latio

n

Time

Logistic function

x

Exponential functions can describe unrestrained population grow th,

such as that of rabbits i f they have no natural enemies. Logistic

functions start out like exponential functions but then level of f.

Logistic functions can model restrained population grow th where

there is a maximum sustainable population in a certain region.

ALGEBRAICALLY

NUMERICALLY

GRAPHICALLY

VERBALLY

In this chapter you will extend what you have already learned about

Mathematical Overview


In addition, you may want to discuss these questions with your class.a. Based on this mathematical model,

how much would you expect a 10-ft snake to weigh?

b. Why does the domain of g contain only nonnegative numbers?

c. Th e equation g (x) 5 0.1x3 has an exponent in it. Why do you suppose it is called a power function rather than an exponential function?

In Problem 3, it is important for students to recognize that the quadratic function, unlike an exponential function, has a vertex. In addition, you may want to discuss these questions with your class.a. Th e quadratic function has the square

of x in it. What other geometrical term do you suppose gives quadratic functions their name?

b. Describe verbally what the quadratic model indicates about the pattern through time in the sales of this product.

c. How do you interpret the fact that the quadratic model gives decimal answers for the number of items sold, even though the actual number of items sold must be an integer?

Problem 4 presents a linear function. Th e graph of a linear function is neither concave up nor concave down.

Additional CAS Problems

1. Explain algebraically why the change in weight in Problem 2 is a quadratic function when g (x) is cubic.

2. For f (x) 5 0.2 2 x , calculate f (x 1 1) 2 f (x) 5 0.2 2 x using your CAS. Verify by hand that the CAS results are not in error.

In Problem 1, the rate of growth could be seen as the change in f-values for consecutive x-values. Here, the change in f is exponential growth, so the rate of growth is increasing. Th is parallels algebraically the graphical concavity at the end of the problem.

Problem 2 asks students what graphical evidence distinguishes the power function graph from the exponential function graph in Problem 1. Make sure students realize

that the graph of the power function passes through the origin, whereas the graph of the exponential function does not. If you graph the two functions on the same axes, students will see that as x gets bigger, the exponential function grows faster than the power function. Th is is an important distinction between the two types of increasing functions.

See page 981 for answers to Problems 1–4 and CAS Problems 1 and 2.

67Section 2-2: Identifying Functions from Graphical Patterns

Quadratic FunctionsGeneral equation: y a x 2 bx c, where a ≠ 0; a, b, and c stand for constants; and the domain is all real numbers

Parent function: y x 2 , where the vertex is at the origin

Transformed function: y k a(x h ) 2 , called the vertex form, with vertex at (h, k). � e value k is the vertical translation, h is the horizontal translation, and a is the vertical dilation. Vertex form can also be written y k a(x h ) 2 , but expressing y explicitly in terms of x makes the equation easier to enter into your grapher.

Graphical properties: � e graph is a parabola (Greek for “along the path of a ball”), as shown in Figure 2-2b. � e graph is concave up if a 0 and concave down if a 0.

Figure 2-2b: Quadratic functions

Power FunctionsGeneral equation: y a x b , where a and b stand for nonzero constants. If b 0, then the domain can be all real numbers. If b 0, then the domain excludes x 0 to avoid division by zero. If b is not an integer, then the domain usually excludes negative numbers to avoid roots of negative numbers. � e domain is also restricted to nonnegative numbers in most applications.

Parent function: y x b

Verbally: For y a x b , “if b 0, then y varies directly with the bth power of x, or y is directly proportional to the bth power of x; if b 0, then y varies inversely with the bth power of x, or y is inversely proportional to the bth power of x.” � e dilation factor a is the proportionality constant.

Translated function: y d a(x c ) b , where c and d are the horizontal and vertical translation, respectively. Compare the translated form with

y y 1 a x x 1 for linear functions

y k a(x h ) 2 for quadratic functions

Unless otherwise stated, “power function” will imply the untranslated form, y a x b .

y

x

5

5

5

Parentquadraticfunctiony x2

5

y 3 0.4(x 1)2

y

x

5

�5

5�5

Vertex (1, 3)

y

x

5

5

y 3 2(x 1)2

5

Vertex (1, 3)

5

� e eruption of Arenal, an active volcano in Costa Rica. � e lava particles follow a parabolic path due to gravitational force.


Identifying Functions from Graphical PatternsOne way to tell what type of function � ts a set of points is by recognizing the properties of the graph of the function.

Given the graph of a function, know whether the function is exponential, power, quadratic, or linear and � nd the particular equation algebraically.

Here is a brief review of the basic functions used in modeling. Some of these appeared in Chapter 1.

Linear and Constant FunctionsGeneral equation: y ax b (o� en written y mx b), where a (or m) and b stand for constants and the domain is all real numbers. � is equation is in the slope-intercept form because a (or m) gives the slope and b gives the y-intercept. If a 0, then y b; this is a constant function.

Parent function: y x

Transformed function: y y 1 a x x 1 , called the point-slope form because the graph contains the point x 1 , y 1 and has slope a. � e slope, a, is the vertical dilation; y 1 is the vertical translation; and x 1 is the horizontal translation. Note that point-slope form can also be written y y 1 a x x 1 , where the coordinates of the � xed point x 1 , y 1 both appear with a sign. � e form y y 1 a x x 1 expresses y explicitly in terms of x and thus is easier to enter into your grapher.

Graphical properties: � e graph is a straight line. � e parent function is shown in the le� graph of Figure 2-2a, the slope-intercept form is shown in the middle graph, and the point-slope form is shown in the right graph.

Verbally: For slope-intercept form: “Start at b on the y-axis, run x, and rise ax.” For point-slope form: “Start at x 1 , y 1 , run x x 1 , and rise a x x 1 .”

Identifying Functions from Graphical Patterns

2-2


Objective

Figure 2-2a: Linear functions

y

x

5

10

5 10Parent functiony x

y

x

5

10

5 10

y-intercept 8Run

Rise(negative)

Point (x, y)

y 0.7x 8

y

x

5

10

5 10

Point (1, 4)

Point (x, y)

Run

Rise

y 4 (x 1) 23


Section 2-2 summarizes the graphs and equations of linear, quadratic, power, and exponential functions. Here is more detailed information about each type of function.

Linear and Constant Functions

Th e name linear comes from the fact that the graph of a linear function is a straight line. Linear functions have no concavity and a constant slope.

Th e slope-intercept form of a linear function is given as y 5 ax 1 b. In this form, a is the slope of the line and b is the y-intercept. Many texts use m rather than a to represent the slope. Th e letter m comes from the French montant, (as in “mountain”), meaning “the rise.”

If a 5 0, the function y 5 ax 1 b becomes the constant function y 5 b. Th e graph of a constant function is a horizontal line.


Class Time1–2 days

Homework AssignmentDay 1: RA, Q1–Q10, Problems 1–25 oddDay 2: Problems 2–10 even, 14–24 even

Teaching Resources Exploration 2-2: Graphical Patterns in

FunctionsSupplementary Problems

Technology Resources

Exploration 2-2: Graphical Patterns in Functions

TE ACH I N G

Important Terms and ConceptsSlope-intercept formPoint-slope formSlopeVertexVertex formParabolaProportionality constantDirectly proportionalInversely proportionalBase-10 exponential functionNatural (base-e) exponential function

Section Notes

In Section 2-1, students plotted equations of exponential, power, quadratic, and linear functions and explored the features of the graphs. In this section, they are given graphs, and they use what they learned in Section 2-1 to identify the type of function each graph represents. Once they identify the type of function, they use information about points on the graph to fi nd a particular equation.


Quadratic FunctionsGeneral equation: y a x 2 bx c, where a ≠ 0; a, b, and c stand for constants; and the domain is all real numbers

Parent function: y x 2 , where the vertex is at the origin

Transformed function: y k a(x h ) 2 , called the vertex form, with vertex at (h, k). � e value k is the vertical translation, h is the horizontal translation, and a is the vertical dilation. Vertex form can also be written y k a(x h ) 2 , but expressing y explicitly in terms of x makes the equation easier to enter into your grapher.

Graphical properties: � e graph is a parabola (Greek for “along the path of a ball”), as shown in Figure 2-2b. � e graph is concave up if a 0 and concave down if a 0.

Figure 2-2b: Quadratic functions

Power FunctionsGeneral equation: y a x b , where a and b stand for nonzero constants. If b 0, then the domain can be all real numbers. If b 0, then the domain excludes x 0 to avoid division by zero. If b is not an integer, then the domain usually excludes negative numbers to avoid roots of negative numbers. � e domain is also restricted to nonnegative numbers in most applications.

Parent function: y x b

Verbally: For y a x b , “if b 0, then y varies directly with the bth power of x, or y is directly proportional to the bth power of x; if b 0, then y varies inversely with the bth power of x, or y is inversely proportional to the bth power of x.” � e dilation factor a is the proportionality constant.

Translated function: y d a(x c ) b , where c and d are the horizontal and vertical translation, respectively. Compare the translated form with

y y 1 a x x 1 for linear functions

y k a(x h ) 2 for quadratic functions

Unless otherwise stated, “power function” will imply the untranslated form, y a x b .

y

x

5

5

5

Parentquadraticfunctiony x2

5

y 3 0.4(x 1)2

y

x

5

�5

5�5

Vertex (1, 3)

y

x

5

5

y 3 2(x 1)2

5

Vertex (1, 3)

5

� e eruption of Arenal, an active volcano in Costa Rica. � e lava particles follow a parabolic path due to gravitational force.


Identifying Functions from Graphical PatternsOne way to tell what type of function � ts a set of points is by recognizing the properties of the graph of the function.


Here is a brief review of the basic functions used in modeling. Some of these appeared in Chapter 1.

Linear and Constant FunctionsGeneral equation: y ax b (o� en written y mx b), where a (or m) and b stand for constants and the domain is all real numbers. � is equation is in the slope-intercept form because a (or m) gives the slope and b gives the y-intercept. If a 0, then y b; this is a constant function.

Parent function: y x

Transformed function: y y 1 a x x 1 , called the point-slope form because the graph contains the point x 1 , y 1 and has slope a. � e slope, a, is the vertical dilation; y 1 is the vertical translation; and x 1 is the horizontal translation. Note that point-slope form can also be written y y 1 a x x 1 , where the coordinates of the � xed point x 1 , y 1 both appear with a sign. � e form y y 1 a x x 1 expresses y explicitly in terms of x and thus is easier to enter into your grapher.

Graphical properties: � e graph is a straight line. � e parent function is shown in the le� graph of Figure 2-2a, the slope-intercept form is shown in the middle graph, and the point-slope form is shown in the right graph.

Verbally: For slope-intercept form: “Start at b on the y-axis, run x, and rise ax.” For point-slope form: “Start at x 1 , y 1 , run x x 1 , and rise a x x 1 .”

Identifying Functions from Graphical Patterns

2-2


Objective

Figure 2-2a: Linear functions

y

x

5

10

5 10Parent functiony x

y

x

5

10

5 10

y-intercept 8Run

Rise(negative)

Point (x, y)

y 0.7x 8

y

x

5

10

5 10

Point (1, 4)

Point (x, y)

Run

Rise

y 4 (x 1) 23


Quadratic Functions

A quadratic function is a second-degree polynomial function (i.e., a polynomial function in which the highest power of x is 2). The graph of a quadratic function is called a parabola.

The left graph in Figure 2-2b shows the parent quadratic function, y 5 x 2 . Any other quadratic function is a transformation of this function. The middle graph shows y 5 3 1 0.4 (x 2 1) 2 , which is a dilation of the parent graph by a factor of 0.4 and a translation of 1 unit horizontally and 3 units vertically. The graph on the right shows y 5 3 2 2 (x 2 1) 2 , which has the same translations but a dilation factor of 22.

If the vertical dilation is positive, the parabola opens up and is said to be concave up (middle, Figure 2-2b). If the vertical dilation is negative, the parabola opens down and is said to be concave down (right, Figure 2-2b).

The equation y 5 3 2 2 (x 2 1) 2 can be written in polynomial form.

y 5 3 2 2( x 2 2 2x 1 1)Expand the squared binomial.

y 5 22 x 2 1 4x 1 1 Distribute the 22 and combine like terms.

This equation is now in the polynomial form y 5 a x 2 1 bx 1 c. When a quadratic function is written in this form, a positive value of a indicates that the parabola is concave up and a negative value of a indicates that the parabola is concave down. The y-intercept is c. The polynomial form y 5 a x 2 1 bx 1 c does not tell us as much about the graph of the quadratic function as does the vertex form y 5 k 1 a (x 2 h) 2 .

If a 0 and b 5 0, the function is of the form y 5 ax, which represents a line through the origin. If this function is translated so the point at the origin moves to the point ( x 1 , y 1 ), then the equation becomes y 5 y 1 1 a(x 2 x 1 ). This form is called point-slope form because it contains both the coordinates of a point on the graph and the slope. It is important for students to master the point-slope form of a line.

This form is more general than the slope-intercept form because any point can be used for ( x 1 , y 1 ), whereas the slope-intercept calls for a specific point, the y-intercept.


Mathematicians usually use one of two particular constants for the base of an exponential function: either 10, which is the base of the decimal system, or the naturally occurring number e, which equals 2.71828…. To make the equation more general, multiply the variable in the exponent by a constant. � e (untranslated) general equations are given in the box.

DEFINITION: Special Exponential Functionsy a 1 0 bx base-10 exponential function

y a e bx natural (base-e) exponential function

where a and b are constants and the domain is all real numbers.

Note: � e equations of these two functions can be generalized by incorporating translations in the x- and y-directions. You’ll get y a 1 0 b(x c) d and y a e b(x c) d.

Base-e exponential functions have an advantage when you study calculus because the rate of change of e x is equal to e x .

In this exploration, you’ll � nd the particular equation of a linear, quadratic, power, or exponential function from a given graph.

1. Identify what kind of function is graphed, and � nd its particular equation.

5 10

(8, 13)

(3, 25)

15

10

20

30

y

x

2. Does your graph agree with the given one?

3. concave down, or neither?

4. What graphical evidence do you have that the function graphed is an exponential function, not a power function? Find its particular equation.

5 10

(8, 9)(3, 4)

15

10

20

30

x

y



continued

1. Identify what kind of function is graphed, 4. What graphical evidence do you have that the

E X P L O R AT I O N 2-2: G r a p h i c a l P a t t e r n s i n Fu n c t i o n s


Graphical properties: Figure 2-2c shows power function graphs for di�erent values of b. In all three cases, a 0. �e shape and concavity of the graph depend on the value of b. �e graph contains the origin if b 0; it has the axes as asymptotes if b 0. �e function is increasing if b 0; it is decreasing if b 0. �e graph is concave up if b 1 or if b 0 and concave down if 0 b 1. �e concavity of the graph describes the rate at which y increases. For b 0, concave up means y is increasing at an increasing rate, and concave down means y is increasing at a decreasing rate.

y 0.02x3Exponent greater than 1

5

Concave up

Origin

Increasing

10

y

5

10

x

Figure 2-2c: Power functions

Exponential FunctionsGeneral equation: y a b x , where a and b are constants, a ≠ 0, b 0, b ≠ 1, and the domain is all real numbers

Parent function: y b x , where the asymptote is the x-axis

Verbally: In the equation y a b x , “y varies exponentially with x.”

Translated function: y a b x c, where the asymptote is the line y c. Unless otherwise stated, “exponential function” will imply the untranslated form, y a b x .

Graphical properties: Figure 2-2d shows exponential functions for di�erent values of a and b. �e constant a is the y-intercept. �e function is increasing if b 1 and decreasing if 0 b 1 (provided a 0). If a 0, the opposite is true. �e graph is concave up if a 0 and concave down if a 0.

y 2 1.2x

Base greater than 1

5

Concave up

Increasing

y-intercept value of a

10

y

5

10

x

y 2.5x0.6Exponent between 0 and 1

5

Concave down

Origin

Increasing

10

y

5

10

x

y 3x–1

Exponent negative

5

Concave up

Decreasing

Origin 10

y

5

10

x

Marie Curie was awarded the Nobel Prize in chemistry for the discovery of radioactive elements (polonium and radium) in 1911. �e breakdown of radioactive elements follows an exponential function.

y 6 0.7x

Base between 0 and 1

5

Concave up

Decreasing


10

y

5

10

x

y x

y a

y

x

Figure 2-2d: Exponential functions


Section Notes (continued)

Power Functions

The power function has the general form of y 5 a x b . Figure 2-2c shows graphs of three power functions. The graph of y 5 0.002 x 3 contains the origin because y 5 0 when x 5 0. The graph is concave up because the y-values increase more and more rapidly as x increases. The graph of y 5 2.5 x 0.6 contains the origin, is increasing, and is concave down because the y-values increase at a slower and slower rate as x increases. The graph of y 5 3 x 21 is undefined when x 5 0 (because 3 x 21 5 3 _ x ) so it has no y-intercept. The y-values decrease as x increases because you are dividing by larger and larger numbers.

This list summarizes the features of graphs of functions of the form y 5 a x b .

• If a . 0 and b . 1, then the graph contains the origin, is increasing, and is concave up in the first quadrant.

• If a . 0 and 0 , b , 1, then the graph contains the origin, is increasing, and is concave down in the first quadrant.

• If a . 0 and b , 0, then the graph does not contain the origin and is decreasing and asymptotic to both axes in the first quadrant.

• If b 5 1, then the function is a linear function.

• If b is a non-integer, then the domain is x 0 because of the ambiguity of the solution. For example, (28) 1/3 5 22, but (28) 2/6 5 2.


Mathematicians usually use one of two particular constants for the base of an exponential function: either 10, which is the base of the decimal system, or the naturally occurring number e, which equals 2.71828…. To make the equation more general, multiply the variable in the exponent by a constant. � e (untranslated) general equations are given in the box.

DEFINITION: Special Exponential Functionsy a 1 0 bx base-10 exponential function

y a e bx natural (base-e) exponential function

where a and b are constants and the domain is all real numbers.

Note: � e equations of these two functions can be generalized by incorporating translations in the x- and y-directions. You’ll get y a 1 0 b(x c) d and y a e b(x c) d.

Base-e exponential functions have an advantage when you study calculus because the rate of change of e x is equal to e x .

In this exploration, you’ll � nd the particular equation of a linear, quadratic, power, or exponential function from a given graph.


5 10

(8, 13)

(3, 25)

15

10

20

30

y

x



4. What graphical evidence do you have that the function graphed is an exponential function, not a power function? Find its particular equation.

5 10

(8, 9)(3, 4)

15

10

20

30

x

y



continued

1. Identify what kind of function is graphed, 4. What graphical evidence do you have that the

E X P L O R AT I O N 2-2: G r a p h i c a l P a t t e r n s i n Fu n c t i o n s


Graphical properties: Figure 2-2c shows power function graphs for di�erent values of b. In all three cases, a 0. �e shape and concavity of the graph depend on the value of b. �e graph contains the origin if b 0; it has the axes as asymptotes if b 0. �e function is increasing if b 0; it is decreasing if b 0. �e graph is concave up if b 1 or if b 0 and concave down if 0 b 1. �e concavity of the graph describes the rate at which y increases. For b 0, concave up means y is increasing at an increasing rate, and concave down means y is increasing at a decreasing rate.

y 0.02x3Exponent greater than 1

5

Concave up

Origin

Increasing

10

y

5

10

x

Figure 2-2c: Power functions

Exponential FunctionsGeneral equation: y a b x , where a and b are constants, a ≠ 0, b 0, b ≠ 1, and the domain is all real numbers

Parent function: y b x , where the asymptote is the x-axis

Verbally: In the equation y a b x , “y varies exponentially with x.”

Translated function: y a b x c, where the asymptote is the line y c. Unless otherwise stated, “exponential function” will imply the untranslated form, y a b x .

Graphical properties: Figure 2-2d shows exponential functions for di�erent values of a and b. �e constant a is the y-intercept. �e function is increasing if b 1 and decreasing if 0 b 1 (provided a 0). If a 0, the opposite is true. �e graph is concave up if a 0 and concave down if a 0.

y 2 1.2x

Base greater than 1

5

Concave up

Increasing


10

y

5

10

x

y 2.5x0.6Exponent between 0 and 1

5

Concave down

Origin

Increasing

10

y

5

10

x

y 3x–1

Exponent negative

5

Concave up

Decreasing

Origin 10

y

5

10

x

Marie Curie was awarded the Nobel Prize in chemistry for the discovery of radioactive elements (polonium and radium) in 1911. �e breakdown of radioactive elements follows an exponential function.

y 6 0.7x

Base between 0 and 1

5

Concave up

Decreasing


10

y

5

10

x

y x

y a

y

x

Figure 2-2d: Exponential functions

69

Exponential Functions

An exponential function has equation y 5 a b x , where a 0, b . 0, and b 1. Figure 2-2d on page 68 shows graphs of three exponential functions. In the function y 5 2 1.2 x , the base is greater than 1. This causes the y-values to increase as x increases. The base in the function y 5 6 0.7 x is greater than 0 but less then 1. Raising such a number to greater and greater powers gives smaller and smaller results, so the y-values decrease as x increases. Both graphs are concave up—the left graph because y increases at a faster and faster rate and the middle graph because y decreases at a slower and slower rate. The graph y 5 23 1.06 x shows what can happen if a is negative.

This list summarizes the features of graphs of functions of the form y 5 a b x , where a 0, b . 0, and b 1.

• If a . 0 and b . 1, then the graph is increasing and concave up. The graph is asymptotic to the x-axis and has y-intercept a.

• If a . 0 and 0 , b , 1, then the graph is decreasing and concave up. The graph is asymptotic to the x-axis and has y-intercept a.

• If a , 0 and b . 1, then the graph is decreasing and concave down. The graph is asymptotic to the x-axis and has y-intercept a.

• If a , 0 and 0 , b , 1, then the graph is increasing and concave down. The graph is asymptotic to the x-axis and has y-intercept a.

Notice that in all four cases the graph is asymptotic to the x-axis and has y-intercept a.

Exploration Notes

Exploration 2-2 provides practice in finding the particular equation of a linear, quadratic, power, or exponential function from a given graph. Allow students 20–25 minutes to complete this activity.

1. Linear. y 5 22.4x 1 32.2

2. The answer checks.

3. Neither

4. The graph is not a power function because the y-intercept is not zero. y 5 2.4589... (1.1760...) x


6. Concave up

Section 2-2: Identifying Functions from Graphical Patterns


19 5a b 6 10a b

Substitute the given values of x and y into the

equation of f.

13 5a a 2.6 Subtract the � rst equation from the second to eliminate b.

6 10( 2.6) b b 32 Substitute 2.6 for a in one of the equations.

f (x) 2.6x 32 Write the particular equation.

e. Figure 2-2f shows the graph of f, which agrees with the given graph. Note that the calculated slope, 2.6, is negative, which corresponds to the fact that f (x) decreases as x increases.

matrices.

5a b 19 10a b 6

� e given system.

5 10 1 1 19 6 Write the system in matrix form.

5 10 1 1

1 19 6 Multiply both sides by the inverse matrix.

2.6 32 Complete the matrix multiplication.

a 2.6 and b 32

You’ll study the matrix solution of linear systems more fully in Section 13-2.

For the function graphed in Figure 2-2g,

a. Identify the kind of function it could be.

b. On what interval or intervals is the function increasing or decreasing? Which way is the graph concave, up or down?

c. Describe something in the real world that a function with this shape graph could model.

d. Find the particular equation of the function, given that points (1, 76),

e. Con� rm by plotting that your equation gives the graph in Figure 2-2g.

6 403020

10

55 10 15

f(x)

x

Figure 2-2f ➤

a b

a b

For the function graphed in Figure 2-2g,EXAMPLE 2 ➤

5

y

(1, 76)

(2, 89)(3, 94)

100

50

x

Figure 2-2g


For the function graphed in Figure 2-2e,

a. Identify the kind of function it is.


c. From your experience, describe something in the real world that a function with this shape graph could model.

d. Find the particular equation of the function, given that points (5, 19) and (10, 6) are on the graph.

e. Con� rm by plotting that your equation gives the graph in Figure 2-2e.

a. Because the graph is a straight line, the function is linear.

b. � e function is decreasing over its entire domain, and the graph is not concave in either direction.

c. � e function could model anything that decreases at a constant rate. � e

history text you have le� to read as a function of the number of minutes you have been reading.

d. f (x) ax b Write the general equation. Use f (x) as shown on the graph, and use a for the slope.

For the function graphed in Figure 2-2e,EXAMPLE 1 ➤

5 5 10 15

f(x)40

x

3020 (5, 19)

(10, 6)10

Figure 2-2e

a. Because the graph is a straight line, the function is linear.SOLUTION

EXPLORATION, continued

7. What graphical evidence do you have that this function is a power function, not an exponential function? Find its particular equation.

5 10

(8, 2)

(3, 15)

15

10

20

30

y

x




5 10 15

10

20

30

40y

x

(3, 26.6)

(8, 38.6)

(14, 13.4)



13. What did you learn as a result of doing this exploration that you did not know before?



In the examples, students identify the type of function from a given graph and then find the equation.

The function in Example 1 is linear. Part d shows how to find the equation by writing and solving a system of equations. The system is solved by the elimination method, but if your students have studied matrices, you can present the alternative solution given at the end of the example. (Students will study matrices in Chapter 13.)

The equation can also be found by calculating the slope and then writing the equation in point-slope form:

slope 5 19 2 6 ______ 5 2 10 5 13 ___ 25 5 22.6

So, using the point (5, 19), the equation is

y 5 19 2 2.6(x 2 5)

7. The y-axis appears to be a vertical asymptote, which indicates a power function with a negative exponent. y 5 143.2961...x 22.0542...


9. Concave up

10. Quadratic; y 5 2 0.6072...x 2 1 9.16x 1 4.1854...


12. Concave down

13. Answers will vary.


19 5a b 6 10a b

Substitute the given values of x and y into the

equation of f.

13 5a a 2.6 Subtract the � rst equation from the second to eliminate b.

6 10( 2.6) b b 32 Substitute 2.6 for a in one of the equations.

f (x) 2.6x 32 Write the particular equation.

e. Figure 2-2f shows the graph of f, which agrees with the given graph. Note that the calculated slope, 2.6, is negative, which corresponds to the fact that f (x) decreases as x increases.

matrices.

5a b 19 10a b 6

� e given system.

5 10 1 1 19 6 Write the system in matrix form.

5 10 1 1

1 19 6 Multiply both sides by the inverse matrix.

2.6 32 Complete the matrix multiplication.

a 2.6 and b 32

You’ll study the matrix solution of linear systems more fully in Section 13-2.

For the function graphed in Figure 2-2g,




d. Find the particular equation of the function, given that points (1, 76),

e. Con� rm by plotting that your equation gives the graph in Figure 2-2g.

6 403020

10

55 10 15

f(x)

x

Figure 2-2f ➤

a b

a b

For the function graphed in Figure 2-2g,EXAMPLE 2 ➤

5

y

(1, 76)

(2, 89)(3, 94)

100

50

x

Figure 2-2g


For the function graphed in Figure 2-2e,

a. Identify the kind of function it is.


c. From your experience, describe something in the real world that a function with this shape graph could model.

d. Find the particular equation of the function, given that points (5, 19) and (10, 6) are on the graph.

e. Con� rm by plotting that your equation gives the graph in Figure 2-2e.

a. Because the graph is a straight line, the function is linear.

b. � e function is decreasing over its entire domain, and the graph is not concave in either direction.

c. � e function could model anything that decreases at a constant rate. � e

history text you have le� to read as a function of the number of minutes you have been reading.

d. f (x) ax b Write the general equation. Use f (x) as shown on the graph, and use a for the slope.

For the function graphed in Figure 2-2e,EXAMPLE 1 ➤

5 5 10 15

f(x)40

x

3020 (5, 19)

(10, 6)10

Figure 2-2e

a. Because the graph is a straight line, the function is linear.SOLUTION

EXPLORATION, continued

7. What graphical evidence do you have that this function is a power function, not an exponential function? Find its particular equation.

5 10

(8, 2)

(3, 15)

15

10

20

30

y

x




5 10 15

10

20

30

40y

x

(3, 26.6)

(8, 38.6)

(14, 13.4)




71

The quadratic equation in Example 2 is found by writing and solving a system of three equations in three variables. Part d shows how to find the solution using matrices. While this example is easy to do algebraically, most systems of three equations in three variables are not easy to solve without matrices. If your students have not learned about matrices, you can present this alternative algebraic solution:

76 5 a 1 b 1 c 89 5 4a 1 2b 1 c 94 5 9a 1 3b 1 c

Subtract the first equation from the second and subtract the second equation from the third to eliminate c and get two equations in two variables:

13 5 3a 1 b 5 5 5a 1 b

Now solve this system of two equations to find a and b:

13 2 5 5 (3a 1 b) 2 (5a 1 b)Subtract the second equation from the first.

8 5 22a Simplify.

a 5 24 Solve for a.

13 5 3(24) 1 bSubstitute –4 for a in the first equation.

b 5 25 Solve for b.

To find c, go back to one of the original equations.

76 5 24 1 25 1 c Substitute –4 for a and 25 for b in the original first equation.

c 5 55 Solve for c.



d. y a x b Write the untranslated general equation.

a b 151.2 a 6 b

Substitute the given x- and y-values into the equation.

151.2 _____ a 6 b _____ a b Divide the second equation by the � rst to eliminate a.

3.375 1. 5 b � e a’s cancel, and 6 b __ b 6 _ b 1.5 b .

log 3.375 log 1. 5 b Take the logarithm of both sides to get b out of

log 3.375 b log 1.5

b log 3.375

________ log 1.5 3

a 3 Substitute 3 for b in one of the equations.

a ____ 3 0.7

y 0.7 x 3 Write the particular equation.

e. value of b is between 0 and 1, which corresponds to the fact that the graph is concave up.

For the function graphed in Figure 2-2i,




d. Find the particular equation of the function, given that the points (2, 10) and (5, 6) are on the graph.

e. Con� rm by plotting that your equation gives the graph in Figure 2-2i.

a. � e function could be exponential or quadratic, but exponential is chosen because the graph appears to approach the x-axis asymptotically.

b. � e function is decreasing and concave up over its entire domain.

c. � e function could model any situation in which a variable quantity starts at some nonzero value and decreases, gradually approaching zero, such as the number of degrees a cup of co� ee is above room temperature as a function of time since it started cooling.

➤

For the function graphed in Figure 2-2i,EXAMPLE 4 ➤

y

(2, 10)(5, 6)

x15105

20

105

Figure 2-2i

a. � e function could be exponential or quadratic, but exponential is chosen because the graph appears to approach the

SOLUTION


a. � e function could be quadratic because it has a vertex.

b. � e function is increasing for x 3 and decreasing for x 3, and it is concave down.

c. � e function could model anything that rises to a maximum and then falls back down again, such as the height of a ball as a function of time or the grade you could earn on a � nal exam as a function of how long you study for it. (Cramming too long might lower your score because of your being sleepy from staying up late!)

d. y a x 2 bx c Write the general equation.

76 a b c89 a 2b c

9a 3b c

Substitute the given x- and y-values.

1

9

1

2 3

1

1 1

1

76

89

25 55

Solve by matrices.

y x 2 25x 55 Write the equation.

e. value of a is negative, which corresponds to the fact that the graph is concave down.

For the function graphed in Figure 2-2h,




d. Find the particular equation of the function you identi� ed in part a, given

e. Con� rm by plotting that your equation gives the graph in Figure 2-2h.

a. � e function could be a power function or an exponential function, but a power function is chosen because the graph appears to contain the origin.

in the y-direction.

b. � e function is increasing and concave up over the entire domain shown.

c. � e function could model anything that starts at zero and increases at an increasing rate, such as the power generated by a windmill as a function of wind speed, when the driver applies the brakes, or the volumes of geometrically similar objects as a function of their lengths.

a. � e function could be quadratic because it has a vertex.SOLUTION

➤

For the function graphed in Figure 2-2h,EXAMPLE 3 ➤

y(6, 151.2)

(4, 44.8)x

654321

150

100

50


SOLUTION

Figure 2-2h



The curve in Example 3 represents a power function. The shape of the curve is similar to that of an exponential function, but it can be distin guished from an exponential function graph because it passes through the origin. Finding the particular equation for the curve requires the use of logarithms. If students are not familiar with logarithms, you can simply give them the equation and explain that they will learn how to find the equation themselves in Section 2-4.

The curve in Example 4 on page 73 represents an exponential function. You may need to remind students that if they plan to use the result of a calculation in subsequent calculations, they should use the unrounded value. One way to do this is to store the result in their grapher’s memory and then use the stored value rather than the rounded value in the calculation. You may need to explain how to store values.

Explain to students that an ellipsis is used to indicate that the digits of a number continue after the last digit displayed. A rounded answer should be preceded with an approximately equal to sign () to indicate that the answer is an approximation. When a problem does not specify how to round a number, students should round appropriately, depending on the real-world context of the problem.

Differentiating Instruction• Make sure ELL students understand

that vertex is singular and vertices is plural. In many languages, vertice is the singular. The same is true about matrix and matrices.

• Some ELL students may not have learned logarithms. Teach them enough to understand Example 3.

• Make sure ELL students know how to use STORE on their grapher.

• Make sure ELL students understand rise over run. Slope is not referred to in this way in other languages—the standard reference is change in y over change in x.

• Some students may avoid using point-slope form because it appears to be more difficult. Point out the advantages of point-slope form over slope-intercept form.

• Students will benefit from drawing samples of the various functions and their equations in their journals, so they can refer to specific examples.

• Consider allowing ELL students to do the Reading Analysis questions in pairs. They will need an extended transition to doing them alone.

• Students should enter Problems 1–8 in their journals. They may also benefit from doing these problems in pairs.


d. y a x b Write the untranslated general equation.

a b 151.2 a 6 b

Substitute the given x- and y-values into the equation.

151.2 _____ a 6 b _____ a b Divide the second equation by the � rst to eliminate a.

3.375 1. 5 b � e a’s cancel, and 6 b __ b 6 _ b 1.5 b .

log 3.375 log 1. 5 b Take the logarithm of both sides to get b out of

log 3.375 b log 1.5

b log 3.375

________ log 1.5 3

a 3 Substitute 3 for b in one of the equations.

a ____ 3 0.7

y 0.7 x 3 Write the particular equation.

e. value of b is between 0 and 1, which corresponds to the fact that the graph is concave up.

For the function graphed in Figure 2-2i,




d. Find the particular equation of the function, given that the points (2, 10) and (5, 6) are on the graph.

e. Con� rm by plotting that your equation gives the graph in Figure 2-2i.

a. � e function could be exponential or quadratic, but exponential is chosen because the graph appears to approach the x-axis asymptotically.

b. � e function is decreasing and concave up over its entire domain.

c. � e function could model any situation in which a variable quantity starts at some nonzero value and decreases, gradually approaching zero, such as the number of degrees a cup of co� ee is above room temperature as a function of time since it started cooling.

➤

For the function graphed in Figure 2-2i,EXAMPLE 4 ➤

y

(2, 10)(5, 6)

x15105

20

105

Figure 2-2i

a. � e function could be exponential or quadratic, but exponential is chosen because the graph appears to approach the

SOLUTION


a. � e function could be quadratic because it has a vertex.

b. � e function is increasing for x 3 and decreasing for x 3, and it is concave down.

c. � e function could model anything that rises to a maximum and then falls back down again, such as the height of a ball as a function of time or the grade you could earn on a � nal exam as a function of how long you study for it. (Cramming too long might lower your score because of your being sleepy from staying up late!)

d. y a x 2 bx c Write the general equation.

76 a b c89 a 2b c

9a 3b c


1

9

1

2 3

1

1 1

1

76

89

25 55

Solve by matrices.

y x 2 25x 55 Write the equation.

e. value of a is negative, which corresponds to the fact that the graph is concave down.

For the function graphed in Figure 2-2h,




d. Find the particular equation of the function you identi� ed in part a, given

e. Con� rm by plotting that your equation gives the graph in Figure 2-2h.


in the y-direction.

b. � e function is increasing and concave up over the entire domain shown.

c. � e function could model anything that starts at zero and increases at an increasing rate, such as the power generated by a windmill as a function of wind speed, when the driver applies the brakes, or the volumes of geometrically similar objects as a function of their lengths.

a. � e function could be quadratic because it has a vertex.SOLUTION

➤

For the function graphed in Figure 2-2h,EXAMPLE 3 ➤

y(6, 151.2)

(4, 44.8)x

654321

150

100

50


SOLUTION

Figure 2-2h

73

Technology Notes

Exploration 2-2 asks students to identify what kinds of functions represent given graphs and to write equations for the graphs. Th e high-resolution graphing capabilities of either Fathom or Sketchpad would be useful for this exploration.

CAS Suggestions

A CAS does not instantly produce the transformed function form of a given general equation of a quadratic function. Students must know the algebraic technique needed to display the desired results. Allowing students to use a CAS provides instant feedback to the students as they apply the techniques they are learning.

Using a CAS to solve systems of equations associated with points on a curve allows diff erentiation between problems intended to assess algebraic manipulation skills from problems intended to determine if students can identify the proper form of a curve from its graph. Consider asking students to compute equations of linear, exponential, or power functions given any two points in Quadrant I.

• For Problems 21–24, ELL students may do better if they write the equation before sketching the graph and answering the questions. You may need to help them express their answers, as they require students to explain.



4. contain the origin but inverse variation power functions do not.

5. reciprocal function f (x) 1 _ x is also a power function.

6. In the de�nition of quadratic function, what is the reason for the restriction a ≠ 0?

7. �e de�nition of exponential function, y a b x , includes the restriction b 0. Suppose that y ( ) x . What would y equal if x were 1 _ 2 ? If x were 1 _ 3 ? Why do you think there is the restriction b 0 for exponential functions?

8. �e vertex form of the quadratic-function equation can be written as

y k a(x h ) 2 or y k a(x h ) 2

are plotting graphs on your grapher and why the second form is more useful for understanding the translations involved.

9. Reading Problem: Clara has been reading her history assignment for 20 min and is now starting page 56 in the text. She reads at a (relatively) constant rate of 0.6 page per minute.

a. Find the particular equation expressing the page number she is on as a function of minutes, using the point-slope form. Transform your answer to the slope-intercept form.

b. Which page was Clara on when she started reading the assignment?

c. �e assignment ends at the top of page 63. When would you expect Clara to �nish?

10. Baseball Problem: Ruth hits a high �y ball to

when she hits it. �ree seconds later it reaches its

a. Write an equation in vertex form of the quadratic function expressing the height of the ball explicitly as a function of time.

b. How high is the ball 5 s a�er it was hit?

c. If nobody catches the ball, how many seconds a�er it was hit will it reach the ground?

function graph is shown.

a. Identify the type of function it could represent.

b. On what interval or intervals is the function increasing or decreasing and which way is the graph concave?

c. From your experience, what relationship in the real world could be modeled by a function with this shape graph?

d. Find the particular equation of the function if the given points are on the graph.

e. Con�rm by plotting that your equation gives the graph shown.

11.

12.

13.

y

(6, 5)

(4, 1) x9

6

y

(0, 20)(1, 16)

x105

20

10

y

(1, 6) (3, 4)

(5, 18)

x5

20

10


d. y a b x Write the untranslated general equation.


Divide the second equation by the � rst to eliminate a.

0.6 b 3

0. 6 1/3 b Raise both sides to the 1 _ 3 power to eliminate the exponent of b.

b Store without rounding.

10 a 2 b in one of the equations.

a Store without rounding.

y x Write the particular equation.

e. value of b is between 0 and 1, which corresponds to the fact that the function is decreasing.

10 a b 2 6 a b 5

6 ___ 10 a b 5 ___ a b 2

Problem Set 2-2

Reading Analysis

From what you have read in this section, what do you consider to be the main idea? What is the di� erence between the parent quadratic function and any other quadratic function? How does the y-intercept of an exponential function di� er from the y-intercept of a power function? Sketch the graph of a function that is increasing but concave down.

Quick Review

Q1. If f (x) x 2 , � nd f (3). Q2. If f (x) x 2 , � nd f (0). Q3. If f (x) x 2 , � nd f ( 3). Q4. If g (x) 2 x , � nd g (3). Q5. If g (x) 2 x , � nd g (0). Q6. If g (x) 2 x , � nd g ( 3). Q7. If h(x) x 1/2 , � nd h(25). Q8. If h(x) x 1/2 , � nd h(0). Q9. If h(x) x 1/2 , � nd h( 9).

Q10. What property of real numbers is illustrated by 3(x 5) 3(5 x)?

A. Associative property of multiplication B. Commutative property of multiplication C. Associative property of addition D. Commutative property of addition E. Distributive property of multiplication over

addition

1. both have exponents. What major algebraic di� erence distinguishes these two types of functions?

2. What graphical feature do quadratic functions have that linear, exponential, and power functions do not have?

3. Write a sentence or two giving the origin of the word concave and explaining how the word applies to graphs of functions.

5min


PRO B LE M N OTES

Supplementary Problems for this section are available at www.keypress.com/keyonline.

Be sure to discuss all the assigned problems, because they contain key concepts important for students to master and understand. Q1. 9Q2. 0Q3. 9Q4. 8Q5. 1Q6. 0.125Q7. 5Q8. 0

In Problem Q9, some students may recognize that h(29) 5 3i.Q9. Undefi nedQ10. D

Problems 1–7 help students review the important features of the graphs and equations of the various types of functions.1. In power functions, the exponent is constant and the independent variable is in the base. In exponential functions, the base is constant and the independent variable is in the exponent.2. Quadratic functions have either a maximum or a minimum point. Exponential, linear, and many power functions do not have these.3. Answers will vary. 4. Direct variation power functions have the form y 5 ax n with n . 0, so y 5 0 when x 5 0. But inverse variation power functions are undefi ned at x 5 0.5. 1 __ x 5 x 21 6. Exclude straight lines from being called quadratic7. (264 ) 1/2 is undefi ned but (264 ) 1/3 5 24. Th e restriction allows the function to be defi ned for all values of x.


4. contain the origin but inverse variation power functions do not.

5. reciprocal function f (x) 1 _ x is also a power function.

6. In the de�nition of quadratic function, what is the reason for the restriction a ≠ 0?

7. �e de�nition of exponential function, y a b x , includes the restriction b 0. Suppose that y ( ) x . What would y equal if x were 1 _ 2 ? If x were 1 _ 3 ? Why do you think there is the restriction b 0 for exponential functions?

8. �e vertex form of the quadratic-function equation can be written as

y k a(x h ) 2 or y k a(x h ) 2

are plotting graphs on your grapher and why the second form is more useful for understanding the translations involved.

9. Reading Problem: Clara has been reading her history assignment for 20 min and is now starting page 56 in the text. She reads at a (relatively) constant rate of 0.6 page per minute.

a. Find the particular equation expressing the page number she is on as a function of minutes, using the point-slope form. Transform your answer to the slope-intercept form.

b. Which page was Clara on when she started reading the assignment?

c. �e assignment ends at the top of page 63. When would you expect Clara to �nish?

10. Baseball Problem: Ruth hits a high �y ball to

when she hits it. �ree seconds later it reaches its

a. Write an equation in vertex form of the quadratic function expressing the height of the ball explicitly as a function of time.

b. How high is the ball 5 s a�er it was hit?

c. If nobody catches the ball, how many seconds a�er it was hit will it reach the ground?

function graph is shown.

a. Identify the type of function it could represent.

b. On what interval or intervals is the function increasing or decreasing and which way is the graph concave?

c. From your experience, what relationship in the real world could be modeled by a function with this shape graph?

d. Find the particular equation of the function if the given points are on the graph.

e. Con�rm by plotting that your equation gives the graph shown.

11.

12.

13.

y

(6, 5)

(4, 1) x9

6

y

(0, 20)(1, 16)

x105

20

10

y

(1, 6) (3, 4)

(5, 18)

x5

20

10


d. y a b x Write the untranslated general equation.


Divide the second equation by the � rst to eliminate a.

0.6 b 3

0. 6 1/3 b Raise both sides to the 1 _ 3 power to eliminate the exponent of b.

b Store without rounding.

10 a 2 b in one of the equations.

a Store without rounding.

y x Write the particular equation.

e. value of b is between 0 and 1, which corresponds to the fact that the function is decreasing.

10 a b 2 6 a b 5

6 ___ 10 a b 5 ___ a b 2

Problem Set 2-2

Reading Analysis

From what you have read in this section, what do you consider to be the main idea? What is the di� erence between the parent quadratic function and any other quadratic function? How does the y-intercept of an exponential function di� er from the y-intercept of a power function? Sketch the graph of a function that is increasing but concave down.

Quick Review

Q1. If f (x) x 2 , � nd f (3). Q2. If f (x) x 2 , � nd f (0). Q3. If f (x) x 2 , � nd f ( 3). Q4. If g (x) 2 x , � nd g (3). Q5. If g (x) 2 x , � nd g (0). Q6. If g (x) 2 x , � nd g ( 3). Q7. If h(x) x 1/2 , � nd h(25). Q8. If h(x) x 1/2 , � nd h(0). Q9. If h(x) x 1/2 , � nd h( 9).

Q10. What property of real numbers is illustrated by 3(x 5) 3(5 x)?

A. Associative property of multiplication B. Commutative property of multiplication C. Associative property of addition D. Commutative property of addition E. Distributive property of multiplication over

addition

1. both have exponents. What major algebraic di� erence distinguishes these two types of functions?

2. What graphical feature do quadratic functions have that linear, exponential, and power functions do not have?

3. Write a sentence or two giving the origin of the word concave and explaining how the word applies to graphs of functions.

5min

75

Problem 8 helps students recognize that there are advantages and disadvantages to both forms of the quadratic-function equation.8. Th e grapher only allows you to enter equations in “y 5” form. Th e second form shows the horizontal translation h and the vertical translation k.

Problems 9 and 10 require students to fi nd equation models for situations based on written descriptions of the situations. Students may use the greatest integer function in Problem 9, because the function is being used to describe the page number she is on, which is a discreet function.9a. y 5 56 1 0.6(x 2 20) 5 0.6x 1 449b. Page 449c. 11 2 _ 3 min from now.10a. y 2 148 5 216(x 2 3) 2 10b. 84 ft 10c. 6.0413... s

Problems 11–20 are trivial if students use a CAS to solve systems.11a. Linear11b. Increasing for all real-number values of x, not concave11c. Answers will vary.11d. y 5 2x 2 711e. Th e graphs match.12a. Linear12b. Decreasing for all real-number values of x, not concave12c. Answers will vary.12d. y 5 24x 1 2012e. Th e graphs match.13a. Quadratic13b. Decreasing for x , 2.25 and increasing for x . 2.25, concave up13c. Answers will vary.13d. y 5 2x 2 2 9x 1 1313e. Th e graphs match.


77Section 2-3: Identifying Functions from Numerical Patterns

Identifying Functions from Numerical PatternsA 16-in. pizza has four times as much area as an 8-in. pizza. A grapefruit whose diameter is 10 cm has eight times the volume of a grapefruit with diameter 5 cm. In general, when you double the linear dimensions of a three-dimensional object, you multiply the surface area by

time you add 1000 mi to the distance you have driven your car, you add a constant amount—say, $300—to the cost of operating that car.

In this section you will use such patterns to identify the type of function that � ts a given set of function values. � en you will � nd more function values, either by following the pattern or by � nding the equation of the function.

x-values and the corresponding y-values, identify which type of function they � t (linear, quadratic, power, or exponential).

equation.

The Add–Add Pattern of Linear Functions

10864

1 3

32

2

25 7

Linear function

f (x)

x

3

3

x f(x)

13 75 107 139 16

Figure 2-3a

Figure 2-3a shows the graph of the linear function f (x) 1.5x 2.5. As you can see from the graph and the adjacent table, each time you add 2 to x, y increases by 3. � is pattern emerges because a linear function has constant slope. Verbally, you can express this property by saying that every time you add a constant to x, you add a constant (not necessarily the same as the constant added to x) to y. � is property is called the add–add property of linear functions.

Identifying Functions from

2-3

identify which type of function they � t (linear, quadratic, power, or Objective

2

2

2

2

3

3

3

3

� e add-add property


14.

15.

16.

17.

18.

19.

20.

21. Suppose that y increases exponentially with x and that z is directly proportional to the square of x. Sketch the graph of each type of function. In what ways are the two graphs similar to each other? What major graphical di� erence would allow you to tell which graph is which if they were not labeled?

22. Suppose that y decreases exponentially with x and that z varies inversely with x. Sketch the graph of each type of function. Give at least three ways in which the two graphs are similar to each other. What major graphical di� erence would allow you to tell which graph is which if they were not labeled?

23. Suppose that y varies directly with x and that z increases linearly with x.direct variation function is a linear function but a linear function is not necessarily a direct variation function.

24. Suppose that y varies directly with the square of x and that z is a quadratic function of x.why the direct-square variation function is a quadratic function but the quadratic function is not necessarily a direct-square variation function.

25. Natural Exponential Function Problem: Figure 2-2j shows the graph of the natural exponential function f (x) 3 e 0.8x .g (x) 3 b x . Find the value of b for which g (x) f (x). Show graphically that the two functions are equivalent.

y (4, 30.6)(2, 28.2)

(1, 25.2)

x10

30

y

(1, 6.5)(2, 8.45)

x8

30

y

(1, 48)(2, 24)

x84

50

y

(1, 5)(2, 8)

x8

20

10

y

(2, 6)(5, 2.4)

x10

10

y

(4, 24)(1, 3) x105

100

50

Figure 2-2j

15

10

5

22 11

y

x

y(10, 8)

(0, 0)

x105

10

5


Problem Notes (continued)14a. Quadratic14b. Increasing for x , 4 and decreasing for x . 4, concave down14c. Answers will vary.14d. y 5 2 11 ___ 15 x 2 1 27 ___ 5 x 1 311 ___ 15 14e. Th e graphs match.15a. Exponential15b. Increasing for all real-number values of x, concave up15c. Answers will vary.15d. y 5 5 (1.3) x 15e. Th e graphs match.16a. Exponential16b. Decreasing for all real-number values of x, concave up16c. Answers will vary.16d. y 5 96 (0.5) x 16e. Th e graphs match.17a. Power17b. Increasing for x 0, concave down17c. Answers will vary.17d. y 5 5 x log 2 1.6 17e. Th e graphs match.18a. Power (inverse)18b. Decreasing for x 0, concave up18c. Answers will vary.18d. y 5 12 x 21 18e. Th e graphs match.

Problems 21–24 help students review the meanings of directly proportional, inverse variation, and direct variation.

Problem 25 gives students a natural exponential function and asks them to fi nd an equivalent exponential function with a diff erent base. 25. b 5 e 0.8 5 2.2255... Th e graphs are equivalent.


1. Given the general equation of any particular linear function, use a CAS to move the constant to the other side of the equation, add a nonzero multiple of the slope to both sides, and factor the result. What is the form of the result? What new information does it give? Repeat the process with a diff erent nonzero multiple of the slope. What new

information does this result give? How can you use this information to confi rm the graph of the original linear function?

2. Find an equation of an exponential function containing the points (2, 7) and (7, 13).

3. Is the power function that contains the points (  

__ 3 , 192) and  4, 27 __ 4 even, odd, or

neither?See pages 981–982 for answers to Problems 19–24 and CAS Problems 1–3.


Identifying Functions from Numerical PatternsA 16-in. pizza has four times as much area as an 8-in. pizza. A grapefruit whose diameter is 10 cm has eight times the volume of a grapefruit with diameter 5 cm. In general, when you double the linear dimensions of a three-dimensional object, you multiply the surface area by

time you add 1000 mi to the distance you have driven your car, you add a constant amount—say, $300—to the cost of operating that car.

In this section you will use such patterns to identify the type of function that � ts a given set of function values. � en you will � nd more function values, either by following the pattern or by � nding the equation of the function.

x-values and the corresponding y-values, identify which type of function they � t (linear, quadratic, power, or exponential).

equation.

The Add–Add Pattern of Linear Functions

10864

1 3

32

2

25 7

Linear function

f (x)

x

3

3

x f(x)

13 75 107 139 16

Figure 2-3a

Figure 2-3a shows the graph of the linear function f (x) 1.5x 2.5. As you can see from the graph and the adjacent table, each time you add 2 to x, y increases by 3. � is pattern emerges because a linear function has constant slope. Verbally, you can express this property by saying that every time you add a constant to x, you add a constant (not necessarily the same as the constant added to x) to y. � is property is called the add–add property of linear functions.

Identifying Functions from

2-3

identify which type of function they � t (linear, quadratic, power, or Objective

2

2

2

2

3

3

3

3

� e add-add property


14.

15.

16.

17.

18.

19.

20.

21. Suppose that y increases exponentially with x and that z is directly proportional to the square of x. Sketch the graph of each type of function. In what ways are the two graphs similar to each other? What major graphical di� erence would allow you to tell which graph is which if they were not labeled?

22. Suppose that y decreases exponentially with x and that z varies inversely with x. Sketch the graph of each type of function. Give at least three ways in which the two graphs are similar to each other. What major graphical di� erence would allow you to tell which graph is which if they were not labeled?

23. Suppose that y varies directly with x and that z increases linearly with x.direct variation function is a linear function but a linear function is not necessarily a direct variation function.

24. Suppose that y varies directly with the square of x and that z is a quadratic function of x.why the direct-square variation function is a quadratic function but the quadratic function is not necessarily a direct-square variation function.

25. Natural Exponential Function Problem: Figure 2-2j shows the graph of the natural exponential function f (x) 3 e 0.8x .g (x) 3 b x . Find the value of b for which g (x) f (x). Show graphically that the two functions are equivalent.

y (4, 30.6)(2, 28.2)

(1, 25.2)

x10

30

y

(1, 6.5)(2, 8.45)

x8

30

y

(1, 48)(2, 24)

x84

50

y

(1, 5)(2, 8)

x8

20

10

y

(2, 6)(5, 2.4)

x10

10

y

(4, 24)(1, 3) x105

100

50

Figure 2-2j

15

10

5

22 11

y

x

y(10, 8)

(0, 0)

x105

10

5



Class Time2 days

Homework AssignmentDay 1: RA, Q1–Q10, Problems 1–23 oddDay 2: Problems 25–27, 29–32, 35

Teaching ResourcesExploration 2-3: Patterns for Quadratic

FunctionsExploration 2-3a: Numerical Patterns in

Function ValuesExploration 2-3b: Equations from Given

Values PracticeSupplementary ProblemsTest 4, Sections 2-1 to 2-3, Forms A and B


Exploration 2-3a: Numerical

Patterns in Function Values

Activity: Moore’s Law

Activity: Population Growth

TE ACH I N G

Important Terms and ConceptsAdd–add propertyAdd–multiply propertyMultiply–multiply propertySecond diff erencesDiscrete dataTh ird diff erencesQuartic function


If you double the x-value from 3 to 6 or from 6 to 12, the corresponding y-values are multiplied by 8, or 2 3 . You can see algebraically why this is true.

h(6) 5 6 3

5 3 2 3 Write the x-value 6 as twice 3.

(5 3 3 ) 2 3 Distribute the exponent over multiplication and then associate.

h(3) 8

In conclusion, if you multiply the x-values by 2, the corresponding y-values are multiplied by 8. � is is called the multiply–multiply property of power

in Figure 2-3c. � ey do belong to the function, but the x-values do not � t the “multiply” pattern.

In this exploration, you’ll � nd patterns for the y-values in quadratic functions similar to the add add property of linear functions.

1. Show by making a table on your grapher that the points in the table � t the quadratic function

q(x) 0.2 x 2 1.3x

x q(x)

2 12.212.0

6 8 10 21.0

2. Find the di� erences between consecutive y-values. � en � nd the second di� erences, that is, the di� erences between the consecutive di� erences. What do you notice?

3. Recall that the general equation of a quadratic function is y a x 2 bx c, where a, b, and c stand for constants. Substitute the � rst three

to get three linear equations involving a, b, and c. Solve this system of equations using matrices. Write the particular equation. Does

4. On the same screen, plot the graph of q(x) and the � ve data points. You may use the stat feature on your grapher. Sketch the result.

5. Trace the graph of q(x) to each value of x in the table. Do the � ve points lie on the graph?

6. Show that a quadratic function � ts the data in this table by � nding second di� erences. Find the particular equation, and show that these values satisfy the equation.

x f(x)

1 12.3

71013 26.7


1. Show by making a table on your grapher 4. On the same screen, plot the graph of ( )

E X P L O R AT I O N 2-3: P a t t e r n s f o r Q u a d r a t i c Fu n c t i o n s

x g(x)

3 135 6 1080 9 12


The Add–Multiply Pattern of Exponential Functions Figure 2-3b shows the graph of the exponential function g (x) 5 3 x . �is time, adding 2 to x results in the corresponding g (x)-values being multiplied by the constant 9. �is is not coincidental. Here’s why the pattern holds.

g (1) 5 3 1 15

g (3) 5 3 3 135 (which equals 9 times 15)

You can see algebraically why this is true.

g (3) 5 3 3

5 3 1 2 Write the exponent as 1 increased by 2.

5 3 1 3 2

(5 3 1 ) 3 2 Associate 5 and 3 1 to get g (1) in the expression.

g (1) 9

�e conclusion is that if you add a constant to x, the corresponding y-value is multiplied by the base raised to that constant. �is is called the add–multiply property of exponential functions.

The Multiply–Multiply Pattern of Power Functions h(x)

x3 6 9 12

8640 8 1080

12 2 6Power function

15

8640

1080

Figure 2-3c

Figure 2-3c shows the graph of the power function h(x) 5 x 3 . As shown in the table, adding a constant, 3, to x does not create a corresponding pattern.

However, a pattern does emerge if you pick values of x that change by being multiplied by a constant.

h(3) 5 3 3 135

h(6) 5 6 3 1080 (which equals 8 times 135)

h(12) 5 12 3

Exponential function1

15

135 15 9

3 1 2

g(x)135

15 x

Figure 2-3b

2

2

2

9

9

9

x g(x)

1 153 1355 12157 10935

�e add-multiply property

�e multiply-multiply property

8 Ignore x 9.

8

2

2


Section Notes

In Section 2-2, students learned to recognize the type of function based on its graph. In this section, students learn to use numerical patterns in the x- and y-values to identify the function type. It is recommended that you spend two days on this section. On Day 1, present the add–multiply pattern of exponential functions and the multiply–multiply pattern of power functions using the section’s graphs and tables. You may prefer to use Exploration 2-3a in the Instructor’s Resource Book to introduce these patterns. Have students do Exploration 2-3 on page 79, which introduces the pattern for the quadratic function. Students usually learn these patterns quickly.

Discuss the add–add, add–multiply, multiply–multiply, and second-differences patterns, using the graphs and tables given in the section. Be sure to emphasize that to use these patterns to identify function type, the x-values must be regularly spaced. The box on page 80 summarizing these four properties should help students as they work on the problems.

Example 1 on page 81 asks students to identify the pattern and then choose the appropriate model, in this case quadratic. Students do not need to find the particular equation for the function.

Example 2 on page 81 gives students two function values and asks them to find a third value in three cases—when the function is linear, when it is a power function, and when it is exponential. To find the three values, students must apply the add–add, add–multiply, and multiply–multiply properties, respectively. This example helps students to distinguish between the various properties. As in Example 1, students do not need to find particular equations for the functions.

For Example 3 on page 82, you may need to remind students what varies directly and varies inversely mean. These ideas were discussed in Section 2-2.

In Example 4 on page 82, you may need to explain what a direct-square power function is. Setting up the example using this table might be helpful.

x f (x)

32

5 1000 3 4 2

20 ?


If you double the x-value from 3 to 6 or from 6 to 12, the corresponding y-values are multiplied by 8, or 2 3 . You can see algebraically why this is true.

h(6) 5 6 3

5 3 2 3 Write the x-value 6 as twice 3.

(5 3 3 ) 2 3 Distribute the exponent over multiplication and then associate.

h(3) 8

In conclusion, if you multiply the x-values by 2, the corresponding y-values are multiplied by 8. � is is called the multiply–multiply property of power

in Figure 2-3c. � ey do belong to the function, but the x-values do not � t the “multiply” pattern.

In this exploration, you’ll � nd patterns for the y-values in quadratic functions similar to the add add property of linear functions.

1. Show by making a table on your grapher that the points in the table � t the quadratic function

q(x) 0.2 x 2 1.3x

x q(x)

2 12.212.0

6 8 10 21.0

2. Find the di� erences between consecutive y-values. � en � nd the second di� erences, that is, the di� erences between the consecutive di� erences. What do you notice?

3. Recall that the general equation of a quadratic function is y a x 2 bx c, where a, b, and c stand for constants. Substitute the � rst three

to get three linear equations involving a, b, and c. Solve this system of equations using matrices. Write the particular equation. Does

4. On the same screen, plot the graph of q(x) and the � ve data points. You may use the stat feature on your grapher. Sketch the result.

5. Trace the graph of q(x) to each value of x in the table. Do the � ve points lie on the graph?

6. Show that a quadratic function � ts the data in this table by � nding second di� erences. Find the particular equation, and show that these values satisfy the equation.

x f(x)

1 12.3

71013 26.7


1. Show by making a table on your grapher 4. On the same screen, plot the graph of ( )

E X P L O R AT I O N 2-3: P a t t e r n s f o r Q u a d r a t i c Fu n c t i o n s

x g(x)

3 135 6 1080 9 12


The Add–Multiply Pattern of Exponential Functions Figure 2-3b shows the graph of the exponential function g (x) 5 3 x . �is time, adding 2 to x results in the corresponding g (x)-values being multiplied by the constant 9. �is is not coincidental. Here’s why the pattern holds.

g (1) 5 3 1 15

g (3) 5 3 3 135 (which equals 9 times 15)

You can see algebraically why this is true.

g (3) 5 3 3

5 3 1 2 Write the exponent as 1 increased by 2.

5 3 1 3 2

(5 3 1 ) 3 2 Associate 5 and 3 1 to get g (1) in the expression.

g (1) 9

�e conclusion is that if you add a constant to x, the corresponding y-value is multiplied by the base raised to that constant. �is is called the add–multiply property of exponential functions.

The Multiply–Multiply Pattern of Power Functions h(x)

x3 6 9 12

8640 8 1080

12 2 6Power function

15

8640

1080

Figure 2-3c

Figure 2-3c shows the graph of the power function h(x) 5 x 3 . As shown in the table, adding a constant, 3, to x does not create a corresponding pattern.

However, a pattern does emerge if you pick values of x that change by being multiplied by a constant.

h(3) 5 3 3 135

h(6) 5 6 3 1080 (which equals 8 times 135)

h(12) 5 12 3

Exponential function1

15

135 15 9

3 1 2

g(x)135

15 x

Figure 2-3b

2

2

2

9

9

9

x g(x)

1 153 1355 12157 10935

�e add-multiply property

�e multiply-multiply property

8 Ignore x 9.

8

2

2

79

Exploration Notes

Exploration 2-3 gives students practice identifying the second differences for a quadratic function and finding a particular equation for the function. Problem 2 suggests that students use the matrix feature on their graphers to find the equation. If your students do not know how to work with matrices, they can solve the system using algebraic methods. You might have students complete this exploration in class, after you have discussed the add–add, add–multiply, and multiply–multiply patterns. Allow students 20–25 minutes to complete this activity.

See page 81 for notes on additional explorations.

1. The values are the same as in the table shown.

2. First differences: 20.2, 1.4, 3.0, 4.6 Second differences: 1.6, 1.6, 1.6 All the second differences are the same!

3. 12.2 5 4a 1 2b 1 c 12.0 5 16a 1 4b 1 c 13.4 5 36a 1 6b 1 c q(x) 5 0.2 x 2 2 1.3x 1 14

4.

5. Yes, the five points lie on the graph.

6. Second differences: 212.6, 212.6, 212.6 f (x) 5 2 0.7x 2 1 11x 1 2


y

x12108642

25

20

15

10

5

Example 5 on page 82 is a radioactive-decay problem. The value of f (12) can be found by extending the add–multiply pattern in the table. However, to find the value of f (25), it is necessary to find the particular equation. Part d explains how to do this.

Be sure to discuss the note after Example 5, which emphasizes that there are many functions that fit a particular set of points. Problem 30 also illustrates this point.

Section 2-3: Identifying Functions from Numerical Patterns


The Constant-Second-Di� erences Pattern of Quadratic Functions

q(x)

x97531

Quadratic function

100 x f(x)

1 153 55 197 579 119

Constant second di� erencesFigure 2-3d

Figure 2-3d shows the graph of the quadratic function q(x) 3 x 2 17x 29. A for linear functions applies to quadratics. For equally spaced x-values, the di� erences between the corresponding y-values are equally spaced. � us the di� erences between these di� erences (the second di� erences) are constant. � is constant is equal to 2 ad 2 , twice the coe� cient of the quadratic term times the square of the di� erence between the x-values.

� ese four properties are summarized in the box.

PROPERTIES: Patterns for Function Values

Add–Add Property of Linear FunctionsIf f is a linear function, adding a constant to x results in adding a constant to the corresponding f (x)-value. � at is,

if f (x) ax b and x 2 c x 1 , then f x 2 ac f x 1

Add–Multiply Property of Exponential FunctionsIf f is an exponential function, adding a constant to x results in multiplying the corresponding f (x)-value by a constant. � at is,

if f (x) a b x and x 2 c x 1 , then f x 2 b c f x 1

Multiply–Multiply Property of Power FunctionsIf f is a power function, multiplying x by a constant results in multiplying the corresponding f (x)-value by a constant. � at is,

if f (x) a x b and x 2 c x 1 , then f x 2 c b f x 1

Constant-Second-Di� erences Property of Quadratic FunctionsIf f is a quadratic function, f (x) a x 2 bx c, and the x-values are spaced d units apart, then the second di� erences between the f (x)-values are constant and equal to 2ad 2 .

2

2

2

2

10

38

62

Constant second di� erences

First di� erences

Identify the pattern in these function values and the kind of function that has this pattern.

the constant-second-di� erences property, as shown in the table. � erefore, a quadratic function � ts the data.

If function f has values f (5) 12 and f (10) 18, � nd f (20) if f is

a. A linear function

b. A power function

c. An exponential function

a. � rst x-value to get the second one and that you add 6 to the � rst f (x)-value to get the second one. Make a table of values ending at x 20. � e answer is f (20) 30.

x f (x)

5 12 10 18 15 20 30

b.

the � rst to the second x- and f (x)-values, notice that you multiply 5 by 2 to get 10 and that you multiply 12 by 1.5 to get 18. Make a table of values ending at x 20. � e answer is f (20) 27.

x f (x)

5 1210 1820 27


EXAMPLE 1 ➤ x f(x)

55 76 117 178 25

SOLUTION x f(x)

55 76 117 178 25

2

6

8

1

1

1

1

2

2

2

➤

If function EXAMPLE 2 ➤

a. � rst

SOLUTION

+5

+5+6

+6

x

f

10 205 15

12

24

6

18

(x)Linear

5

5

5

6

6

6

2

2

1.5

1.5


The Constant-Second-Di�erences Pattern of Quadratic Functions

q(x)

x97531

Quadratic function

100 x f(x)

1 153 55 197 579 119

Constant second di�erencesFigure 2-3d

Figure 2-3d shows the graph of the quadratic function q(x) 3 x 2 17x 29. A for linear functions applies to quadratics. For equally spaced x-values, the di�erences between the corresponding y-values are equally spaced. �us the di�erences between these di�erences (the second di�erences) are constant. �is constant is equal to 2 ad 2 , twice the coe�cient of the quadratic term times the square of the di�erence between the x-values.

�ese four properties are summarized in the box.


Add–Add Property of Linear FunctionsIf f is a linear function, adding a constant to x results in adding a constant to the corresponding f (x)-value. �at is,


Add–Multiply Property of Exponential FunctionsIf f is an exponential function, adding a constant to x results in multiplying the corresponding f (x)-value by a constant. �at is,


Multiply–Multiply Property of Power FunctionsIf f is a power function, multiplying x by a constant results in multiplying the corresponding f (x)-value by a constant. �at is,


Constant-Second-Di�erences Property of Quadratic FunctionsIf f is a quadratic function, f (x) a x 2 bx c, and the x-values are spaced d units apart, then the second di�erences between the f (x)-values are constant and equal to 2ad 2 .

2

2

2

2

10

38

62

Constant second di�erences

First di�erences


Differentiating Instruction• Have students, in pairs, write each of

these properties in their own words: add–add property, add–multiply property, multiply–multiply property, constant-second-differences-property.

• Monitor students as they do the exploration. If you want ELL students to use the STAT feature on their grapher, you may need to teach them how.

• For Problems 25–27, ELL students may not be familiar with the relationship between areas and volumes of similar objects.

• Proofs as in Problems 33–36 may be difficult for ELL students, as the concept of proof varies from language to language. You may want to delay assigning them or make them extra credit.


The Constant-Second-Di� erences Pattern of Quadratic Functions

q(x)

x97531

Quadratic function

100 x f(x)

1 153 55 197 579 119

Constant second di� erencesFigure 2-3d

Figure 2-3d shows the graph of the quadratic function q(x) 3 x 2 17x 29. A for linear functions applies to quadratics. For equally spaced x-values, the di� erences between the corresponding y-values are equally spaced. � us the di� erences between these di� erences (the second di� erences) are constant. � is constant is equal to 2 ad 2 , twice the coe� cient of the quadratic term times the square of the di� erence between the x-values.

� ese four properties are summarized in the box.


Add–Add Property of Linear FunctionsIf f is a linear function, adding a constant to x results in adding a constant to the corresponding f (x)-value. � at is,


Add–Multiply Property of Exponential FunctionsIf f is an exponential function, adding a constant to x results in multiplying the corresponding f (x)-value by a constant. � at is,


Multiply–Multiply Property of Power FunctionsIf f is a power function, multiplying x by a constant results in multiplying the corresponding f (x)-value by a constant. � at is,


Constant-Second-Di� erences Property of Quadratic FunctionsIf f is a quadratic function, f (x) a x 2 bx c, and the x-values are spaced d units apart, then the second di� erences between the f (x)-values are constant and equal to 2ad 2 .

2

2

2

2

10

38

62

Constant second di� erences

First di� erences


the constant-second-di� erences property, as shown in the table. � erefore, a quadratic function � ts the data.

If function f has values f (5) 12 and f (10) 18, � nd f (20) if f is

a. A linear function

b. A power function

c. An exponential function

a. � rst x-value to get the second one and that you add 6 to the � rst f (x)-value to get the second one. Make a table of values ending at x 20. � e answer is f (20) 30.

x f (x)

5 12 10 18 15 20 30

b.

the � rst to the second x- and f (x)-values, notice that you multiply 5 by 2 to get 10 and that you multiply 12 by 1.5 to get 18. Make a table of values ending at x 20. � e answer is f (20) 27.

x f (x)

5 1210 1820 27


EXAMPLE 1 ➤ x f(x)

55 76 117 178 25

SOLUTION x f(x)

55 76 117 178 25

2

6

8

1

1

1

1

2

2

2

➤

If function EXAMPLE 2 ➤

a. � rst

SOLUTION

+5

+5+6

+6

x

f

10 205 15

12

24

6

18

(x)Linear

5

5

5

6

6

6

2

2

1.5

1.5


The Constant-Second-Di�erences Pattern of Quadratic Functions

q(x)

x97531

Quadratic function

100 x f(x)

1 153 55 197 579 119

Constant second di�erencesFigure 2-3d

Figure 2-3d shows the graph of the quadratic function q(x) 3 x 2 17x 29. A for linear functions applies to quadratics. For equally spaced x-values, the di�erences between the corresponding y-values are equally spaced. �us the di�erences between these di�erences (the second di�erences) are constant. �is constant is equal to 2 ad 2 , twice the coe�cient of the quadratic term times the square of the di�erence between the x-values.

�ese four properties are summarized in the box.


Add–Add Property of Linear FunctionsIf f is a linear function, adding a constant to x results in adding a constant to the corresponding f (x)-value. �at is,


Add–Multiply Property of Exponential FunctionsIf f is an exponential function, adding a constant to x results in multiplying the corresponding f (x)-value by a constant. �at is,


Multiply–Multiply Property of Power FunctionsIf f is a power function, multiplying x by a constant results in multiplying the corresponding f (x)-value by a constant. �at is,


Constant-Second-Di�erences Property of Quadratic FunctionsIf f is a quadratic function, f (x) a x 2 bx c, and the x-values are spaced d units apart, then the second di�erences between the f (x)-values are constant and equal to 2ad 2 .

2

2

2

2

10

38

62

Constant second di�erences

First di�erences

81

Additional Exploration Notes

Exploration 2-3a provides practice in finding a pattern in the y-values for regularly spaced x-values for exponential, power, or linear functions. You might assign this at the beginning of either Day 1 or Day 2 to help students review what they learned the day before. Or you might assign this activity as a review before the chapter test. Allow students 20–25 minutes to complete this activity.

Exploration 2-3b provides practice in finding a particular equation and using it to find other values when a set of regularly spaced x- and y-values is given. You might assign this exploration to groups on Day 2 or use it as an extra homework assignment or a review sheet. The problems in the exploration also make excellent quiz or test questions. Allow students 20–25 minutes to complete this activity.



d. Find a particular equation of f (x). Show by plotting that all the f (x)-values in the table satisfy the equation.

e. Use the equation to calculate f (25). Interpret the solution.

a. Follow the add pattern in the x-values until you reach 12, and follow the multiply pattern in the corresponding f (x)-values.

f (12) 0.15625 µCi

b.

c. x-values.

� e x-values skip over 25, so f (25) cannot be found using the pattern.

d. f (x) a b x General equation of an exponential function.

5 a b 2 2.5 a b

Substitute any two of the ordered pairs.

2.5 ___ 5 a b ___ a b 2

0.5 b 2 Simplify.

0. 5 1/2 b Raise both sides to the 1 _ 2 power.

b 0.7071… Store without rounding.

5 a(0.7071… ) 2 Substitute the value for b into one of the equations.

a 5 _________ 0.7071 … 2 10 Solve for a.

f (x) 10(0.7071… ) x Write the particular equation.

Figure 2-3e shows the graph of f passing through all four given points.

e. f (25) 10(0.7071… ) 25 0.0017…

� is means that there was about 0.0017 µCi of 18-FDG a� er 25 h.

possible for other functions to � t this set of points, such as the function

g (x) 10(0.7071… ) x sin __ 2 x See Chapter 5 for the meaning of the sine function.

which also � ts the given points, as shown in Figure 2-3f. Deciding which function � ts better will depend on the situation you are modeling. Also, you can test further to see whether your model is supported by data. For example, to test the second model, you could collect measurements over shorter time intervals and see if the data have a wavy pattern.

a. Follow the add pattern in the multiply pattern in the corresponding

SOLUTION

x (h) f (x) (µCi)

10 0.3125

12 0.15625

Divide the equations. Have the larger exponent in the numerator.

2 4 6 8

f (x) (µCi)

Graph �tspoints.

x (h)

108642

Figure 2-3e

➤

2 4 6 8

f(x) (µCi)

�is graph also�ts the points.

x (h)

108642

Figure 2-3f


c. adding 5 to x results in multiplying the corresponding f (x)-value by 1.5. Make a table of values ending at x 20. � e answer is f (20)

x f(x)

5 1210 1815 2720

Describe the e� ect on y of doubling x if

a. y varies directly with x.

b. y varies inversely with the square of x.

c. y varies directly with the cube of x.

a. y is doubled that is, multiplied by 2 1 .

b. y is multiplied by 1 _ that is, multiplied by 2 2 .

c. y is multiplied by 8 that is, multiplied by 2 3 .

Suppose that f is a direct-square power function and that f (5) 1000. Find f (20).

Because fx 20 as x 5. Multiplying x f (x)-value by 2 because f is a direct square, so

f (20) f ( 5) 2 f (5) 16 1000 16,000

Radioactive Tracer Problem: � e compound 18-� uorodeoxyglucose (18-FDG) is composed of radioactive � uorine (18-F) and a sugar (deoxyglucose). It is used to trace glucose metabolism in the heart. 18-F has a half-life of about 2 h, which means that at the end of each 2-hour time period, only half of the 18-F that was there at the beginning of the time period remains. Suppose a dose of

f (x) be the number of microcuries (µCi) of 18-FDG that remains over time x, in hours, as shown in the table.

a. Find the number of microcuries that remains a� er 12 h.

b. Identify the pattern these data points follow. What type of function shows this pattern?

c. Why can’t you use the pattern to � nd f (25)?

5

5

5

1.5

1.5

1.5x

10 205 15

105

152025

+5

+5 1.5

1.5f (x)

Exponential

➤

Describe the e� ect on EXAMPLE 3 ➤

a. y is doubled y is doubled ySOLUTION

➤

Suppose that EXAMPLE 4 ➤

Because fffx 20 as x

SOLUTION

➤

Radioactive Tracer Problem18-� uorodeoxyglucose (18-FDG) is composed

EXAMPLE 5 ➤

x (h) f (x) (µCi)

2 52.5

6 1.25

8 0.652


Technology Notes

Exploration 2-3a: Numerical Patterns in Function Values in the Instructor’s Resource Book focuses on finite differences. Students might benefit from using Fathom’s tools for making tables of finite differences.

Activity: Moore’s Law in Teaching Mathematics with Fathom has students use Fathom to fit an exponential curve to data on the number of transistors in Intel processors since 1974. Students are then asked to determine the truth of Moore’s Law based on their findings. This would make an excellent project to use now and then to revisit during Section 3-4. Allow 35–50 minutes.

Activity: Population Growth in Teaching Mathematics with Fathom gives experience with exponential functions based on numerical patterns. Students begin by fitting an exponential growth model to population data, and then they introduce a crowding effect to obtain a logistic function. You could ask students to do only the exponential part, or you could have them do the whole activity to foreshadow Section 2-7. Allow 50 minutes.


d. Find a particular equation of f (x). Show by plotting that all the f (x)-values in the table satisfy the equation.

e. Use the equation to calculate f (25). Interpret the solution.

a. Follow the add pattern in the x-values until you reach 12, and follow the multiply pattern in the corresponding f (x)-values.

f (12) 0.15625 µCi

b.

c. x-values.

� e x-values skip over 25, so f (25) cannot be found using the pattern.

d. f (x) a b x General equation of an exponential function.

5 a b 2 2.5 a b

Substitute any two of the ordered pairs.

2.5 ___ 5 a b ___ a b 2

0.5 b 2 Simplify.

0. 5 1/2 b Raise both sides to the 1 _ 2 power.

b 0.7071… Store without rounding.

5 a(0.7071… ) 2 Substitute the value for b into one of the equations.

a 5 _________ 0.7071 … 2 10 Solve for a.

f (x) 10(0.7071… ) x Write the particular equation.

Figure 2-3e shows the graph of f passing through all four given points.

e. f (25) 10(0.7071… ) 25 0.0017…

� is means that there was about 0.0017 µCi of 18-FDG a� er 25 h.

possible for other functions to � t this set of points, such as the function

g (x) 10(0.7071… ) x sin __ 2 x See Chapter 5 for the meaning of the sine function.

which also � ts the given points, as shown in Figure 2-3f. Deciding which function � ts better will depend on the situation you are modeling. Also, you can test further to see whether your model is supported by data. For example, to test the second model, you could collect measurements over shorter time intervals and see if the data have a wavy pattern.

a. Follow the add pattern in the multiply pattern in the corresponding

SOLUTION

x (h) f (x) (µCi)

10 0.3125

12 0.15625

Divide the equations. Have the larger exponent in the numerator.

2 4 6 8

f (x) (µCi)

Graph �tspoints.

x (h)

108642

Figure 2-3e

➤

2 4 6 8

f(x) (µCi)

�is graph also�ts the points.

x (h)

108642

Figure 2-3f


c. adding 5 to x results in multiplying the corresponding f (x)-value by 1.5. Make a table of values ending at x 20. � e answer is f (20)

x f(x)

5 1210 1815 2720

Describe the e� ect on y of doubling x if

a. y varies directly with x.

b. y varies inversely with the square of x.

c. y varies directly with the cube of x.

a. y is doubled that is, multiplied by 2 1 .

b. y is multiplied by 1 _ that is, multiplied by 2 2 .

c. y is multiplied by 8 that is, multiplied by 2 3 .

Suppose that f is a direct-square power function and that f (5) 1000. Find f (20).

Because fx 20 as x 5. Multiplying x f (x)-value by 2 because f is a direct square, so

f (20) f ( 5) 2 f (5) 16 1000 16,000

Radioactive Tracer Problem: � e compound 18-� uorodeoxyglucose (18-FDG) is composed of radioactive � uorine (18-F) and a sugar (deoxyglucose). It is used to trace glucose metabolism in the heart. 18-F has a half-life of about 2 h, which means that at the end of each 2-hour time period, only half of the 18-F that was there at the beginning of the time period remains. Suppose a dose of

f (x) be the number of microcuries (µCi) of 18-FDG that remains over time x, in hours, as shown in the table.

a. Find the number of microcuries that remains a� er 12 h.

b. Identify the pattern these data points follow. What type of function shows this pattern?

c. Why can’t you use the pattern to � nd f (25)?

5

5

5

1.5

1.5

1.5x

10 205 15

105

152025

+5

+5 1.5

1.5f (x)

Exponential

➤

Describe the e� ect on EXAMPLE 3 ➤

a. y is doubled y is doubled ySOLUTION

➤


Because fffx 20 as x

SOLUTION

➤

Radioactive Tracer Problem18-� uorodeoxyglucose (18-FDG) is composed

EXAMPLE 5 ➤

x (h) f (x) (µCi)

2 52.5

6 1.25

8 0.652

83

CAS Suggestions

Numerical pattern properties for various functions can be verifi ed with a CAS without overwhelming students with intermediate algebra. Th is is where the power of a CAS is most useful. Once students learn how to ask the general algebraic form of the questions, the CAS handles the diffi cult algebra, and the student is left to interpret and use the results.

Encourage students to fi nd ways to prove numerical pattern properties on their own using a CAS.

Example 3 can be solved using a CAS by fi rst defi ning the functions. In this case, doubling x in parts b and c multiplies the function by 1 _ 4 and 8, respectively.

At this level, students should be able to look at the results in these screens and read from the algebraic form that doubling x multiplies the corresponding y-values by 1 _ 4 and 8, respectively. If you want to be more explicit, enter f (2x)

____ f (x) to see the multipliers in isolation.



11. x f (x)

1 3523 13657 1369 352

12. x f (x)

1 25 5 85 9 113

13 10917 73

For value if f is

a. A linear function b. A power function c. An exponential function

13. Given f (2) 5 and f (6) 20, �nd f (18).14. Given f (3) 80 and f (6) 120, �nd f15. Given f (10) 100 and f (20) 90, �nd f16. Given f (1) 1000 and f (3) 100, �nd f (9).

the other values specified.17. Given f is a linear function with f (2) 1 and

f (5) 7, �nd f (8), f (11), and f18. Given f is a direct-cube power function with

f (3) 0.7, �nd f (6) and f (12).19. Given that f (x) varies inversely with the square

of x and that f (5) 1296, �nd f (10) and f (20).20. Given that f (x) varies exponentially with x and

that f (1) 100 and f 90, �nd f (7), f (10), and f (16).

f (x) if you double the value of x.21. Direct-square power function22. Direct fourth-power function23. Inverse variation power function24. Inverse-square variation power function25. Volume Problem: �e volumes of similarly

shaped objects are directly proportional to the cube of a linear dimension.

a. Recall from geometry that the volume, V, of a sphere equals _ 3 r 3 , where r is the radius.

V _ 3 r 3 shows that the volume of a sphere varies directly with the cube of the radius. If a baseball has volume 200 c m 3 , what is the volume of a volleyball that has three times the radius of the baseball (Figure 2-3g)?

b. King Kong is depicted as having the same proportions as a normal gorilla but as being 10 times as tall. How would his volume (and thus his weight) compare to that of a normal gorilla? If a normal gorilla weighs

weigh? Is this surprising? c. A great white shark 20 � long weighs about

of years ago suggest that there were once great whites 100 � long. How much would you expect such a shark to weigh?

d. were 1 __ 10 as tall as normal people. If Gulliver weighed 200 lb, how much would you expect a

26. Area Problem: �e areas of similarly shaped objects are directly proportional to the square of a linear dimension.

a. Give the formula for the area of a circle.

square of the radius. b. If a grapefruit has twice the diameter of

an orange, how do the areas of their rinds compare?VolleyballBaseball

Figure 2-3g

Iris Weddell White’s illustration Visits Gulliver in Jonathan Swi�’s Gulliver’s Travels. (�e Granger Collection, New York)


Reading Analysis

From what you have read in this section, what do you consider to be the main idea? How is the

the concept of slope? How do the properties of exponential and power functions di� er? If you triple the diameter of a circle, what e� ect does this have on the circle’s area? What type of function has this property? What numerical pattern do quadratic functions have?

Quick Review

Q1. Write the general equation of a linear function. Q2. Write the general equation of a power function. Q3. Write the general equation of an exponential

function. Q4. Write the general equation of a quadratic

function. Q5. f (x) 3 x 5 is the equation of a particular

? function. Q6. f (x) 3 5 x is the equation of a particular

? function. Q7. Name the transformation of f (x) that gives

g (x) f (x). Q8. � e function g (x) 3 f (5(x 6)) is a

vertical dilation of function f by a factor of A. 3 B. C. 5 D. 6 E. 6

Q9. Sketch the graph of a linear function with negative slope and positive y-intercept.

Q10. Sketch the graph of an exponential function with base greater than 1.

constant-second-di� erences pattern. Identify the type of function that has the pattern.

1. x f (x)

2 2700

6 2100

8 1800

10 1500

2. x f (x)

2 1500750

6 500 8 375 10 300

3. x f (x)

2 12

6 108 8 192 10 300

4. x f (x)

2 12

6 192 8 768 10 3072

5. x f (x)

2 2652

6 78 8 10 130

6. x f (x)

26.0

6 8 8.8 10 10.2

7. x f (x)

2 1800

6 200 8 112.5 10 72

8. x f (x)

2 100

6 200 8 500 10 800

9. x f (x)

2 900 100

6 111.111 . . . 8 10 0.1371…

10. x f (x)

2 5.6

6 151.2 8 10 700.0

5min

Problem Set 2-3


PRO B LE M N OTES


Using a CAS to solve the problems in this problem set should not be seen as disabling their intent. Knowing what question to ask of a CAS and how to ask it is a complex mathematical skill. Q1. y 5 ax 1 b Q2. y 5 ax b , a 0, b 0 Q3. y 5 ax b , b . 0, b 1, a 0 Q4. y 5 ax 2 1 bx 1 c, a 0Q5. PowerQ6. ExponentialQ7. Vertical dilation by 4Q8. BQ9.

Q10.

Problems 1–24 follow the examples and should be routine for students. 1. Add–add property: linear2. Multiply–multiply property: power, inverse variation3. Multiply–multiply property: power; and constant-second-diff erences property: quadratic

y

x

y

x

4. Add–multiply property: exponential5. Add–add property: linear; multiply–multiply property: power6. Add–add property: linear7. Multiply–multiply property: power, inverse variation8. Add–add property: linear

9. Add–multiply property: exponential10. Multiply–multiply property: power11. Constant-second-diff erences property: quadratic12. Constant-second-diff erences property: quadratic


11. x f (x)

1 3523 13657 1369 352

12. x f (x)

1 25 5 85 9 113

13 10917 73

For value if f is

a. A linear function b. A power function c. An exponential function

13. Given f (2) 5 and f (6) 20, �nd f (18).14. Given f (3) 80 and f (6) 120, �nd f15. Given f (10) 100 and f (20) 90, �nd f16. Given f (1) 1000 and f (3) 100, �nd f (9).

the other values specified.17. Given f is a linear function with f (2) 1 and

f (5) 7, �nd f (8), f (11), and f18. Given f is a direct-cube power function with

f (3) 0.7, �nd f (6) and f (12).19. Given that f (x) varies inversely with the square

of x and that f (5) 1296, �nd f (10) and f (20).20. Given that f (x) varies exponentially with x and

that f (1) 100 and f 90, �nd f (7), f (10), and f (16).

f (x) if you double the value of x.21. Direct-square power function22. Direct fourth-power function23. Inverse variation power function24. Inverse-square variation power function25. Volume Problem: �e volumes of similarly

shaped objects are directly proportional to the cube of a linear dimension.

a. Recall from geometry that the volume, V, of a sphere equals _ 3 r 3 , where r is the radius.

V _ 3 r 3 shows that the volume of a sphere varies directly with the cube of the radius. If a baseball has volume 200 c m 3 , what is the volume of a volleyball that has three times the radius of the baseball (Figure 2-3g)?

b. King Kong is depicted as having the same proportions as a normal gorilla but as being 10 times as tall. How would his volume (and thus his weight) compare to that of a normal gorilla? If a normal gorilla weighs

weigh? Is this surprising? c. A great white shark 20 � long weighs about

of years ago suggest that there were once great whites 100 � long. How much would you expect such a shark to weigh?

d. were 1 __ 10 as tall as normal people. If Gulliver weighed 200 lb, how much would you expect a

26. Area Problem: �e areas of similarly shaped objects are directly proportional to the square of a linear dimension.

a. Give the formula for the area of a circle.

square of the radius. b. If a grapefruit has twice the diameter of

an orange, how do the areas of their rinds compare?VolleyballBaseball

Figure 2-3g

Iris Weddell White’s illustration Visits Gulliver in Jonathan Swi�’s Gulliver’s Travels. (�e Granger Collection, New York)


Reading Analysis

From what you have read in this section, what do you consider to be the main idea? How is the

the concept of slope? How do the properties of exponential and power functions di� er? If you triple the diameter of a circle, what e� ect does this have on the circle’s area? What type of function has this property? What numerical pattern do quadratic functions have?

Quick Review

Q1. Write the general equation of a linear function. Q2. Write the general equation of a power function. Q3. Write the general equation of an exponential

function. Q4. Write the general equation of a quadratic

function. Q5. f (x) 3 x 5 is the equation of a particular

? function. Q6. f (x) 3 5 x is the equation of a particular

? function. Q7. Name the transformation of f (x) that gives

g (x) f (x). Q8. � e function g (x) 3 f (5(x 6)) is a

vertical dilation of function f by a factor of A. 3 B. C. 5 D. 6 E. 6

Q9. Sketch the graph of a linear function with negative slope and positive y-intercept.

Q10. Sketch the graph of an exponential function with base greater than 1.

constant-second-di� erences pattern. Identify the type of function that has the pattern.

1. x f (x)

2 2700

6 2100

8 1800

10 1500

2. x f (x)

2 1500750

6 500 8 375 10 300

3. x f (x)

2 12

6 108 8 192 10 300

4. x f (x)

2 12

6 192 8 768 10 3072

5. x f (x)

2 2652

6 78 8 10 130

6. x f (x)

26.0

6 8 8.8 10 10.2

7. x f (x)

2 1800

6 200 8 112.5 10 72

8. x f (x)

2 100

6 200 8 500 10 800

9. x f (x)

2 900 100

6 111.111 . . . 8 10 0.1371…

10. x f (x)

2 5.6

6 151.2 8 10 700.0

5min

Problem Set 2-3

85

Problems 13–20 can be solved easily with a CAS system solver, aft er which the additional function values are trivial to compute.13a. 65 13b. 8013c. 1280 14a. 36014b. 270 14c. 1366.87515a. 70 15b. 8115c. 72.9 16a. 2260016b. 10 16c. 0.117. f(8) 5 13, f(11) 5 19, f(14) 5 2518. f(6) 5 5.6, f(12) 5 44.819. f(10) 5 324, f(20) 5 8120. f(7) 5 81, f(10) 5 72.9, f(16) 5 59.04921. Multiply y by 4.22. Multiply y by 16.23. Divide y by 2.24. Divide y by 4.

Problems 25–27 require students to apply the facts that the volumes of similarly shaped objects are directly proportional to the cube of a linear dimension and that the areas of similarly shaped objects are directly proportional to the square of a linear dimension.25a. V(r) has the form V 5 ar 3 where a 5 4 _ 3 ; 5400 cm 3 . 25b. 400,000 lb 5 200 tons25c. 500,000 lb25d. 0.2 lb26a. A 5 r 2 . A(r) has the form A 5 ar 2 where a 5 .26b. Th e grapefruit’s rind would have four times as much area as that of the orange.



x f (x)

2 12

6 1088 192

10 300

f(x)

x108642

300

200

100

Figure 2-3i

a. Show that the function f (x) 3 x 2 �ts the data, as shown in Figure 2-3i.

b. Select radian mode, and then plot f 1 (x) 3 x 2 and f 2 (x) 3 x 2 100 sin __ 2 (x) , where “sin” is the sine function (see Chapter 5). Sketch the result. Does the equation of f 2 (x) also �t the given data?

c. Deactivate f 2 (x) from part b and plot f 3 (x) 3 x 2 cos( x), where “cos” is the cosine function (see Chapter 5). Sketch the result. What do the results tell you about �tting functions to discrete data points?

31. Incorrect Point Problem: By considering second di�erences, show that a quadratic function does not �t the values in this table.

x y

55 76 117 178 27

What would the last y-value have to be for a quadratic function to fit the values exactly?

32. Cubic Function Problem: Figure 2-3j shows the graph of the cubic function

f (x) x 3 6 x 2 5x 20

a. Make a table of values of f (x) for each integer value of x from 1 to 6.

b. Show that the third di�erences between the values of f (x) are constant. You can calculate the third di�erences in a time-e�cient way using the list and delta list features of your grapher. If you do it by pencil and paper, be sure to subtract (value previous value) in each case.

c. Make a conjecture about how you could determine whether a quartic function (fourth degree) �ts a set of points.

33. �e Add–Add Property Proof Problem:for a linear function, adding a constant to x adds a constant to the corresponding value of f (x). Do this by showing that if x 2 x 1 c, then f x 2 equals a constant plus f x 1 . Start by writing the equations of f x 1 and f x 2 , and then make the appropriate substitutions and algebraic manipulations.

34. �e Multiply–Multiply Property Proof Problem: x

by a constant multiplies the corresponding value of f (x) by a constant as well. Do this by showing that if x 2 c x 1 , then f x 2 equals a constant times f x 1 . Start by writing the equations of f x 1 and f x 2 , and then make the appropriate substitutions and algebraic manipulations.

35. �e Add–Multiply Property Proof Problem:

constant to x multiplies the corresponding value of f (x) by a constant. Do this by showing that if x 2 c x 1 , then f x 2 equals a constant times f x 1 . Start by writing the equations of f x 1 and f x 2 , and then make the appropriate substitutions and algebraic manipulations.

36. �e Constant-Second-Di�erences Property Proof Problem: f (x) a x 2 bx c. d be the constant di�erence between successive x-values. Find f (x d), f (x 2d), and f (x 3d). Simplify. By subtracting consecutive f (x)-values, �nd the three �rst di�erences. By subtracting consecutive �rst di�erences, show that the two second di�erences equal the constant 2a d 2 .Figure 2-3j

f (x)

x32 1 4 5 6

100

50


c. When Gutzon Borglum designed the reliefs he carved into Mount Rushmore in South Dakota, he started with models 1 __ 12 the lengths of the actual reliefs. How does the area of each model compare to the area of each of the �nal

in the linear dimension results in a relatively large decrease in the surface area to be carved.

d. Gulliver traveled to Brobdingnag, where

people were 10 times as tall as normal people. If Gulliver had 2 m 2 of skin, how much skin surface would you expect a Brobdingnagian to have?

27. Airplane Weight and Area Problem: In 1896,

an airplane he was designing. In 1903, he tried unsuccessfully to �y the full-size airplane.

length of the model (Figure 2-3h).

a. �e wing area, and thus the li�, of similarly shaped airplanes is directly proportional to the square of the length of each plane. How many times more wing area did the full-size plane have than the model?

b. �e volume, and thus the weight, of similarly shaped airplanes is directly proportional to the cube of the length. How many times heavier was the full-size plane than the model?

c. Why do you think the model was able to �y but the full-size plane was not?

28. Compound Interest Problem: Money le� in a savings account grows exponentially with time. Suppose that you invest $1000 and �nd that a year later you have $1100 in your account.

a.

b. In how many years will your investment double?

29. Archery Problem: Ann Archer shoots an arrow into the air. �e table lists its height at various times a�er she shoots it.

Time (s) Height (ft)

1 79

2 121

3 131 109

5 55

a. Show that the second di�erences between consecutive height values in the table are constant.

b. Use the �rst three ordered pairs to �nd the particular equation of the quadratic function that �ts these points. Show that the function contains all of the points.

c. Based on the graph you �t to the points, how high was the arrow at 2.3 s? Was it going up or going down? How do you tell?

d. At what two times was the arrow 100 � high? How do you explain the fact that there were two times?

e. When was the arrow at its highest? How high was that?

f. At what time did the arrow hit the ground? 30. �e Other Function Fit Problem: It is possible for

di�erent functions to �t the same set of discrete data points. Suppose that the data in the table have been given.

ActualModelFigure 2-3h


Problem Notes (continued)26c. 1 ___ 144 . The proportion of the original length is squared to find the proportion of the original area. 26d. 200 m 2 27a. 16 times more wing area27b. 64 times heavier27c. The full-sized plane had four times as much weight per unit of wing area as the model.

Problems 28 and 29 are application problems that depend on students being able to write the particular equation that meets specific conditions.28a. A(2) 5 $1210, A(3) 5 $1331, A(4) 5 $1464.1028b. After about 7 years29a. [H(3) 2 H(2)] 2 [H(2) 2 H(1)] 5 232 ft; [H(4) 2 H(3)] 2 [H(3) 2 H(2)] 5 232 ft; [H(5) 2 H(4)] 2 [H(4) 2 H(3)] 5 232 ft29b. H(t) 5 216 t 2 1 90t 1 5; H(4) 5 109; H(5) 5 5529c. H(2.3) 5 127.36 ft; going up; the height seems to peak at about t 5 3 s.29d. 1.4079... s (going up) or 4.2170… s (coming down); there are two solutions to the equation at y 5 100.29e. 2.8125 s; 131.5625 ft29f. 5.6800... s

Problem 30 emphasizes the point that more than one function fits a set of data points.30a. x f (x)

2 3(2 ) 2 5 124 3(4 ) 2 5 486 3(6 ) 2 5 1088 3(8 ) 2 5 192

10 3(10 ) 2 5 300

30b.

f 2 (x) also fits the data.

30c.

Many functions can fit a set of discrete data points.

2 4 6 8 10

100

200

300

x

y

2 8 10

100

200

300

x

y


x f (x)

2 12

6 1088 192

10 300

f(x)

x108642

300

200

100

Figure 2-3i

a. Show that the function f (x) 3 x 2 �ts the data, as shown in Figure 2-3i.

b. Select radian mode, and then plot f 1 (x) 3 x 2 and f 2 (x) 3 x 2 100 sin __ 2 (x) , where “sin” is the sine function (see Chapter 5). Sketch the result. Does the equation of f 2 (x) also �t the given data?

c. Deactivate f 2 (x) from part b and plot f 3 (x) 3 x 2 cos( x), where “cos” is the cosine function (see Chapter 5). Sketch the result. What do the results tell you about �tting functions to discrete data points?

31. Incorrect Point Problem: By considering second di�erences, show that a quadratic function does not �t the values in this table.

x y

55 76 117 178 27

What would the last y-value have to be for a quadratic function to fit the values exactly?

32. Cubic Function Problem: Figure 2-3j shows the graph of the cubic function

f (x) x 3 6 x 2 5x 20

a. Make a table of values of f (x) for each integer value of x from 1 to 6.

b. Show that the third di�erences between the values of f (x) are constant. You can calculate the third di�erences in a time-e�cient way using the list and delta list features of your grapher. If you do it by pencil and paper, be sure to subtract (value previous value) in each case.

c. Make a conjecture about how you could determine whether a quartic function (fourth degree) �ts a set of points.

33. �e Add–Add Property Proof Problem:for a linear function, adding a constant to x adds a constant to the corresponding value of f (x). Do this by showing that if x 2 x 1 c, then f x 2 equals a constant plus f x 1 . Start by writing the equations of f x 1 and f x 2 , and then make the appropriate substitutions and algebraic manipulations.

34. �e Multiply–Multiply Property Proof Problem: x

by a constant multiplies the corresponding value of f (x) by a constant as well. Do this by showing that if x 2 c x 1 , then f x 2 equals a constant times f x 1 . Start by writing the equations of f x 1 and f x 2 , and then make the appropriate substitutions and algebraic manipulations.

35. �e Add–Multiply Property Proof Problem:

constant to x multiplies the corresponding value of f (x) by a constant. Do this by showing that if x 2 c x 1 , then f x 2 equals a constant times f x 1 . Start by writing the equations of f x 1 and f x 2 , and then make the appropriate substitutions and algebraic manipulations.

36. �e Constant-Second-Di�erences Property Proof Problem: f (x) a x 2 bx c. d be the constant di�erence between successive x-values. Find f (x d), f (x 2d), and f (x 3d). Simplify. By subtracting consecutive f (x)-values, �nd the three �rst di�erences. By subtracting consecutive �rst di�erences, show that the two second di�erences equal the constant 2a d 2 .Figure 2-3j

f (x)

x32 1 4 5 6

100

50


c. When Gutzon Borglum designed the reliefs he carved into Mount Rushmore in South Dakota, he started with models 1 __ 12 the lengths of the actual reliefs. How does the area of each model compare to the area of each of the �nal

in the linear dimension results in a relatively large decrease in the surface area to be carved.

d. Gulliver traveled to Brobdingnag, where

people were 10 times as tall as normal people. If Gulliver had 2 m 2 of skin, how much skin surface would you expect a Brobdingnagian to have?

27. Airplane Weight and Area Problem: In 1896,

an airplane he was designing. In 1903, he tried unsuccessfully to �y the full-size airplane.

length of the model (Figure 2-3h).

a. �e wing area, and thus the li�, of similarly shaped airplanes is directly proportional to the square of the length of each plane. How many times more wing area did the full-size plane have than the model?

b. �e volume, and thus the weight, of similarly shaped airplanes is directly proportional to the cube of the length. How many times heavier was the full-size plane than the model?

c. Why do you think the model was able to �y but the full-size plane was not?

28. Compound Interest Problem: Money le� in a savings account grows exponentially with time. Suppose that you invest $1000 and �nd that a year later you have $1100 in your account.

a.

b. In how many years will your investment double?

29. Archery Problem: Ann Archer shoots an arrow into the air. �e table lists its height at various times a�er she shoots it.

Time (s) Height (ft)

1 79

2 121

3 131 109

5 55

a. Show that the second di�erences between consecutive height values in the table are constant.

b. Use the �rst three ordered pairs to �nd the particular equation of the quadratic function that �ts these points. Show that the function contains all of the points.

c. Based on the graph you �t to the points, how high was the arrow at 2.3 s? Was it going up or going down? How do you tell?

d. At what two times was the arrow 100 � high? How do you explain the fact that there were two times?

e. When was the arrow at its highest? How high was that?

f. At what time did the arrow hit the ground? 30. �e Other Function Fit Problem: It is possible for

di�erent functions to �t the same set of discrete data points. Suppose that the data in the table have been given.

ActualModelFigure 2-3h

87

32b. Th e third diff erences equal 6. 32c. A quartic function will have constant fourth diff erences.

Problems 33–36 guide students through proofs of the properties from this section.33. If f(x) 5 ax 1 b, then f( x 2 ) 5 f( x 1 1 c) 1 b 5 ax 1 1 ac 1 b 5 ( ax 1 1 b) 1 ac 5 f( x 1 ) 1 ac.34. If f(x) 5 ab x , then f( x 2 ) 5 f( cx 1 ) 5 a( cx 1 ) b 5 a( c b x 1

b ) 5 c b a x 1 b 5 c b f( x 1 ).

35. If f(x) 5 ax b , then f( x 2 ) 5 f(c 1 x 1 ) 5 ab c1 x 1 5 a( b c b x 1 ) 5 b c ab x 1 5 b c f( x 1 ).


1. What is the eff ect on each of the following functions of multiplying x by 2.5?a. f (x) 5 a 4 x b. y varies inversely with the square

root of x.c. y varies directly with the fi ft h

power of x.

2. Cubic functions are constant on the third common diff erence. If the general form of a cubic function is f (x) 5 a x 3 1 b x 2 1 cx 1 d and w is the distance between successive x-values, then what is the constant value of the third diff erences in the f (x) values? (Note: Problem 32 does this for a particular cubic function, but a CAS makes the general solution just as accessible as Problem 32 is without a CAS.)

3. Without using any statistical functions, compute the particular equations of the functions represented by the coordinate pairs in Problems 1–12.

31. If y(8) were 25, then a quadratic function would fi t.

Problem 32 extends the concept of second diff erences and quadratics to third diff erences and cubics.

32a. x f (x)1 202 143 84 85 206 50


See page 982 for answers to Problem 36 and CAS Problems 1–3.

89Section 2-4: Properties of Logarithms

De� nition and Properties of Base-10 LogarithmsTo gain con� dence in the meaning of logarithm, press LOG 3 on your calculator. You will get

log 3

� en, without rounding, raise 10 to this power. You will get

1 0 3

� e powers of 10 have the normal properties of exponentiation. For instance,

15 (3)(5) 1 0 1 0 0.6989…

1 0 0.6989… Add the exponents; keep the same base.

1 0 1.1760…

You can check by calculator that 1 0 1.1760… really does equal 15.

From this example you can infer that logarithms have the same properties as exponents. � is is not surprising, because logarithms are exponents. For instance,

log(3 5) log 3 log 5 � e log of a product equals the sum of the logsof the factors.

From the values given earlier, you can also show that

log 15 ___ 3 log 15 log 3 � e log of a quotient.

� is property is reasonable because you divide powers of equal bases by subtracting the exponents.

15 ___ 3 1 0 1.1760… _______ 1 0 10 1.1760… 1 0 0.6989… 5

Because a power can be written as a product, you can � nd the logarithm of a power like this:

log(3 3 3 3) log 3 log 3 log 3 log 3

Combine like terms.

� e logarithm of a power equals the exponent of that power times the logarithm of the base. To verify this result, observe that 3 81 4 LOG 3 on your

LOG 81. You get the same answer,

A decibel, which measures the relative intensity of sounds, has a logarithmic scale. Prolonged exposure to noise intensity exceeding 85 decibles can lead to hearing loss.


Properties of LogarithmsAny positive number can be written as a power of 10. For instance,

3 1 0

5 1 0 0.6989…

15 1 0 1.1760…

logarithms of 3, 5, and 15, respectively.

log 3

log 5 0.6989…

log 15 1.1760…

Learn the properties of base-10 logarithms.

you an algebraic way to solve exponential equations such as 1.0 6 x 2.


2- 4

Learn the properties of base-10 logarithms.Objective

1. � e LOG key on your calculator � nds the

logarithms:

log 10 ?

log 100 ?

log 1000 ?

log 10000 ?

log 10 5 ?

2. From what log x means. Write what you discover.

3. Based should log 10 1.8 equal?

4. Test the value of 10 1.8 on your calculator and then � nding the logarithm of the (unrounded) answer.

5. Test your conjecture again by � nding

of the answer you get. 6. Find log 2 and log 32. Show numerically

that log 32 is � ve times log 2. 7. Note that 32 2 5 . � us, log 2 5

5 log 2. Show numerically that log 1 7

8. Complete the property of the log of a power: log b x ?


1. � e key on your calculator � nds the 4. Test

E X P L O R AT I O N 2- 4 : I n t r o d u c t i o n t o L o g a r i t h m s


S e c t i o n 2- 4S e c t i o n 2- 4S e c t i o n 2- 4S e c t i o n 2- 4S e c t i o n 2- 4S e c t i o n 2- 4PL AN N I N G

Class Time1 day

Homework AssignmentRA, Q1–Q10, Problems 1–47 odd

Teaching ResourcesExploration 2-4: Introduction to

LogarithmsExploration 2-4a: Introduction to

Logarithmic Functions


CAS Activity 2-4a: Dilations of Exponential Functions

TE ACH I N G

Important Terms and ConceptsBase-10 logarithmLogarithmLogarithm of a productLogarithm of a quotientLogarithm of a powerA logarithm is an exponent

Exploration Notes

Exploration 2-4 introduces students to the properties of logarithms and using logarithms to solve exponential equations.

Note that TI-Nspire handhelds allow you to enter a base. If you do not enter a base, it will automatically enter base 10.

1. log 10 5 1log 100 5 2log 1000 5 3log 10000 5 4log 10 5 5 5

2. log x means the exponent in the power of 10 that equals x.

3. Conjecture: log( 10 1.8 ) 5 1.8

4. 10 1.8 5 63.0957...log 63.0957... 5 1.8, confi rming the conjecture.

5. log 347 5 2.5403... 10 2.5403... 5 347, again confi rming the conjecture.

6. log 2 5 0.3010...log 32 5 1.5051...5 log 2 5 5(0.3010...) 5 1.5051..., which equals log 32.

7. log 17 2.34 5 log 757.2695... 5 2.8792...2.34 log 17 5 2.34(1.2304...) 5 2.8792..., which equals log 17 2.34 .

8. log b x 5 x log b



De� nition and Properties of Base-10 LogarithmsTo gain con� dence in the meaning of logarithm, press LOG 3 on your calculator. You will get

log 3

� en, without rounding, raise 10 to this power. You will get

1 0 3

� e powers of 10 have the normal properties of exponentiation. For instance,

15 (3)(5) 1 0 1 0 0.6989…

1 0 0.6989… Add the exponents; keep the same base.

1 0 1.1760…

You can check by calculator that 1 0 1.1760… really does equal 15.

From this example you can infer that logarithms have the same properties as exponents. � is is not surprising, because logarithms are exponents. For instance,

log(3 5) log 3 log 5 � e log of a product equals the sum of the logsof the factors.

From the values given earlier, you can also show that

log 15 ___ 3 log 15 log 3 � e log of a quotient.

� is property is reasonable because you divide powers of equal bases by subtracting the exponents.

15 ___ 3 1 0 1.1760… _______ 1 0 10 1.1760… 1 0 0.6989… 5

Because a power can be written as a product, you can � nd the logarithm of a power like this:

log(3 3 3 3) log 3 log 3 log 3 log 3

Combine like terms.

� e logarithm of a power equals the exponent of that power times the logarithm of the base. To verify this result, observe that 3 81 4 LOG 3 on your

LOG 81. You get the same answer,

A decibel, which measures the relative intensity of sounds, has a logarithmic scale. Prolonged exposure to noise intensity exceeding 85 decibles can lead to hearing loss.



3 1 0

5 1 0 0.6989…

15 1 0 1.1760…

logarithms of 3, 5, and 15, respectively.

log 3

log 5 0.6989…

log 15 1.1760…

Learn the properties of base-10 logarithms.

you an algebraic way to solve exponential equations such as 1.0 6 x 2.


2- 4

Learn the properties of base-10 logarithms.Objective

1. � e LOG key on your calculator � nds the

logarithms:

log 10 ?

log 100 ?

log 1000 ?

log 10000 ?

log 10 5 ?

2. From what log x means. Write what you discover.

3. Based should log 10 1.8 equal?

4. Test the value of 10 1.8 on your calculator and then � nding the logarithm of the (unrounded) answer.

5. Test your conjecture again by � nding

of the answer you get. 6. Find log 2 and log 32. Show numerically

that log 32 is � ve times log 2. 7. Note that 32 2 5 . � us, log 2 5

5 log 2. Show numerically that log 1 7

8. Complete the property of the log of a power: log b x ?


1. � e key on your calculator � nds the 4. Test

E X P L O R AT I O N 2- 4 : I n t r o d u c t i o n t o L o g a r i t h m s

89

Section Notes

When discussing the defi nition and properties of logarithms, the most important idea to stress is that a logarithm is an exponent, so logarithms obey the same laws exponents do. Refer to the Additional Class Examples, which can help students see the connection between exponents and logarithms.

Examples 1 and 2 on pages 90 and 91 are direct applications of the defi nition of logarithm. Examples 3–5 and 7 on pages 91 and 92 apply the properties of logarithms, and Example 6 on page 92 gives the proof of the “logarithm of a product” property. As you work through the examples with students, keep coming back to the idea that a logarithm is an exponent.

Th is section discusses only base-10 logarithms, which are also called common logarithms. Th e next section introduces base-e logarithms, which are called natural logarithms, as well as the change-of-base property.

If your students use TI-Nspire, consider asking them to enter diff erent values for b as the base of a logarithm. What appears to be the range of acceptable values for b to determine real values for a logarithm? Explain why this restriction is reasonable.

Additional Exploration Notes

Exploration 2-4a enforces the notion that logarithms are exponents by comparing base-10 logarithms with powers of 10. It fi nishes by considering other bases, so it makes a good transitional activity from Section 2-4 to Section 2-5.

Section 2-4: Properties of Logarithms


Find x if 0.258 10 x . Verify your solution numerically.

By de� nition, x, the exponent of 10, is the logarithm of 0.258.

x log 0.258 0.5883… By calculator. Do not round.

1 0 0.5883… 0.258

which checks.

logarithms.

Show numerically that log(7 9) log 7 with the de� nition of logarithm.

log(7 9) log 63 1.7993…

log 7 log 9 1.7993… Calculate without rounding.

log(7 9) log 7 log 9

� is equality agrees with the de� nition because

(7 9) 10 … 10

10 Add the exponents. Keep the same base.

10 1.7993

log(7 9) 1.7993… � e logarithm is the exponent of 10.

Show numerically that log 51 __ 17 log 51 with the de� nition of logarithm.

log 51 ___ 17 log 3


log 51 ___ 17 log 51 log 17


51 ___ 17 10 1.7075 ______ 10 1 1.7075… Subtract the exponents. Keep the same base.

10

log 51 ___ 17 � e logarithm is the exponent of 10.

Find EXAMPLE 2 ➤

By de� nition, By de� nition, SOLUTION

1 0 0.5883…CHECK

➤

Show numerically that log(7 with the de� nition of logarithm.

EXAMPLE 3 ➤

log(7 SOLUTION

➤

Show numerically that log with the de� nition of logarithm.

EXAMPLE 4 ➤

log 51___log ___log 17 SOLUTION

➤


The definition and three properties of logarithms are summarized in this box.

DEFINITION AND PROPERTIES: Base-10 LogarithmsDe� nition

log x y if and only if 1 0 y x

Verbally: log x is the exponent in the power of 10 that gives x

Properties

log xy log x log y

Verbally: � e log of a product equals the sum of the logs of the factors.

log x __ y log x log y

Verbally: � e log of a quotient equals the log of the numerator minus the log of the denominator.

log x y y log x

Verbally: � e log of a power equals the exponent times the log of the base.

� e reason for the name logarithm is historical. Before there were calculators, base-10 logarithms, calculated approximately using in� nite series, were recorded

could then be calculated by adding their logarithms (exponents) column-wise in one step rather than by tediously multiplying several pairs of numbers.

credited with inventing this “logical way to do arithmetic” that you will explore in logarithm actually comes from the Greek words logos, which

here means “ratio,” and arithmos, which means “number.”

� e most important thing to remember about logarithms is this:

A logarithm is an exponent.

Find x if log 10 3.721 x. Verify your solution numerically.

By de� nition, the logarithm is the exponent of 10. So x 3.721.

10 3.721 5260.1726… By calculator. Do not round.

log 5260.1726… 3.721

which checks.

Slide rules, used by engineers in the 19th and early 20th centuries, employ the principle of logarithms for performing complicate calculations.

Find EXAMPLE 1 ➤

By de� nition, the logarithm is the exponent of 10. So By de� nition, the logarithm is the exponent of 10. So SOLUTION

10 3.721CHECK


Differentiating Instruction• If you assign Problems 45 and 46, you

may need to help ELL students with the proof.

• ELL students may struggle with the language in Problem 47.

• Have students put the base-10 logarithmic properties in their journal, both symbolically and verbally, for ELL students, perhaps in their primary language.

• Verify that ELL students have done the Reading Analysis questions, perhaps in pairs.


Find x if 0.258 10 x . Verify your solution numerically.

By de� nition, x, the exponent of 10, is the logarithm of 0.258.

x log 0.258 0.5883… By calculator. Do not round.

1 0 0.5883… 0.258

which checks.

logarithms.

Show numerically that log(7 9) log 7 with the de� nition of logarithm.

log(7 9) log 63 1.7993…


log(7 9) log 7 log 9


(7 9) 10 … 10

10 Add the exponents. Keep the same base.

10 1.7993

log(7 9) 1.7993… � e logarithm is the exponent of 10.

Show numerically that log 51 __ 17 log 51 with the de� nition of logarithm.

log 51 ___ 17 log 3


log 51 ___ 17 log 51 log 17


51 ___ 17 10 1.7075 ______ 10 1 1.7075… Subtract the exponents. Keep the same base.

10

log 51 ___ 17 � e logarithm is the exponent of 10.

Find EXAMPLE 2 ➤

By de� nition, By de� nition, SOLUTION

1 0 0.5883…CHECK

➤

Show numerically that log(7 with the de� nition of logarithm.

EXAMPLE 3 ➤

log(7 SOLUTION

➤

Show numerically that log with the de� nition of logarithm.

EXAMPLE 4 ➤

log 51___log ___log 17 SOLUTION

➤


The definition and three properties of logarithms are summarized in this box.

DEFINITION AND PROPERTIES: Base-10 LogarithmsDe� nition

log x y if and only if 1 0 y x

Verbally: log x is the exponent in the power of 10 that gives x

Properties

log xy log x log y

Verbally: � e log of a product equals the sum of the logs of the factors.

log x __ y log x log y

Verbally: � e log of a quotient equals the log of the numerator minus the log of the denominator.

log x y y log x

Verbally: � e log of a power equals the exponent times the log of the base.

� e reason for the name logarithm is historical. Before there were calculators, base-10 logarithms, calculated approximately using in� nite series, were recorded

could then be calculated by adding their logarithms (exponents) column-wise in one step rather than by tediously multiplying several pairs of numbers.

credited with inventing this “logical way to do arithmetic” that you will explore in logarithm actually comes from the Greek words logos, which

here means “ratio,” and arithmos, which means “number.”

� e most important thing to remember about logarithms is this:

A logarithm is an exponent.

Find x if log 10 3.721 x. Verify your solution numerically.

By de� nition, the logarithm is the exponent of 10. So x 3.721.

10 3.721 5260.1726… By calculator. Do not round.

log 5260.1726… 3.721

which checks.

Slide rules, used by engineers in the 19th and early 20th centuries, employ the principle of logarithms for performing complicate calculations.

Find EXAMPLE 1 ➤

By de� nition, the logarithm is the exponent of 10. So By de� nition, the logarithm is the exponent of 10. So SOLUTION

10 3.721CHECK

91

Technology Notes

CAS Activity 2-4a: Dilations of Exponential Functions in the Instructor’s Resource Book has students use a CAS to explore how horizontally translated exponential functions relate to dilations of exponential functions. Allow 20–25 minutes.

Additional Class Examples

1. Find x 5 log10 37. Check your answer by showing that 10x 5 37 for the value of x you found.Solution

log 10 37 5 1.5682…Press log (37) on your calculator.

101.5682… 5 37, which checksFind 10 answer without round-off .

2. Find x 5 10 20.375 . Check your answer by showing the log 10 x 5 20.375 for the value of x you found.Solution

1020.375 5 0.4216….Use your calculator.

log 0.4216… 5 20.375Take log (answer) without round-off .

Th e logarithm is the same as the exponent of 10.



Reading AnalysisFrom what you have read in this section, what

statement “A logarithm is an exponent” and support it with examples.

Quick Review

Q1. In the expression 7 5 , the number 7 is called the ? .


Q3. � e entire expression 7 5 is called a(n) ? .

Q4. Write x 5 x 7 as a single exponential expression.

Q5. Write x 5 __ x 7 as a single exponential expression.

Q6. Write the expression ( x 5 ) 7 without parentheses.

Q7. For the expression (xy ) 7 , you ? the exponent 7 to get x 7 y 7 .

Q8. 5 2 .

Q9. 9 1/2 .

Q10. � e function y 5 x is called a(n) A.

B.

C.

D.

E. Inverse of a power function

1. What does it mean to say that equals log 0.7?

2. What does it mean to say that 0.9030… equals log 8?

3. By the de� nition of logarithm, if a log b, then 1 0 ? ? .

4. By the de� nition of logarithm, if 1 0 a b, then ? log ? .

write the value of x. � en con� rm that your solution is correct by raising 10 to the given power, taking the logarithm of the result, and showing that the � nal result agrees with your answer. 5. log 10 x 6. log 10 2.803 x

7. log 10 0.981 x 8. log 10 23.58 x

to write x as a logarithm. � en evaluate the logarithm by calculator and show that raising 10 to that power gives a result that agrees with the given equality. 9. 57 10 x 10. 359 10 x 11. 0.85 10 x 12. 0.0321 10 x

calculator. � en show numerically that raising 10 to that power, gives the argument of the logarithm. 13. log 1066 14. log 2001 15. log 0.0596 16.

show that the logarithm of the answer is equal to the original exponent of 10. 17. 10 2.7 18. 10 3.5 19. 10 15.2 20. 10 21.

log 20 log 5 logarithms does this equality illustrate? What property of exponentiation does this property come from?

22. log 120 log 30 logarithms does this equality illustrate? What property of exponentiation does this property come from?

23. Find log 35, log 7, and log 5. Show that log 5 log 35 log 7. What property of logarithms does this equality illustrate? What property of exponentiation does this property come from?

5min

Problem Set 2-4


Show numerically that log 5 3 from the log of a product property.

log 5 3 log 125 2.0969…

3 log 5 3 0.6989… 2.0969… Calculate without rounding.

log 5 3 3 log 5 Combine like terms.

� is equality derives from the product of a log property because

log 5 3 log(5 5 5)

log 5 log 5 log 5 � e log of a product equals the sum of the logs of the factors.

3 log 5 Combine like terms.

of two numbers equals the sum of the logarithms of the factors.

xy log x log y.

Proof

c log x and let d log y.

� en 10 c x and 10 d y. By the de� nition of logarithm.

xy 1 0 c 10 d Multiply x times y.

xy 10 c d Add the exponents. Keep the same base.

log xy c d � e logarithm is the exponent of 10.

log xy log x log y, q.e.d Substitute for c and d.

logarithms algebraically.

expressions that contain logarithms.

Use the properties of logarithms to � nd the number that goes in the blank: log 3 log 7 log 5 log ? . Check your answer numerically.

log 3 log 7 log 5 log 3 7 ____ 5

log 3 log 7 log 5 0.6232… By calculator.

0.6232…

log 3 log 7 log 5

Show numerically that log 5 from the log of a product property.

EXAMPLE 5 ➤

log 5 3SOLUTION

➤

EXAMPLE 6 ➤

ProofProofSOLUTION

➤

Use the properties of logarithms to � nd the number that goes in the blank: log 3

EXAMPLE 7 ➤

log 3 SOLUTION

log 3 log 7 CHECK

➤


CAS Suggestions

Consider having students use a CAS to discover the properties of logarithms instead of telling the students what the properties are. Th e fi gure below shows some sample calculations a student might use to explore the addition property.

Th e diff erence property is a bit trickier for consistent results for students using a TI-Nspire because the programming sometimes produces an unexpected result (see the last line of the fi gure below).

Additionally, some students are surprised when the log of a power property shows up unexpectedly while they are computing data for the log of a product property. Lead them to recognize that the addends could be thought of as the result of the log of a power property to ease the cognitive dissonance. Alternatively, ask students to verify their understanding of logarithmic properties by using Boolean operators.

Th ese suggestions provide motivation for Problems 45 and 46 in the Problem Set.


Reading AnalysisFrom what you have read in this section, what

statement “A logarithm is an exponent” and support it with examples.

Quick Review



Q3. � e entire expression 7 5 is called a(n) ? .

Q4. Write x 5 x 7 as a single exponential expression.

Q5. Write x 5 __ x 7 as a single exponential expression.

Q6. Write the expression ( x 5 ) 7 without parentheses.

Q7. For the expression (xy ) 7 , you ? the exponent 7 to get x 7 y 7 .

Q8. 5 2 .

Q9. 9 1/2 .

Q10. � e function y 5 x is called a(n) A.

B.

C.

D.

E. Inverse of a power function

1. What does it mean to say that equals log 0.7?

2. What does it mean to say that 0.9030… equals log 8?

3. By the de� nition of logarithm, if a log b, then 1 0 ? ? .

4. By the de� nition of logarithm, if 1 0 a b, then ? log ? .

write the value of x. � en con� rm that your solution is correct by raising 10 to the given power, taking the logarithm of the result, and showing that the � nal result agrees with your answer. 5. log 10 x 6. log 10 2.803 x

7. log 10 0.981 x 8. log 10 23.58 x

to write x as a logarithm. � en evaluate the logarithm by calculator and show that raising 10 to that power gives a result that agrees with the given equality. 9. 57 10 x 10. 359 10 x 11. 0.85 10 x 12. 0.0321 10 x

calculator. � en show numerically that raising 10 to that power, gives the argument of the logarithm. 13. log 1066 14. log 2001 15. log 0.0596 16.

show that the logarithm of the answer is equal to the original exponent of 10. 17. 10 2.7 18. 10 3.5 19. 10 15.2 20. 10 21.

log 20 log 5 logarithms does this equality illustrate? What property of exponentiation does this property come from?

22. log 120 log 30 logarithms does this equality illustrate? What property of exponentiation does this property come from?


5min

Problem Set 2-4


Show numerically that log 5 3 from the log of a product property.

log 5 3 log 125 2.0969…

3 log 5 3 0.6989… 2.0969… Calculate without rounding.

log 5 3 3 log 5 Combine like terms.

� is equality derives from the product of a log property because

log 5 3 log(5 5 5)

log 5 log 5 log 5 � e log of a product equals the sum of the logs of the factors.

3 log 5 Combine like terms.

of two numbers equals the sum of the logarithms of the factors.

xy log x log y.

Proof

c log x and let d log y.

� en 10 c x and 10 d y. By the de� nition of logarithm.

xy 1 0 c 10 d Multiply x times y.

xy 10 c d Add the exponents. Keep the same base.

log xy c d � e logarithm is the exponent of 10.

log xy log x log y, q.e.d Substitute for c and d.

logarithms algebraically.

expressions that contain logarithms.

Use the properties of logarithms to � nd the number that goes in the blank: log 3 log 7 log 5 log ? . Check your answer numerically.

log 3 log 7 log 5 log 3 7 ____ 5

log 3 log 7 log 5 0.6232… By calculator.

0.6232…

log 3 log 7 log 5

Show numerically that log 5 from the log of a product property.

EXAMPLE 5 ➤

log 5 3SOLUTION

➤

EXAMPLE 6 ➤

ProofProofSOLUTION

➤

Use the properties of logarithms to � nd the number that goes in the blank: log 3

EXAMPLE 7 ➤

log 3 SOLUTION

log 3 log 7 CHECK

➤

93

PRO B LE M N OTES

Problems 1–44 are like the examples and give students practice applying the defi nition and properties of logarithms.1. 10 20.1549... 5 0.7 2. 10 0.9030... 5 83. 1 0 a 5 b 4. a 5 log b5. x 5 1.574; 10 1.574 5 37.4973…, log 37.4973… 5 1.5746. x 5 2.803 ; 10 2.803 5 635.3309…, log 635.3309… 5 2.8037. x 5 20.981; 10 20.981 5 0.1044…, log 0.1044… 5 20.9818. x 5 23.58; 1 0 23.58 5 0.0002630…, log 0.0002630… 5 23.589. x 5 log 57 5 1.7558…; 10 1.7558... 5 5710. x 5 log 359 5 2.5550…; 10 2.5550... 5 359 11. x 5 log 0.85 5 20.0705…; 1 0 20.0705… 5 0.8512. x 5 log 0.0321 5 21.4934…; 1 0 21.4934… 5 0.032113. 3.0277…; 1 0 3.0277… 5 106614. 3.3012…; 1 0 3.3012… 5 200115. 21.2247…; 1 0 21.2247… 5 0.0596 16. 20.5030…; 1 0 20.5030… 5 0.314 17. 0.001995…; log 0.001995… 5 22.7 18. 3162.2776…; log 3162.2776… 5 3.5 19. 1.5848… 3 10 15 ; log(1.5848… 3 10 15 ) 5 15.220. 1 0 24 5 0.0001; log 0.0001 5 2421. log(5 4) 5 log 20 5 1.3010… 5 0.6989… 1 0.6020… 5 log 5 1 log 4; log xy 5 log x 1 log y; b c b d 5 b c1d 22. log(30 4) 5 log 120 5 2.0791… 5 1.4771… 1 0.6020… 5 log 30 1 log 4; log xy 5 log x 1 log y; b c b d 5 b c1d 23. log(35 7) 5 log 5 5 0.6989… 5 1.5440… 2 0.8450… 5 log 35 2 log 7; log x __ y 5 log x 2 log y; b c __ b d 5 b c2d

Q1. BaseQ2. ExponentQ3. Exponential expressionQ4. x 12 Q5. x 22 5 1 __ x 2

Q6. x 35 Q7. DistributeQ8. 1 __ 5 2

Q9.  __

9 Q10. B


95Section 2-5: Logarithms: Equations and Other Bases

Logarithms: Equations and Other BasesIn the previous section you learned that a logarithm is an exponent of 10. In this section you will learn that it is possible to � nd logarithms using other positive numbers as the base. � en you will learn how to use the properties of logarithms to solve an equation for an unknown exponent or to solve an equation involving logarithms.

Use logarithms with base 10 or other bases to solve exponential or logarithmic equations.

Logarithms with Any Base: The Change-of-Base PropertyIf x 1 0 y , then y is the base-10 logarithm of x. Similarly, if x 2 y , then y is the base-2 logarithm of x. � e only di� erence is the number that is the base. To distinguish among logarithms with di� erent bases, the base is written as a subscript a� er the abbreviation log. For instance,

3 lo g 2 8 2 3 8

lo g 3 81 3 81

2 lo g 10 100 10 2 100

� e symbol lo g 2 8 is pronounced “log to the base 2 of 8.” � e symbol lo g 10 100 is, of course, equivalent to log 100, as de� ned in the previous section. Note that in all cases a logarithm is an exponent.

DEFINITION: Logarithm with Any BaseAlgebraically:

lo g b x y if and only if b y x, where b > 0, b ≠ 1, and x > 0

Verbally: lo g b x y means that y is the exponent of b that gives x as the answer.

� e way you pronounce the symbol for logarithm gives you a way to remember the

Logarithms: Equations and Other Bases

2-5


Objective



25. Find log 32 and log 2. Show that log 32 5 log 2. What property of logarithms does this equality illustrate? What property of exponentiation does this property come from?

26.

this equality illustrate? What property of exponentiation does this property come from?

properties of logarithms. � en explain how each result agrees with the de� nition of logarithm.

27. log(0.3 0.7) log 0.3 log 0.7

28. log(7 8) log 7 log 8

29. log(30 ÷ 5) log 30 log 5

30. log 2 __ 8 log 2 log 8

31. log 2 5 5 log 2

32. log 5 3 3 log 5

33. log 1 __ 7 log 7

34. log 1 ____ 1000 log 1000

35. log 7 log 3 log ?


37. log 12 log ?

38. log 20 log ?

39. log 8 log 5 log 35 log ?

40. log 2000 log 2 log ?

41. 7 log 2 log ?

42. 5 log 3 log ?

43. log 125 ? log 5

44. ? log 2

45. Logarithm of a Power Property Proof Problem: x n n log x.

46. Logarithm of a Quotient Property Proof Problem: x _ y log x log y.

47. � e Name “Logarithm” Problem: a. Before there were calculators, if you had to

multiply 27 356 592, you would have to use long multiplication three times to

then that answer by 592. Simulate this process on your calculator by multiplying 27 356 and writing the result, then multiplying that

multiplying that answer by 592 and writing the � nal result.

b. Before there were calculators, if you had to add 27 356 592, you could write the numbers in a column and add without writing down any intermediate results. Do this addition column-wise, without using a calculator:

27 356

592

c. Base-10 logarithms were invented so that strings of numbers could be multiplied by adding their logarithms column-wise. You would look up the logarithms in tables, add these column-wise, and then use the tables backward to � nd the answer. � e computation would look something like this:

log 27

log 356

1.6335

log 592 2.7723 Add these logarithms column-wise, without

using a calculator. Simulate � nding the product by raising 10 to the exponent you found from adding the logarithms and rounding to four signi� cant digits. Does the result agree with your result in part a?


Problem Notes (continued)28. log 56 5 1.7481… 5 0.8450… 1 0.9030… 5 log 7 1 log 8; 56 5 10 1.7481… 5 10 0.8450… 1 0.9030… 5 10 0.8450… 10 0.9030… 5 7 829. log 6 5 0.7781… 5 1.4771… 2 0.6989…5 log 30 2 log 5; 6 5 10 0.7781… 5 10 1.4771… 2 0.6989… 5 10 1.4771… 10 0.6989… 5 30 530. log 1 __ 4 5 20.6020… 5 0.3010… 2 0.9030… 5 log 2 2 log 8; 0.25 5 10 20.6020… 5 10 0.3010… 2 0.9030… 5 10 0.3010… _______ 10 0.9030… 5 2 __ 8 31. log 32 5 1.5051… 5 5(0.3010…) 5 5 log 2; 32 5 10 1.5051… 5 10 5(0.3010…) 5 ( 10 0.3010… ) 5 5 2 5 32. log 125 5 2.0969… 5 3(0.6989…) 5 3 log 5; 125 5 10 2.0969… 5 10 3(0.6989…) 5 ( 10 0.6989… ) 3 5 5 3

33. log 1 __ 7 5 20.8450… 5 2log 7;

1 __ 7 5 10 20.8450… 5 1 _______ 10 0.8450… 5 1 __ 7

34. log 0.001 5 23 5 2log 1000; 0.001 5 10 23 5 1 ___ 10 3 5 1 ____ 1000

35. 21 36. 4037. 4 38. 1 __ 5 39. 56 40. 2541. 128 42. 3 5 5 24343. 3 44. 6

Problems 45 and 46 ask students to prove some of the properties of logarithms.45. Let c 5 log x, so x 5 10 c . Th en x n 5 ( 10 c ) n 5 1 0 cn , so log x n 5 cn 5 nc 5 n log x.46. Let c 5 log x and d 5 log y, so x 5 1 0 c and y 5 1 0 d . Th en x _ y 5 10 c __ 10 d 5 10 c2d , so log x _ y 5 c 2 d 5 log x 2 log y.

Problem 47 gives students an opportunity to see how logarithms were useful in doing long multiplication before calculators.

47a. 27 3356 9,612 343 415,316 3592 244,683,07247b. 101847c. 8.3886; 10 8.3886 245,700,000, which agrees (to four signifi cant digits) with the answer from part a.

Additional CAS Problem

Investigate the log of a product, log of a quotient, and log of a power properties using diff erent number bases in your logarithms. Which rules, if any, also apply for logarithms with bases other than 10? Can you state generic product, quotient, and power properties for logarithms?

See page 983 for answers to Problems 24–27 and the CAS Problem.


Logarithms: Equations and Other BasesIn the previous section you learned that a logarithm is an exponent of 10. In this section you will learn that it is possible to � nd logarithms using other positive numbers as the base. � en you will learn how to use the properties of logarithms to solve an equation for an unknown exponent or to solve an equation involving logarithms.


Logarithms with Any Base: The Change-of-Base PropertyIf x 1 0 y , then y is the base-10 logarithm of x. Similarly, if x 2 y , then y is the base-2 logarithm of x. � e only di� erence is the number that is the base. To distinguish among logarithms with di� erent bases, the base is written as a subscript a� er the abbreviation log. For instance,

3 lo g 2 8 2 3 8

lo g 3 81 3 81

2 lo g 10 100 10 2 100

� e symbol lo g 2 8 is pronounced “log to the base 2 of 8.” � e symbol lo g 10 100 is, of course, equivalent to log 100, as de� ned in the previous section. Note that in all cases a logarithm is an exponent.

DEFINITION: Logarithm with Any BaseAlgebraically:

lo g b x y if and only if b y x, where b > 0, b ≠ 1, and x > 0

Verbally: lo g b x y means that y is the exponent of b that gives x as the answer.

� e way you pronounce the symbol for logarithm gives you a way to remember the

Logarithms: Equations and Other Bases

2-5


Objective



25. Find log 32 and log 2. Show that log 32 5 log 2. What property of logarithms does this equality illustrate? What property of exponentiation does this property come from?

26.

this equality illustrate? What property of exponentiation does this property come from?

properties of logarithms. � en explain how each result agrees with the de� nition of logarithm.

27. log(0.3 0.7) log 0.3 log 0.7

28. log(7 8) log 7 log 8

29. log(30 ÷ 5) log 30 log 5

30. log 2 __ 8 log 2 log 8

31. log 2 5 5 log 2

32. log 5 3 3 log 5

33. log 1 __ 7 log 7

34. log 1 ____ 1000 log 1000



37. log 12 log ?

38. log 20 log ?

39. log 8 log 5 log 35 log ?

40. log 2000 log 2 log ?

41. 7 log 2 log ?

42. 5 log 3 log ?

43. log 125 ? log 5

44. ? log 2

45. Logarithm of a Power Property Proof Problem: x n n log x.

46. Logarithm of a Quotient Property Proof Problem: x _ y log x log y.

47. � e Name “Logarithm” Problem: a. Before there were calculators, if you had to

multiply 27 356 592, you would have to use long multiplication three times to

then that answer by 592. Simulate this process on your calculator by multiplying 27 356 and writing the result, then multiplying that

multiplying that answer by 592 and writing the � nal result.

b. Before there were calculators, if you had to add 27 356 592, you could write the numbers in a column and add without writing down any intermediate results. Do this addition column-wise, without using a calculator:

27 356

592

c. Base-10 logarithms were invented so that strings of numbers could be multiplied by adding their logarithms column-wise. You would look up the logarithms in tables, add these column-wise, and then use the tables backward to � nd the answer. � e computation would look something like this:

log 27

log 356

1.6335

log 592 2.7723 Add these logarithms column-wise, without

using a calculator. Simulate � nding the product by raising 10 to the exponent you found from adding the logarithms and rounding to four signi� cant digits. Does the result agree with your result in part a?

95


Class Time1 day

Homework AssignmentRA, Q1–Q10, Problems 1, 2, 3–49 odd

Teaching ResourcesSupplementary Problems


CAS Activity 2-5a: Dilations of Logarithmic Functions

TE ACH I N G

Important Terms and ConceptsCommon logarithmNatural logarithmeChange-of-base propertyExponential equationLogarithmic equation

Section 2-5: Logarithms: Equations and Other Bases


Find lo g 5 17. Check your answer by an appropriate numerical method.

x lo g 5 17.

5 x 17 Use the de� nition of logarithm.

lo g 10 5 x lo g 10 17 Take lo g 10 of both sides.

x lo g 10 5 lo g 10 17 Use the log of a power property to “peel o� ” the exponent.

x lo g 10 17

_______ lo g 10 5 1.7603… Divide both sides by the coe� cient of x.

lo g 5 17 1.7603… Substitute for x.

5 1.7603… 17 Do not round the 1.7603….

to the base-10 logarithm of that number. � e conclusion of the example can be written this way:

lo g 5 17 1 ______ lo g 10 5 lo g 10 17 10 17

To � nd the base-5 logarithm of any number, simply multiply its base-10 logarithm 10 5).

� is proportional relationship is called the change-of-base property. From the

lo g 5 17 lo g 10 17

_______ lo g 10 5

Notice that the logarithm with the desired base is by itself on the le� side of the equation and that the two logarithms on the right side have the same base, presumably one available on your calculator. � e box shows this property for bases a and b and argument x.

PROPERTY: The Change-of-Base Property of Logarithms

lo g a x log b x _____ lo g b a

or lo g a x 1 _____ lo g b a (lo g b x)

Find ln 29 using the change-of-base property with base-10 logarithms. Check your answer directly by pressing ln 29 on your calculator.

ln 29 log 29

_____ log e ________ 3.3672… e).

Directly: ln 29 3.3672…

which agrees with the answer you got using the change-of-base property.

Find lo g appropriate numerical method.

EXAMPLE 3 ➤

x lo g SOLUTION

x = log5 17

exponent

base

the “answer”

5 1.7603…CHECK ➤


EXAMPLE 4 ➤

ln 29 SOLUTION

➤


Write lo g 5 c a in exponential form.

� ink this:

5 …” is read “log base 5…,” so 5 is the base.

exponent. Because the log equals a, a must be the exponent.

a is the argument of the logarithm, c.

Write only this:

5 a c

Write z m in logarithmic form.

lo g z m

Two bases of logarithms are used frequently enough to have their own key on most calculators. One is base-10 logarithms, or common logarithms, as you saw in the previous section. � e other is base-e logarithms, called natural logarithms, where e 2.71828…, a naturally occurring number (like ) that you will � nd advantageous later in your mathematical studies. � e symbol ln x (pronounced “el en of x”) is used for the natural logarithms: ln x lo g e x.

DEFINITION: Common Logarithm and Natural LogarithmCommon: � e symbol log x means lo g 10 x.

Natural: � e symbol ln x means lo g e x, where e is a constant equal to

To � nd the value of a base-e logarithm, just press the key on your grapher. For instance,

ln 30

To show what this answer means, raise e

e 30 Use the e x

your calculator.

Write lo g EXAMPLE 1 ➤

log5 c = a

5

a = c

� ink this: � ink this:SOLUTION

➤

Write EXAMPLE 2 ➤

logz m = 4

z4 = m lo g z m lo g SOLUTION

➤

Nautilus shells have a logarithmic spiral pattern.


Section Notes

This section discusses logarithms log b a, where the base b must satisfy the conditions b . 0 and b 1. Emphasize to students that the number log b a (pronounced “log base b of a”) is the exponent to which you raise b in order to get a, a statement that would not make sense if b did not satisfy b . 0 and b 1.

Examples 1 and 2 are direct applications of the definition of a logarithm with any base.

Examples 5–8 on pages 98–100 demonstrate how to use the logarithmic properties to solve logarithmic equations. Examples 6 and 8 on pages 98–100 emphasize the importance of checking the solutions you find in the original logarithmic equation. Emphasize to students that the work they do in solving a logarithmic (or other type of) equation only gives them candidates for solutions and that they need to check these solutions in the original equation.

There are two built-in logarithm bases on a grapher, base 10 and base e. Base-10 logarithms are called common logarithms, and base-e logarithms are called natural logarithms. Natural logarithms are very important in calculus, so it is necessary for students to become familiar with this new base. The natural logarithm is denoted “ln” and pronounced “el, en.” To calculate logarithms with bases other than 10 or e, students will need to use the change-of-base property. Example 3 shows how to calculate log 5 17. Example 4 on page 97 is another application of the change-of-base property.


Find lo g 5 17. Check your answer by an appropriate numerical method.

x lo g 5 17.

5 x 17 Use the de� nition of logarithm.

lo g 10 5 x lo g 10 17 Take lo g 10 of both sides.

x lo g 10 5 lo g 10 17 Use the log of a power property to “peel o� ” the exponent.

x lo g 10 17

_______ lo g 10 5 1.7603… Divide both sides by the coe� cient of x.

lo g 5 17 1.7603… Substitute for x.

5 1.7603… 17 Do not round the 1.7603….

to the base-10 logarithm of that number. � e conclusion of the example can be written this way:

lo g 5 17 1 ______ lo g 10 5 lo g 10 17 10 17

To � nd the base-5 logarithm of any number, simply multiply its base-10 logarithm 10 5).

� is proportional relationship is called the change-of-base property. From the

lo g 5 17 lo g 10 17

_______ lo g 10 5

Notice that the logarithm with the desired base is by itself on the le� side of the equation and that the two logarithms on the right side have the same base, presumably one available on your calculator. � e box shows this property for bases a and b and argument x.

PROPERTY: The Change-of-Base Property of Logarithms

lo g a x log b x _____ lo g b a

or lo g a x 1 _____ lo g b a (lo g b x)


ln 29 log 29

_____ log e ________ 3.3672… e).

Directly: ln 29 3.3672…

which agrees with the answer you got using the change-of-base property.

Find lo g appropriate numerical method.

EXAMPLE 3 ➤

x lo g SOLUTION

x = log5 17

exponent

base

the “answer”

5 1.7603…CHECK ➤


EXAMPLE 4 ➤

ln 29 SOLUTION

➤


Write lo g 5 c a in exponential form.

� ink this:

5 …” is read “log base 5…,” so 5 is the base.

exponent. Because the log equals a, a must be the exponent.

a is the argument of the logarithm, c.

Write only this:

5 a c

Write z m in logarithmic form.

lo g z m

Two bases of logarithms are used frequently enough to have their own key on most calculators. One is base-10 logarithms, or common logarithms, as you saw in the previous section. � e other is base-e logarithms, called natural logarithms, where e 2.71828…, a naturally occurring number (like ) that you will � nd advantageous later in your mathematical studies. � e symbol ln x (pronounced “el en of x”) is used for the natural logarithms: ln x lo g e x.

DEFINITION: Common Logarithm and Natural LogarithmCommon: � e symbol log x means lo g 10 x.

Natural: � e symbol ln x means lo g e x, where e is a constant equal to

To � nd the value of a base-e logarithm, just press the key on your grapher. For instance,

ln 30

To show what this answer means, raise e

e 30 Use the e x

your calculator.

Write lo g EXAMPLE 1 ➤

log5 c = a

5

a = c

� ink this: � ink this:SOLUTION

➤

Write EXAMPLE 2 ➤

logz m = 4

z4 = m lo g z m lo g SOLUTION

➤

Nautilus shells have a logarithmic spiral pattern.

97

One of the main uses of logarithms is in solving exponential equations—equations in which the unknown is an exponent. (In Section 2-2, logarithms were used in this way to find the particular equation for a power function.) When solving an exponential equation, the base of the logarithm used does not affect the answer. However, sometimes choosing a logarithm with a particular base can make solving the equation easier. Show students that the same solution is obtained whether 1000 5 50 3 x is solved using common logarithms or natural logarithms. Then show them that solving 1000 5 50 e x is easier if natural logarithms are used, because ln e 5 1.

Differentiating Instruction• Note that the concept of log b a being a

number is difficult for many students.• Have students write the definition of

a logarithm in their journals, both symbolically and verbally. For ELL students, perhaps in their primary language.

• Make sure ELL students understand what b . 0 and b 1 mean. Give examples of bases that fit these restrictions, such as 2 and 1 _ 2 .

• Because some countries use only log e to represent natural logarithms, clarify for ELL students that the terms ln, natural log, and log e mean the same thing and are interchangeable.

• Clarify that log a x 5 ( log 10 x)

_____ ( log 10 a) 5 ln x ___ ln a .



8 x 2 x 3

x 2 x 5 0 Reduce one side to zero. Use the symmetric property of equality.

(x 5)(x 1) 0 Solve by factoring.

x 5 or x 1 Solutions of the quadratic equation.

You need to be cautious here because the solutions in the � nal step are the solutions of the quadratic equation, and you must make sure they are also solutions of the original logarithmic equation. Check by substituting your solutions into the original equation.

If x 5, then If x 1, then

lo g 2 (5 1) lo g 2 (5 3) lo g 2 ( 1 1) lo g 2 ( 1 3)

lo g 2 lo g 2 2 lo g 2 ( 2) lo g 2 (

2 1 3 which is unde� ned.

x 5 is a solution, but x 1 is not.

Solve the equation and check your solutions.

e 2x 3 e x 2 0

e 2x 3 e x 2 0

( e x ) 2 3 e x 2 0 Apply the properties of exponents.

You can realize that this is a quadratic equation in the variable e x . Using the quadratic formula, you get

e x 3 ________

9 _______________ 2 3 1 _____ 2

e x 2 or e x 1

You now have to solve these two equations.

e x 2 e x 1

x ln 2 0.6931… x 0

e 2 ln 2 3 e ln 2 2 e 0 2 3 e 0 2

e ln 2 2 3 e ln 2 2 1 2 3(1) 2 0

2 2 3(2) 2 0

Both solutions are correct.

Solve the logarithmic equation ln(x 3) ln(x 5) 0 and check your solution(s).

ln(x 3) ln(x 5) 0

ln[(x 3)(x 5)] 0 Use the logarithm of a product property.

➤

Solve the equation and check your solutions.EXAMPLE 7 ➤

e 2xSOLUTION

e 2 ln 2CHECK

➤

Solve the logarithmic equation ln(your solution(s).

EXAMPLE 8 ➤

ln(xSOLUTION


� e properties of base-10 logarithms presented in the previous section are generalized here for any base.

Properties of LogarithmsThe Logarithm of a Power:

lo g b x y y lo g b x

Verbally: � e logarithm of a power equals the product of the exponent and the logarithm of the base

The Logarithm of a Product:lo g b (xy) lo g b x log b y

Verbally: � e logarithm of a product equals the sum of the logarithms of the factors.

The Logarithm of a Quotient:lo g b

x __ y log b x lo g b y

Verbally: � e logarithm of a quotient equals the logarithm of the numerator minus the logarithm of the denominator.

Solving Exponential and Logarithmic Equations

how you can do this.

Solve the exponential equation 7 3x 983 algebraically, using logarithms.

7 3x 983

log 7 3x log 983 Take the base-10 logarithm of both sides.

3x log 7 log 983 Use the logarithm of a power property.

x log 983

______ 3 log 7 Divide both sides by the coe� cient of x.

x 1.1803…

Solve the equation

l og 2 (x 1) lo g 2 (x 3) 3

lo g 2 (x 1) lo g 2 (x 3) 3

lo g 2 [(x 1)(x 3)] 3 Apply the logarithm of a product property.

2 3 (x 1)(x 3) Use the de� nition of logarithm.

� e spiral arms of galaxies follow a logarithmic pattern.

Solve the exponential equation 7 EXAMPLE 5 ➤

7 3x 983SOLUTION

➤

Solve the equationEXAMPLE 6 ➤

log2(xSOLUTION


Technology Notes

CAS Activity 2-5a: Dilations of Logarithmic Functions in the Instructor’s Resource Book has students use a CAS to explore how horizontally translated logarithmic functions relate to dilations of logarithmic functions. Allow 20–25 minutes.

Additional Class Examples

1. Find the missing value.

ln 72 2 ln 8 5 ln ?

Solution

72 ___ 8 5 9

2. Estimate log2 17. Find log2 17 using the properties you have learned in this section.

Solution log 2 17 4 because 2 4 5 16 and 16 is close to 17. Use your calculator to get

log 2 17 5 log 17

_____ log 2 5 4.0874….

3. Solve log2(x 2 2) 1 log2(x 1 3) 5 4.

Solution

log 2 (x 2 2) 1 log 2 (x 1 3) 5 4

log 2 (x 2 2)(x 1 3) 5 4

(x 2 2)(x 1 3) 5 2 4

x 2 1 x 2 22 5 0

x 5 4.2169… or x 5 25.2169…Check both solutions in the original equation to see that 25.2169… is extraneous and that the only solution is 4.2169….


8 x 2 x 3

x 2 x 5 0 Reduce one side to zero. Use the symmetric property of equality.

(x 5)(x 1) 0 Solve by factoring.

x 5 or x 1 Solutions of the quadratic equation.

You need to be cautious here because the solutions in the � nal step are the solutions of the quadratic equation, and you must make sure they are also solutions of the original logarithmic equation. Check by substituting your solutions into the original equation.

If x 5, then If x 1, then

lo g 2 (5 1) lo g 2 (5 3) lo g 2 ( 1 1) lo g 2 ( 1 3)

lo g 2 lo g 2 2 lo g 2 ( 2) lo g 2 (

2 1 3 which is unde� ned.

x 5 is a solution, but x 1 is not.

Solve the equation and check your solutions.

e 2x 3 e x 2 0

e 2x 3 e x 2 0

( e x ) 2 3 e x 2 0 Apply the properties of exponents.

You can realize that this is a quadratic equation in the variable e x . Using the quadratic formula, you get

e x 3 ________

9 _______________ 2 3 1 _____ 2

e x 2 or e x 1

You now have to solve these two equations.

e x 2 e x 1

x ln 2 0.6931… x 0

e 2 ln 2 3 e ln 2 2 e 0 2 3 e 0 2

e ln 2 2 3 e ln 2 2 1 2 3(1) 2 0

2 2 3(2) 2 0

Both solutions are correct.

Solve the logarithmic equation ln(x 3) ln(x 5) 0 and check your solution(s).

ln(x 3) ln(x 5) 0

ln[(x 3)(x 5)] 0 Use the logarithm of a product property.

➤

Solve the equation and check your solutions.EXAMPLE 7 ➤

e 2xSOLUTION

e 2 ln 2CHECK

➤

Solve the logarithmic equation ln(your solution(s).

EXAMPLE 8 ➤

ln(xSOLUTION


� e properties of base-10 logarithms presented in the previous section are generalized here for any base.

Properties of LogarithmsThe Logarithm of a Power:

lo g b x y y lo g b x

Verbally: � e logarithm of a power equals the product of the exponent and the logarithm of the base

The Logarithm of a Product:lo g b (xy) lo g b x log b y

Verbally: � e logarithm of a product equals the sum of the logarithms of the factors.

The Logarithm of a Quotient:lo g b

x __ y log b x lo g b y

Verbally: � e logarithm of a quotient equals the logarithm of the numerator minus the logarithm of the denominator.

Solving Exponential and Logarithmic Equations

how you can do this.

Solve the exponential equation 7 3x 983 algebraically, using logarithms.

7 3x 983

log 7 3x log 983 Take the base-10 logarithm of both sides.

3x log 7 log 983 Use the logarithm of a power property.

x log 983

______ 3 log 7 Divide both sides by the coe� cient of x.

x 1.1803…

Solve the equation

l og 2 (x 1) lo g 2 (x 3) 3

lo g 2 (x 1) lo g 2 (x 3) 3

lo g 2 [(x 1)(x 3)] 3 Apply the logarithm of a product property.

2 3 (x 1)(x 3) Use the de� nition of logarithm.

� e spiral arms of galaxies follow a logarithmic pattern.

Solve the exponential equation 7 EXAMPLE 5 ➤

7 3x 983SOLUTION

➤

Solve the equationEXAMPLE 6 ➤

log2(xSOLUTION

99

CAS Suggestions

Consider having students explore the properties they learned about base-10 logarithms using logarithms with diff erent bases before presenting them to the class.

Th e TI-Nspire CAS operates as easily with bases e and 10 as it does with any other positive number as its base. Note, however, that if a decimal value is entered for either the base or object of the logarithm, the TI-Nspire will default to a non-CAS numerical approximation for the expression unless the calculator is set in exact mode.



Check your answer by raising the appropriate number to the appropriate power. 7. log 7 29 8. log 8 352 9. l og 3 729 10. log 32 2 11. log 2 32 12. l og 5 125 13. log 6 0.3 14. log 15 0.777

15. ln 8 ln 7 ln ? 16. ln 10 ln 20 ln ? 17. ln 3 5 ? ln 3 18. 2 ln ? ln 81 19. ln 36 ln ? ln 9 20. ln ? ln 7 ln 2 21. ln

__ x ? ln x 22. ln 5

__ x ? ln x

23. ln 1 ? 24. ln e ? 25. log 10 ? 26. log 1 ?

27. lo g 7 33 lo g 10 33

______ ? 28. log 0.07 53 ln 53 _____ ?

29. lo g 0.6 x _____ lo g 0.6 3

lo g ? x 30. lo g 13 n

_______ log 13 0.5 lo g ? n

31. ln x _____ ln 10 ? x 32. log x

_____ log e ? x

33. lo g k k 3 ?

34. If x lo g k 2 and y lo g k 5, then lo g k ? .

and check your solution. 35. log(3x 7) 0 36. 2 log(x 3) 1 5 37. lo g 2 (x 3) lo g 2 (x 3 38. lo g 2 (2x 1) lo g 2 (x 2) 1 39. ln(x 9 ) 8 40. ln(x 2) ln(x 2) 0 41. 5 3x 786 42. 8 0.2x 98.6 43. 0.8 x 2001 44. 6 5x 0.007

45. 3e x 5 10 46. e 2x 3 7 47. 2 e 2x 5e x 3 0 48. 5 2 2x 3 2 x 2 0 49. Compound Interest Problem: If you invest $10,000

in a savings account that pays interest at the rate

amount M in the account a�er x years is given by the exponential function

M 10,000 1.0 7 x a. Make a table of values of M for each year

from 0 to 6. b. How can you conclude that the values in the

c. Suppose that you want to cash in the savings account when the amount M reaches $27,000. Set M equal to 27,000 and solve the resulting exponential equation algebraically using logarithms. Convert the solution to months, and round appropriately to �nd how many whole months must elapse before M �rst exceeds $27,000.

50. Population of the United States Problem: Based on the 1990 and 2000 U.S. censuses, the population

that time period. �at is, the population at the end of any one year was population at the beginning of that year.

a. How do you tell that the population function

b. �e population in 1990 was about

expressing population, P, as a function of the n years that have elapsed since 1990.

c. Assume that the population continues to grow

the year in which the population �rst reached 300 million. In �nding the real-world answer, use the fact that the 1990 census was taken as of April 1. How does your prediction compare with that of the U.S. Census Bureau, which placed the date as October 2006?


(x 3)(x 5) e 0 1 De� nition of (natural) logarithm.

x 2 8x 15 1

x 2 8x 0

x 2.5857… or x By the quadratic formula.

x 2.5857…:

ln( 2.5857… 3) ln( 2.5857… 5) Substitute without rounding.

0.8813… 0.8813… 0

which checks.

x

ln( 3) ln( 5)

ln( ln( No logarithms of negative numbers.

which is unde� ned.

� e only valid solution is x 2.5857….

x 2.5857…:CHECK

➤

Reading AnalysisFrom what you have read in this section, what do you consider to be the main idea? What two bases of logarithms are found on most calculators? How do they di� er? How do you � nd a logarithm with a base other than these? How do you use the logarithm of a power property to solve an equation that has an unknown exponent?

Quick Review

Q1. log 5 log 8 log ?

Q2. log 36 log ?

Q3. log 5 2 2 log ?

Q4. log 7 3 ? log 7

Q5. 1.5 log 13 log ?

Q6. � e name “logarithm” comes from the words ? and ? .

Q7. A logarithm is a(n) ? .

Q8. log 6 log ?

Q9. log 0.5 log 20 log ?

Q10. log( 2) is A. log(1/2) B. 1 ____ log 2

C. log 2 D. log(1/2)

E. Unde� ned

1. Write the definition of base-b logarithms. 2. State the change-of-base property.

3. Write in exponential form: l og 7 p c 4. Write in exponential form: lo g v 6 x 5. Write in logarithmic form: k 5 9 6. Write in logarithmic form: m d 13

5min

Problem Set 2-5


PRO B LE M N OTES


Th e problems in this section are trivial for students using a CAS. For an added challenge have students use a CAS to confi rm their algebraic steps. Th is also provides them with feedback on their problem-solving techniques. Note that a CAS may use a diff erent notational format when making calculations. Students need to be able to adapt to the challenge of unexpected formats. Understanding algebraic equivalence is a key aspect of using a CAS eff ectively.Q1. 40Q2. 9Q3. 5Q4. 3Q5. 1 3 1.5 5 46.8721…Q6. Logos, arithmos Q7. Exponent Q8. 8Q9. 40Q10. E

Problems 1–34 give students practice applying the defi nition and properties of logarithms.1. log b x 5 y if and only if b y 5 x for x . 0, b . 0, b 1

2. log a x 5 log b x _____ log b a

for x . 0, a . 0, a 1,

b . 0, b 1 3. 7 c 5 p5. log k 9 5 57. 1.7304…9. 611. 513. 20.6719…15. 5617. 5

4. v x 5 66. lo g m 13 5 d8. 2.8198…10. 1 __ 5 12. 314. 20.0931…16. 20018. 9

19. 421. 1 _ 2 23. 025. 127. log 10 729. 331. log (or lo g 10 )33. 3

20. 1422. 1 _ 5 24. 126. 028. ln 0.07 30. 0.532. ln34. x 2 y


Check your answer by raising the appropriate number to the appropriate power. 7. log 7 29 8. log 8 352 9. l og 3 729 10. log 32 2 11. log 2 32 12. l og 5 125 13. log 6 0.3 14. log 15 0.777

15. ln 8 ln 7 ln ? 16. ln 10 ln 20 ln ? 17. ln 3 5 ? ln 3 18. 2 ln ? ln 81 19. ln 36 ln ? ln 9 20. ln ? ln 7 ln 2 21. ln

__ x ? ln x 22. ln 5

__ x ? ln x

23. ln 1 ? 24. ln e ? 25. log 10 ? 26. log 1 ?

27. lo g 7 33 lo g 10 33

______ ? 28. log 0.07 53 ln 53 _____ ?

29. lo g 0.6 x _____ lo g 0.6 3

lo g ? x 30. lo g 13 n

_______ log 13 0.5 lo g ? n

31. ln x _____ ln 10 ? x 32. log x

_____ log e ? x

33. lo g k k 3 ?

34. If x lo g k 2 and y lo g k 5, then lo g k ? .

and check your solution. 35. log(3x 7) 0 36. 2 log(x 3) 1 5 37. lo g 2 (x 3) lo g 2 (x 3 38. lo g 2 (2x 1) lo g 2 (x 2) 1 39. ln(x 9 ) 8 40. ln(x 2) ln(x 2) 0 41. 5 3x 786 42. 8 0.2x 98.6 43. 0.8 x 2001 44. 6 5x 0.007

45. 3e x 5 10 46. e 2x 3 7 47. 2 e 2x 5e x 3 0 48. 5 2 2x 3 2 x 2 0 49. Compound Interest Problem: If you invest $10,000

in a savings account that pays interest at the rate

amount M in the account a�er x years is given by the exponential function

M 10,000 1.0 7 x a. Make a table of values of M for each year

from 0 to 6. b. How can you conclude that the values in the

c. Suppose that you want to cash in the savings account when the amount M reaches $27,000. Set M equal to 27,000 and solve the resulting exponential equation algebraically using logarithms. Convert the solution to months, and round appropriately to �nd how many whole months must elapse before M �rst exceeds $27,000.

50. Population of the United States Problem: Based on the 1990 and 2000 U.S. censuses, the population

that time period. �at is, the population at the end of any one year was population at the beginning of that year.

a. How do you tell that the population function

b. �e population in 1990 was about

expressing population, P, as a function of the n years that have elapsed since 1990.

c. Assume that the population continues to grow

the year in which the population �rst reached 300 million. In �nding the real-world answer, use the fact that the 1990 census was taken as of April 1. How does your prediction compare with that of the U.S. Census Bureau, which placed the date as October 2006?


(x 3)(x 5) e 0 1 De� nition of (natural) logarithm.

x 2 8x 15 1

x 2 8x 0

x 2.5857… or x By the quadratic formula.

x 2.5857…:

ln( 2.5857… 3) ln( 2.5857… 5) Substitute without rounding.

0.8813… 0.8813… 0

which checks.

x

ln( 3) ln( 5)

ln( ln( No logarithms of negative numbers.

which is unde� ned.

� e only valid solution is x 2.5857….

x 2.5857…:CHECK

➤

Reading AnalysisFrom what you have read in this section, what do you consider to be the main idea? What two bases of logarithms are found on most calculators? How do they di� er? How do you � nd a logarithm with a base other than these? How do you use the logarithm of a power property to solve an equation that has an unknown exponent?

Quick Review

Q1. log 5 log 8 log ?

Q2. log 36 log ?

Q3. log 5 2 2 log ?

Q4. log 7 3 ? log 7

Q5. 1.5 log 13 log ?

Q6. � e name “logarithm” comes from the words ? and ? .

Q7. A logarithm is a(n) ? .

Q8. log 6 log ?

Q9. log 0.5 log 20 log ?

Q10. log( 2) is A. log(1/2) B. 1 ____ log 2

C. log 2 D. log(1/2)

E. Unde� ned

1. Write the definition of base-b logarithms. 2. State the change-of-base property.

3. Write in exponential form: l og 7 p c 4. Write in exponential form: lo g v 6 x 5. Write in logarithmic form: k 5 9 6. Write in logarithmic form: m d 13

5min

Problem Set 2-5

101

Problems 47 and 48 can be solved on a CAS without using the Solve command by completing the square. Notice that the CAS automatically rewrites ln(1/2) as 2ln 2.

47. x 5 20.6931…48. x 5 0

Problems 49 and 50 are real-world problems that can be solved using logarithms.49a.

49b. Whenever you add 1 to x, you multiply M by 1.07.49c. 177 mo, or 14 yr 9 mo50a. Every time you add one year, the population is multiplied by 1.0124.50b. P(n) 5 248.7 1.0124 n , with n in years and P(n) in millions of people.

50c. 15 yr 79 d; around June 19, 2005; Th is prediction is earlier than the actual date identifi ed by the U.S. Census Bureau.


1. Use the Expand command on the expression log(1025). Use the properties of logarithms to explain the result. What do all of the terms of the result have in common? Why?

2. Solve log B (A) 5 log 1 __ B (x) for x. State any

restrictions on A and B. Why does this property suggest that it is appropriate to restrict the bases of logarithms to values greater than 1?

x M0 10,0001 10,7002 11,4493 12,2504 13,1085 14,0266 15,007

Problems 35–48 require students to solve logarithmic and exponential equations.35. x 5 2236. x 5 10337. x 1 5 5, x 2 5 24Th e equation is undefi ned for x 5 24 38. x 5 4 __ 3 39. x 2 9 5 16.3890…

40. x 5  __

5 5 2.2360…Th e equation is undefi ned for x 5 2  

__ 5 .

41. x 5 1.3808…42. x 5 11.0391…43. x 5 285.1626…44. x 5 0.5538…

45. x 5 4.5108…46. no solution


See page 983 for answers to CAS Problems 1 and 2.

103Section 2-6: Logarithmic Functions

DEFINITION: Logarithmic FunctionsGeneral equation: y a b log c x Base-c logarithmic function

where a, b, and c are constants, with b 0, c 0, and c 1. � e domain is all positive real numbers.

Transformed function: y a b lo g c (x d)

where a is the vertical translation, b is the vertical dilation, and d is the horizontal translation.

Note: Remember that log stands for the base-10 logarithm and ln stands for the base-e logarithm.

Multiply–Add Property of Logarithmic Functions� ese x- and y-values have the multiply–add property. Multiplying x by 3 results in adding 1 to the corresponding y-value.

x y

6 1 18 2

3 162

By interchanging the variables, you can notice that x is an exponential function of y. You can � nd its particular equation by algebraic calculations.

x 2 3 y

� is equation can be solved for y as a function of x with the help of logarithms.

ln x ln 2 3 y Take the natural logarithm (ln) of both sides.

ln x ln 2 y ln 3 Use the product and power properties of logarithms.

y ln 3 ln x ln 2 Isolate the term with y.

y 1 ____ ln 3 ln x ln 2 ____ ln 3 Solve for y.

y 0.9102… ln x 0.6309… Calculate constants by calculator.

� is equation is a logarithmic function with a 0.6309… and b 0.9102… .

of a logarithmic function. Reversing the steps lets you conclude that logarithmic functions have this property in general.

3

3

3

1

1

1


Logarithmic FunctionsYou have already learned about identifying properties of several types of functions.

In this section you’ll learn that logarithmic functions have the multiply–add property.

Show that logarithmic functions have the multiply-add property, and � nd particular equations algebraically.

Logarithmic FunctionsFigures 2-6a and 2-6b show the natural logarithmic function y ln x and the common logarithmic function y log x (solid graphs). � ese functions are inverses of the corresponding exponential functions (dashed graphs), as shown by the fact that the graphs are re� ections of the graphs of y e x and y 10 x across the line y x. Both logarithmic graphs are concave down. Notice also that the y-values are increasing at a decreasing rate as x increases. In both cases the y-axis is a vertical asymptote for the logarithmic graph. In addition, you can tell that the domain of these basic logarithmic functions is the set of positive real numbers

Natural logarithm: y ln x

10

y

5

5

x

y x

ln x

ex

y

xlog x

10x

Common logarithm: y log x

10

5

5y x

Figure 2-6a Figure 2-6b

� e general equation of a logarithmic function on most graphers has constants to allow for vertical translation and dilation.

Logarithmic FunctionsYou have already learned about identifying properties of several types of

2- 6


Objective



Class Time1 day

Homework AssignmentRA, Q1–Q10, Problems 1–13 odd, 14

Teaching ResourcesSupplementary Problems

TE ACH I N G

Important Terms and ConceptsLogarithmic functionMultiply–add property

Section Notes

Th is section focuses on the graphs and numerical patterns associated with logarithmic functions.

Emphasize that a logarithmic function is the inverse of an exponential function with the same base. In an exponential function, the base must be greater than zero and not equal to 1, and the same is true of a logarithmic function. Th e range of an exponential function is y . 0, so the domain of a logarithmic function is x . 0. Th is makes it easy to explain that one cannot take the log of 0 or a negative number. An exponential function has the add–multiply property, so a logarithmic function has the multiply–add property.

Discuss the graphs of the natural and common logarithmic functions shown in Figures 2-6a and 2-6b. Emphasize that both of the functions have the same domain, x . 0. Moreover, the graphs of both functions are increasing, concave down, asymptotic to the y-axis, and contain the point (1, 0). It is important that students learn to visualize these graphs so that they can sketch translations and dilations of these parent functions.

Th e fi gure to the right shows the eff ect of varying the constant a in the natural log function y 5 a 1 ln x. As you might expect, the graphs are vertical translations of y 5 ln x. Note that while y 5 ln x has the expected x-intercept of 1, the vertical translations give the other graphs diff erent x-intercepts. Because there was no horizontal translation, the vertical asymptote is still at the y-axis.

x

y y = 2 + ln x

y = –2 + ln x10

–5

5y = ln x


DEFINITION: Logarithmic FunctionsGeneral equation: y a b log c x Base-c logarithmic function

where a, b, and c are constants, with b 0, c 0, and c 1. � e domain is all positive real numbers.

Transformed function: y a b lo g c (x d)

where a is the vertical translation, b is the vertical dilation, and d is the horizontal translation.

Note: Remember that log stands for the base-10 logarithm and ln stands for the base-e logarithm.

Multiply–Add Property of Logarithmic Functions� ese x- and y-values have the multiply–add property. Multiplying x by 3 results in adding 1 to the corresponding y-value.

x y

6 1 18 2

3 162

By interchanging the variables, you can notice that x is an exponential function of y. You can � nd its particular equation by algebraic calculations.

x 2 3 y

� is equation can be solved for y as a function of x with the help of logarithms.

ln x ln 2 3 y Take the natural logarithm (ln) of both sides.

ln x ln 2 y ln 3 Use the product and power properties of logarithms.

y ln 3 ln x ln 2 Isolate the term with y.

y 1 ____ ln 3 ln x ln 2 ____ ln 3 Solve for y.

y 0.9102… ln x 0.6309… Calculate constants by calculator.

� is equation is a logarithmic function with a 0.6309… and b 0.9102… .

of a logarithmic function. Reversing the steps lets you conclude that logarithmic functions have this property in general.

3

3

3

1

1

1


Logarithmic FunctionsYou have already learned about identifying properties of several types of functions.

In this section you’ll learn that logarithmic functions have the multiply–add property.


Logarithmic FunctionsFigures 2-6a and 2-6b show the natural logarithmic function y ln x and the common logarithmic function y log x (solid graphs). � ese functions are inverses of the corresponding exponential functions (dashed graphs), as shown by the fact that the graphs are re� ections of the graphs of y e x and y 10 x across the line y x. Both logarithmic graphs are concave down. Notice also that the y-values are increasing at a decreasing rate as x increases. In both cases the y-axis is a vertical asymptote for the logarithmic graph. In addition, you can tell that the domain of these basic logarithmic functions is the set of positive real numbers

Natural logarithm: y ln x

10

y

5

5

x

y x

ln x

ex

y

xlog x

10x

Common logarithm: y log x

10

5

5y x

Figure 2-6a Figure 2-6b

� e general equation of a logarithmic function on most graphers has constants to allow for vertical translation and dilation.

Logarithmic FunctionsYou have already learned about identifying properties of several types of

2- 6


Objective

103

The figure below shows the effect of varying the multiplicative constant b in y 5 b ln x. The x-intercept remains at x 5 1, and the vertical asymptote remains at the y-axis. Note that if b is negative, the graph is reflected across the x-axis.

x

yy = 2 ln x

y = –2 ln x10

–5

5y = ln x

The figure below shows that changing the base of the logarithmic function has the same effect as a vertical dilation. It is for this reason that it is not necessary to use different bases in order to fit logarithmic functions to various data sets.

x

yy = log1.5 x

y = log 0.5 x

10

–5

5y = ln x

The table on page 103 illustrates the multiply–add property of logarithmic functions. The particular equation for the function is found by first writing x as an exponential function of y and then using logarithms to write y in terms of x. (Part b of Example 1 on page 104 shows a more direct method for finding a particular logarithmic function that fits a set of points.)

The figure to the right shows the effect of varying the constant d for the natural log function y 5 ln(x 2 d). As you might expect, the graphs are horizontal translations of y 5 ln x. Note that while y 5 ln x is asymptotic to the y-axis, the horizontal translations give the other graphs different vertical asymptotes.

x

y

y = ln(x – 2)

y = ln(x + 2)

–5

5

y = ln x

10

Section 2-6: Logarithmic Functions


a. f (x) 3 log(x 1)

b. f (x) ln(x 3)

c. f (x) lo g 2 ( x 2 1)

a. You can get the graph of the function f (x) 3 log(x 1) through transformations of the parent logarithmic function: a horizontal translation by 1 unit and a vertical dilation by a factor of 3. Figure 2-6c shows the resulting graph.

You know that the domain of a logarithmic function is positive real numbers, so the argument of a logarithmic function has to be positive.

x 1 0

� e domain of the function is x 1. Add 1 to both sides of the inequality.

b. Figure 2-6d shows the graph of the function f (x) ln(x 3). You can get this graph by re� ecting the graph of the function y ln x across the x-axis and translating it horizontally by 3 units.

Domain:

x 3 0 Argument of a logarithmic function is positive.

� e domain of the function is x 3.

c. In order to graph this function on your grapher, use the change-of-base property.

f (x) lo g 2 x 2 1 log x 2 1

_________ log 2

Figure 2-6e shows the resulting graph.

Domain:

x 2 1 0 Argument of a logarithmic function is positive.

You can solve this inequality graphically. Graph the quadratic function and look for those values of x for which the function value is greater than zero or the graph is above the x-axis (see Figure 2-6f).

y

y x2 1

xx 1x 15 5

5

Figure 2-6f

� e domain is x 1 or x 1.

f

EXAMPLE 2

a. You can get the graph of the function transformations of the parent logarithmic function: a horizontal

SOLUTION

5 10

5

5 f(x)

x

Figure 2-6c

1055

f (x)

5

5

x

Figure 2-6d

f(x)

x5 5

5

Figure 2-6e

➤


PROPERTY: Multiply–Add Property of Logarithmic FunctionsIf f is a logarithmic function, then multiplying x by a constant results in adding a constant to the value of f (x). � at is,

for f (x) a b log c x, if x 2 k x 1 , then f x 2 b lo g c k f x 1

Particular Equations of Logarithmic FunctionsYou can � nd the particular equation of a logarithmic function algebraically by substituting two points that are on the graph of the function and evaluating the

Suppose that f is a logarithmic function with values f (3) 7 and f (6) 10.

a. Without � nding the particular equation, � nd f (12) and f

b. Find the particular equation algebraically using natural logarithms.

c. Con� rm that your equation gives the value of f

a.

x y

3 7 6 10 12 13

16

f (12) 13 and f 16

b. f (x) a b ln x Write the general equation.

7 a b ln 3 10 a b ln 6

Substitute the given points.

3 b ln 6 b ln 3 Subtract the equations to eliminate a.

b 3 _________ ln 6 ln 3 Factor out b and then divide by ln 6 ln 3

7 a b.

a 7 Store a and b without rounding.

f (x) x Write the particular equation. Graph it on your grapher.

c. By calculator, f 16, which checks.


a. SOLUTION

2

2

2

3

3

3

➤



Example 1 demonstrates how to use the multiply–add property to find missing values and how to find the particular equation of a logarithmic function algebraically.

Example 2 involves graphing three logarithmic functions and finding their domains. In parts a and b, have students sketch the graphs by applying transformations to the parent functions and then check their answers with their graphers. The function in part c is more difficult to sketch by hand, so students will want to use their graphers. Because the function involves base-2 logarithms, it must be rewritten in terms of base-10 logarithms (or base-e logarithms) before it can be graphed. (Refer to the Additional Class Example.)

Finding the domain of each function in Example 2 involves finding the values where the argument of the logarithmic function is greater than zero. For part c, this involves solving a quadratic inequality. You may need to remind students how to solve such inequalities. The solution in the text uses the graph of the equation to find out where the quadratic function has positive values (in other words, where the graph of the parabola is above the x-axis). An alternate solution to the problem is to factor x 2 2 1 into (x 2 1)(x 1 1). Then the domain of f is when (x 2 1)(x 1 1) . 0 or when both factors have the same sign.

Differentiating Instruction• Have ELL students write the Multiply–

Add Property in their journals and note that this is the inverse of the Add–Multiply Property of Exponential Functions. Also have them include a sketch of y 5 log

2 x and y 5 2 x (similar

to Figure 2-6a) and a labeled sketch of a transformed logarithmic function, so that they have a visual reference.

• Have ELL students write a summary of the section, perhaps in pairs, in place of answering the Reading Analysis questions.

• ELL students may need help understanding the language in Problems 3 and 4. However, being introduced to concepts such as Carbon-14 dating and the Richter scale may be useful in their general education.

• If you assign Problem 14, let ELL students write in their primary language if they wish.


a. f (x) 3 log(x 1)

b. f (x) ln(x 3)

c. f (x) lo g 2 ( x 2 1)

a. You can get the graph of the function f (x) 3 log(x 1) through transformations of the parent logarithmic function: a horizontal translation by 1 unit and a vertical dilation by a factor of 3. Figure 2-6c shows the resulting graph.

You know that the domain of a logarithmic function is positive real numbers, so the argument of a logarithmic function has to be positive.

x 1 0

� e domain of the function is x 1. Add 1 to both sides of the inequality.

b. Figure 2-6d shows the graph of the function f (x) ln(x 3). You can get this graph by re� ecting the graph of the function y ln x across the x-axis and translating it horizontally by 3 units.

Domain:

x 3 0 Argument of a logarithmic function is positive.

� e domain of the function is x 3.

c. In order to graph this function on your grapher, use the change-of-base property.

f (x) lo g 2 x 2 1 log x 2 1

_________ log 2

Figure 2-6e shows the resulting graph.

Domain:

x 2 1 0 Argument of a logarithmic function is positive.

You can solve this inequality graphically. Graph the quadratic function and look for those values of x for which the function value is greater than zero or the graph is above the x-axis (see Figure 2-6f).

y

y x2 1

xx 1x 15 5

5

Figure 2-6f

� e domain is x 1 or x 1.

f

EXAMPLE 2

a. You can get the graph of the function transformations of the parent logarithmic function: a horizontal

SOLUTION

5 10

5

5 f(x)

x

Figure 2-6c

1055

f (x)

5

5

x

Figure 2-6d

f(x)

x5 5

5

Figure 2-6e

➤


PROPERTY: Multiply–Add Property of Logarithmic FunctionsIf f is a logarithmic function, then multiplying x by a constant results in adding a constant to the value of f (x). � at is,

for f (x) a b log c x, if x 2 k x 1 , then f x 2 b lo g c k f x 1

Particular Equations of Logarithmic FunctionsYou can � nd the particular equation of a logarithmic function algebraically by substituting two points that are on the graph of the function and evaluating the

Suppose that f is a logarithmic function with values f (3) 7 and f (6) 10.

a. Without � nding the particular equation, � nd f (12) and f

b. Find the particular equation algebraically using natural logarithms.

c. Con� rm that your equation gives the value of f

a.

x y

3 7 6 10 12 13

16

f (12) 13 and f 16

b. f (x) a b ln x Write the general equation.

7 a b ln 3 10 a b ln 6

Substitute the given points.

3 b ln 6 b ln 3 Subtract the equations to eliminate a.

b 3 _________ ln 6 ln 3 Factor out b and then divide by ln 6 ln 3

7 a b.

a 7 Store a and b without rounding.

f (x) x Write the particular equation. Graph it on your grapher.

c. By calculator, f 16, which checks.


a. SOLUTION

2

2

2

3

3

3

➤

105

Additional Class Example

Plot f (x) 5 5x and g (x) 5 log5 x on the same screen. Use equal scales on both axes. Show that the graphs are refl ections of each other across the line y 5 x.

Solution

log 5 x 5 log x

____ log 5 5 1.4306… log x

Use the change-of-base property.

Enter f(x) 5 5 x and g(x) 5 1.4306… log x.Th e graphs are refl ections of each other across the line y 5 x, as shown in the fi gure.

1 x

f(x) = 5x

g(x) = log5 x

y = x

1

CAS Suggestions

Proving the general case of the multiply–add property of logarithms is straightforward on a CAS. Th e fi gure below shows that if the x-coordinate in f (x) 5 a 1 b log c (x) is multiplied by a positive constant k, the diff erence in the resulting y-coordinates is constant. Students need to be aware of potential domain diffi culties in order to obtain the desired result with a CAS. In this case, k needs to be restricted in order to give the desired result. Students oft en expand log(AB) to log(A) 1 log(B) without considering the signs of A and B; using a CAS encourages this awareness.



3. Carbon-14 Dating Problem: �e ages of things, such as wood, bone, and cloth, that are made from materials that were once living can be determined by measuring the percentage

in them. �is table contains data on age as a function of the remaining percentage of

Percentage Remaining

Age (yr)

100 0 90 871 80 70 60 50 5730

�e skull of the saber-toothed cat, which lived in the Pleistocene more than 11,000 years ago.

a. Based on theoretical considerations, it is

remaining is an exponential function of the age. How does this fact indicate that the age should be a logarithmic function of the percentage?

b. Using the �rst and last data points, �nd the particular equation of the logarithmic function that goes through the points. Show that the equation gives values of other points close to those in the table.

c. You can use your mathematical model to interpolate between the given data points to �nd fairly precise ages. Suppose that a piece

content 73.9%. What would you predict its age to be?

d. How old would you predict a piece of wood to

e. Search on the Internet or in some other resource to �nd out about early hominid

Give the source of your information.

4. Earthquake Problem: You can gauge the amount of energy released by an earthquake by its Richter magnitude, a scale devised by seismologist Charles F. Richter in 1935. �e Richter magnitude is a base-10 logarithmic function of the energy released by the earthquake. �ese data show the Richter magnitude m for earthquakes that release energy equivalent to the explosion of x tons of TNT (tri-nitro-toluene).

x (tons) m (Richter magnitude)

1,000 1,000,000 6.0

�e director of the National Earthquake Service in Golden,

Colorado, studies the seismograph display of a magnitude 7.5 earthquake.

a. Find the particular equation of the common logarithmic function m a b log x that �ts the two data points.


Reading AnalysisFrom what you have read in this section, what do you consider to be the main idea? What is the general equation of a logarithmic function, and how is a logarithmic function related to an exponential function? What numerical pattern do regularly spaced values of a logarithmic function follow?

Quick Review

Q1. Name the kind of function graphed in Figure 2-6g.

y

x

Figure 2-6g

y

x

Figure 2-6h

Q2. Name the kind of function graphed in Figure 2-6h.

Q3. Name the kind of function graphed in Figure 2-6i.

y

x

Figure 2-6i

y

x

Figure 2-6j

Q4. Name the kind of function graphed in Figure 2-6j.

Q5. Name a real-world situation that could be modeled by the function in Figure 2-6k.

y

x

Figure 2-6k

Q6. Sketch a reasonable graph: � e population of a city depends on time.

Q7. � e graph of a quadratic function is called a ? .

Q8. x 7 ) 2

Q9. Write the next three terms in this sequence:

Q10. ? functions.

a. Show that the values in the table have the

b. Use the � rst and last points to � nd algebraically the particular equation of the natural logarithmic function that � ts the points.

c. Verify that the equation in part b gives the other points in the table.

1. x y

3.6 12

57.6 3

921.6 5

2. x y

1 2

10 3

1001000 5

5min

Problem Set 2-6


PRO B LE M N OTES

Supplementary problems for this section are available at www.keypress.com/keyonline.

Q1. LinearQ2. ExponentialQ3. Inverse powerQ4. QuadraticQ5. Answers will vary.Q6.

Q7. ParabolaQ8. 9 x 2 2 42x 1 49Q9. 48, 96, 192Q10. Exponential

Problems 1–4 require students to fi nd particular equations for logarithmic functions that fi t given data sets. Th ey then use the particular equations as mathematical models to make predictions of y for given values of x, and vice versa.

Problems 1 and 2 can be solved with a system solution on a CAS. Once the function is defi ned, function notation can be used to show that the remaining values hold. Here you can see the steps for Problem 1b and c. Consider demonstrating this approach before asking students to use it.

t

P

1a. 14.4 ____ 3.6 5 57.6 ____ 14.4 5 230.4 _____ 57.6 5 921.6 _____ 230.4 5 4

1b. y 5 1 2 4 ln 3.6 ______ ln 256 1 4 ______ ln 256 ln x

5 0.0760… 1 0.7213…ln x1c. Th e equation fi ts the data.

2a. 10 ___ 1 5 100 ___ 10 5 1000 ____ 100 5 10

2b. y 5 2 1 1 _____ ln 10 ln x 5 2 1 0.4342…ln x2c. Th e equation fi ts the data.


3. Carbon-14 Dating Problem: �e ages of things, such as wood, bone, and cloth, that are made from materials that were once living can be determined by measuring the percentage

in them. �is table contains data on age as a function of the remaining percentage of

Percentage Remaining

Age (yr)

100 0 90 871 80 70 60 50 5730

�e skull of the saber-toothed cat, which lived in the Pleistocene more than 11,000 years ago.

a. Based on theoretical considerations, it is

remaining is an exponential function of the age. How does this fact indicate that the age should be a logarithmic function of the percentage?

b. Using the �rst and last data points, �nd the particular equation of the logarithmic function that goes through the points. Show that the equation gives values of other points close to those in the table.

c. You can use your mathematical model to interpolate between the given data points to �nd fairly precise ages. Suppose that a piece

content 73.9%. What would you predict its age to be?

d. How old would you predict a piece of wood to

e. Search on the Internet or in some other resource to �nd out about early hominid

Give the source of your information.

4. Earthquake Problem: You can gauge the amount of energy released by an earthquake by its Richter magnitude, a scale devised by seismologist Charles F. Richter in 1935. �e Richter magnitude is a base-10 logarithmic function of the energy released by the earthquake. �ese data show the Richter magnitude m for earthquakes that release energy equivalent to the explosion of x tons of TNT (tri-nitro-toluene).

x (tons) m (Richter magnitude)

1,000 1,000,000 6.0

�e director of the National Earthquake Service in Golden,

Colorado, studies the seismograph display of a magnitude 7.5 earthquake.

a. Find the particular equation of the common logarithmic function m a b log x that �ts the two data points.


Reading AnalysisFrom what you have read in this section, what do you consider to be the main idea? What is the general equation of a logarithmic function, and how is a logarithmic function related to an exponential function? What numerical pattern do regularly spaced values of a logarithmic function follow?

Quick Review

Q1. Name the kind of function graphed in Figure 2-6g.

y

x

Figure 2-6g

y

x

Figure 2-6h

Q2. Name the kind of function graphed in Figure 2-6h.

Q3. Name the kind of function graphed in Figure 2-6i.

y

x

Figure 2-6i

y

x

Figure 2-6j

Q4. Name the kind of function graphed in Figure 2-6j.

Q5. Name a real-world situation that could be modeled by the function in Figure 2-6k.

y

x

Figure 2-6k

Q6. Sketch a reasonable graph: � e population of a city depends on time.

Q7. � e graph of a quadratic function is called a ? .

Q8. x 7 ) 2

Q9. Write the next three terms in this sequence:

Q10. ? functions.

a. Show that the values in the table have the

b. Use the � rst and last points to � nd algebraically the particular equation of the natural logarithmic function that � ts the points.

c. Verify that the equation in part b gives the other points in the table.

1. x y

3.6 12

57.6 3

921.6 5

2. x y

1 2

10 3

1001000 5

5min

Problem Set 2-6

107

Problems 3–5 allow students to focus on the question while the CAS does the algebra.3a. Th e inverse of an exponential function is a logarithmic function.3b. y 5 238,069.2959… 2 8,266.6425… ln p; r 5 20.999…, very close to 213c. y(73.9) 5 2500.3068… 2500 years old3d. y(20) 5 13,304.6479… 13,300 years old3e. Answers will vary.4a. m 5 2 1 2 __ 3 log x



� is problem prepares you for the next section. 13. � e De� nition of e Problem: Figure 2-6o shows

the graph of y (1 2x) 1/x . If x 0, then y is unde� ned because of division by zero. If x is close to zero, then 1 _ x is very large. For instance,

(1 0.0001 ) 1/0.0001 1.000 1 10000

11 2 3 4

12

Limit e4

y

x

Figure 2-6o

a. Reproduce the graph in Figure 2-6o on your grapher. Use a window that has a grid point at x 0. Trace to values close to zero, and record the corresponding values of y.

b. Two competing properties in� uence the expression (1 x) 1/x as x approaches zero. A number greater than 1 raised to a large power is very large, but 1 raised to any power is still 1. Which of these competing properties “wins”? Or is there a “compromise” at some number larger than 1?

c. Call up the number e on your grapher. If it does not have an e key, calculate e 1 . What do you notice about the answer to part b and the number e?

14. Research Project: On the Internet or in some other reference source, � nd out about

contributions to the mathematics of logarithms. See if you can � nd out why natural logarithms are sometimes called Napierian logarithms. Give the source of your information.

8

1 62 43 24 04 85 66 47 2

2468

1 01 21 41 61 8

123456789

91 82 73 64 55 46 37 28 1

51 01 52 02 53 03 54 04 5

61 21 82 43 03 64 24 85 4

48

1 21 62 02 42 83 23 6

71 42 12 83 54 24 95 66 3

369

1 21 51 82 12 42 7

0 0

0

0

0

0

0

0

0

� ese rods are called “Napier’s bones.” Invented in the early 1600s, they made multiplication, division, and the extraction of square roots easier.


b. Use the equation to predict the Richter magnitude for

strongest on record, which released the energy of 5 billion tons of TNTAn earthquake that would release the

day, 160 trillion tons of TNT Blasting done at a construction site that releases the energy of about 30 lb of TNT

c. �e earthquake that caused a tsunami in the

about 9.0. How many tons of TNT would it take to produce a shock of this magnitude?

d. True or false? “Doubling the energy released by an earthquake doubles the Richter magnitude.” Give evidence to support your answer.

e.in some other resource. Name one thing you learned that is not mentioned in this problem. Give the source of your information.

5. Logarithmic Function Vertical Dilation and Translation Problem:

y

xf

g5

105

Figure 2-6l

y

x

g

f

h

5

105

Figure 2-6m

a. Figure 2-6l shows the graph of the common logarithmic function f (x) lo g 10 x (dashed) and a vertical dilation of this graph by a factor of 6, y g (x) (solid). Write an equation of g (x), considering it as a vertical dilation. Write another equation of g (x) in terms of l og b x, where b is a number other than 10. Identify the base.

b. Figure 2-6m shows the graph of f (x) ln x (dashed). Two vertical translations are also shown, g (x) 3 ln x and h(x) 1 ln x. Find algebraically or numerically the x-intercepts of the graphs of g and h.

6. Logarithmic and Exponential Function Graphs Problem: Figure 2-6n shows the graph of an exponential function y f (x) and its inverse function y g (x).

y

x

f

g

1

1

Figure 2-6n

a. �e base of the exponential function is an integer. Which integer?

b. Write the particular equation of the inverse function y g (x) f 1 (x).

c. Con�rm that your answers to parts a and b are correct by plotting on your grapher.

d. With your grapher in parametric mode, plot these parametric equations:

x 1 (t) f (t) y 1 (t) t What do you notice about the resulting graph? e. From your answer to part d, explain how you

could plot on your grapher the inverse of any given function. Show that your method works by plotting the inverse of the function y x 3 9 x 2 23x 15.

their domains. 7. f (x) 2 log(x 3) 8. f (x) log(3 2x) 9. f (x) lo g 3 x 2 10. f (x) ln x 2 11. f (x) ln 3x 12. f (x) 2 (3x 5)


Problem Notes (continued)4b. m(5 3 10 9 ) 5 8.4659… 8.5; m(160 3 10 12 ) 5 11.4694… 11.5; m   3 ____ 2000 5 0.7840… 0.84c. 31.6 billion tons4d. False. Doubling the energy increases the Richter magnitude linearly by 2 _ 3 log 2 5 0.2006… points. This is not surprising, because logarithmic functions have the multiply–add property.4e. Answers will vary.

Problem 5 asks students to express the same function both as a common logarithm function with a dilation and as a log function with a base other than 10.5a. g (x) 5 6 lo g 10 x; log 6  

___ 10 x 5 log 1.4677... 5 x

5b. The x-intercept of g is e 23 , and the x-intercept of h is e.

Problem 6 asks students to explore the relationship between an exponential function and its inverse.6a. 26b. y 5 lo g 2 x or

log x _____ log 2 or ln x ____ ln 2

6c. The graph matches the dotted function.6d. This graph also matches the dotted function.6e. In parametric mode, graph x(t) 5 f(t), y(t) 5 t. y 5 f(x) 5 x 3 2 9 x 2 1 23x 2 15:

y

x

10

�10

5

y 5 f 21 (x), given by x(t) 5 t 3 2 9 t 2 1 23t 2 15, y(t) 5 t:

y

x10�10

6

7. Domain: x . 23y

x

3

3

�3

�3

8. Domain: x , 1.5y

x

3

3

�3

�3


� is problem prepares you for the next section. 13. � e De� nition of e Problem: Figure 2-6o shows

the graph of y (1 2x) 1/x . If x 0, then y is unde� ned because of division by zero. If x is close to zero, then 1 _ x is very large. For instance,

(1 0.0001 ) 1/0.0001 1.000 1 10000

11 2 3 4

12

Limit e4

y

x

Figure 2-6o

a. Reproduce the graph in Figure 2-6o on your grapher. Use a window that has a grid point at x 0. Trace to values close to zero, and record the corresponding values of y.

b. Two competing properties in� uence the expression (1 x) 1/x as x approaches zero. A number greater than 1 raised to a large power is very large, but 1 raised to any power is still 1. Which of these competing properties “wins”? Or is there a “compromise” at some number larger than 1?

c. Call up the number e on your grapher. If it does not have an e key, calculate e 1 . What do you notice about the answer to part b and the number e?

14. Research Project: On the Internet or in some other reference source, � nd out about

contributions to the mathematics of logarithms. See if you can � nd out why natural logarithms are sometimes called Napierian logarithms. Give the source of your information.

8

1 62 43 24 04 85 66 47 2

2468

1 01 21 41 61 8

123456789

91 82 73 64 55 46 37 28 1

51 01 52 02 53 03 54 04 5

61 21 82 43 03 64 24 85 4

48

1 21 62 02 42 83 23 6

71 42 12 83 54 24 95 66 3

369

1 21 51 82 12 42 7

0 0

0

0

0

0

0

0

0

� ese rods are called “Napier’s bones.” Invented in the early 1600s, they made multiplication, division, and the extraction of square roots easier.


b. Use the equation to predict the Richter magnitude for

strongest on record, which released the energy of 5 billion tons of TNTAn earthquake that would release the

day, 160 trillion tons of TNT Blasting done at a construction site that releases the energy of about 30 lb of TNT

c. �e earthquake that caused a tsunami in the

about 9.0. How many tons of TNT would it take to produce a shock of this magnitude?

d. True or false? “Doubling the energy released by an earthquake doubles the Richter magnitude.” Give evidence to support your answer.

e.in some other resource. Name one thing you learned that is not mentioned in this problem. Give the source of your information.

5. Logarithmic Function Vertical Dilation and Translation Problem:

y

xf

g5

105

Figure 2-6l

y

x

g

f

h

5

105

Figure 2-6m

a. Figure 2-6l shows the graph of the common logarithmic function f (x) lo g 10 x (dashed) and a vertical dilation of this graph by a factor of 6, y g (x) (solid). Write an equation of g (x), considering it as a vertical dilation. Write another equation of g (x) in terms of l og b x, where b is a number other than 10. Identify the base.

b. Figure 2-6m shows the graph of f (x) ln x (dashed). Two vertical translations are also shown, g (x) 3 ln x and h(x) 1 ln x. Find algebraically or numerically the x-intercepts of the graphs of g and h.

6. Logarithmic and Exponential Function Graphs Problem: Figure 2-6n shows the graph of an exponential function y f (x) and its inverse function y g (x).

y

x

f

g

1

1

Figure 2-6n

a. �e base of the exponential function is an integer. Which integer?

b. Write the particular equation of the inverse function y g (x) f 1 (x).

c. Con�rm that your answers to parts a and b are correct by plotting on your grapher.

d. With your grapher in parametric mode, plot these parametric equations:

x 1 (t) f (t) y 1 (t) t What do you notice about the resulting graph? e. From your answer to part d, explain how you

could plot on your grapher the inverse of any given function. Show that your method works by plotting the inverse of the function y x 3 9 x 2 23x 15.

their domains. 7. f (x) 2 log(x 3) 8. f (x) log(3 2x) 9. f (x) lo g 3 x 2 10. f (x) ln x 2 11. f (x) ln 3x 12. f (x) 2 (3x 5)

109

to the left of x 5 21. (Th e grapher is not familiar with our restriction that if a base is raised to a power, then the base must be positive.) Th e function has asymptotes at x 5 21 and y 5 1 and a removable discontinuity (a hole) at (0, e). Consider talking informally about the behavior of the function as x approaches 21 and as x approaches infi nity. Another interesting aspect of this function is what’s happening as x approaches 0. Th is is a good opportunity for an informal discussion of limits. One of the defi nitions of e is

e 5 lim x→0

(1 1 x) 1/x

13b. Th e two properties balance out, so that as x approaches 0, y approaches 2.7182….13c. e 5 2.7182… ; they are the same.

Problem 14 is a research project that can be assigned for extra credit.14. Answers will vary.


1. A given logarithmic function contains the points (3, 5) and (7, 13).a. If the function is a common

logarithm, determine its general form.

b. If the function is a natural logarithm with a vertical asymptote at x 5 2, determine its translated form.

c. What is the base of the function if only dilations were applied to make the function contain the given points?

2. Th e graph of a non-transformed exponential function intersects the graph of a non-transformed logarithmic function twice. If each function has the same base, what can be said about the value of the base of the functions?

See page 983 for answers toProblems 11–13a and CAS Problems 1–2.

9. Domain: x 0

y

x

3

�3 3

�3

10. Domain: x . 2 or x , 22

y

x

3

�3

�3

3

Problem 13 can be used as the basis for an interesting class discussion. Th e domain of the function is x . 21 and x 0. However, some graphers may show discrete points


111Section 2-7: Logistic Functions for Restrained Growth

Suppose that this table lists the population of a small community, in thousands of people. � e � gure shows a scatter plot of the data.

x (years)y (thousands

of people)

1 22 33 5

95 136 197 278 329 36

10 39

1020304050

5 10 15 20

y (thousands of people)

x (years)

1.

2. At � rst the population seems to be increasing exponentially with time. On a copy of the given graph, sketch the graph of an exponential function that would � t the � rst six data points reasonably well.

3. Toward the end of the 10-yr period, the function seems to be leveling o� . A function that models such population growth is the logistic function. Its general equation is

y c ________ 1 a e bx

where x and y are the variables, e is the base of the natural logarithm, and a, b, and c stand for constants. � e community has room

c Calculate a and b using the � rst and the tenth points. Write the particular equation, and plot it on the same screen as the data. Sketch the result.

4. What does the logistic function indicate the population was at time x 0 yr?

5. What graphical evidence do you have that the maximum population in the community

6. logistic in a dictionary, and � nd the origin of the word.


Suppose that this table lists the population of a 3. Toward the end of the 10-yr period, the

E X P L O R AT I O N 2-7: T h e L o g i s t i c Fu n c t i o n f o r P o p u l a t i o n G r o w t h

Figure 2-7c shows the graphs of

f (x) 2 x and g (x) 2 x ______ 2 x 1

Function f is an exponential function, and function g is a logistic function. For large positive values of x, the graph of g levels o� to y 1. � is is because, for large values of x, 2 x is large compared to the 1 in the denominator. So the denominator is not much di� erent from the 2 x in the numerator, and the fraction representing g (x) approaches 1.

g (x) 2 x __ 2 x 1

For large negative values of x, the 2 x in the denominator is close to zero. So the denominator is close to 1. � us the fraction representing g (x) approaches 2 x .

g (x) 2 x _____ 0 1 2 x

y

x

gf

55

1

Figure 2-7c


Logistic Functions for Restrained GrowthSuppose that the population of a new subdivision is growing rapidly. � is table shows monthly population � gures.

x (mo) y (houses)

2 103117

6 132

8 10 167

Figure 2-7a shows the plot of points and a smooth (dashed) curve that goes through them. You can tell that it is increasing, is concave up, and has a positive y-intercept, suggesting that an exponential function � ts the points. Using the � rst and last points gives the function y 91.2782… (1.0622… ) x , the curve shown in the � gure, which � ts the points almost exactly. Suppose that there are only 1000 lots in the subdivision. � e actual number of houses will level o� , approaching 1000 gradually, as shown in Figure 2-7b.

10 155

250 Concave up

200

150

100

50

y

x

Figure 2-7a

y

x

Restrained growthFigure 2-7b

In this section you will learn about logistic functions that are useful as mathematical models of restrained growth.

In this exploration, you’ll � t the logistic function to restricted population growth.

Logistic Functions for Restrained Growth

2-7

Objective



Class Time 1 day

Homework AssignmentRA, Q1–Q10, Problems 1, 3, 4, 5, 7

Teaching ResourcesExploration 2-7: Th e Logistic Function

for Population GrowthBlackline Master

Exploration Problem 2Supplementary ProblemsTest 5, Sections 2-4 to 2-7 Forms A and B


Presentation Sketch: Logistic

Present.gsp

Exploration 2-7, Th e Logistic

Function for Population Growth

TE ACH I N G

Important Terms and ConceptsLogistic functionRestrained growth

Section Notes

A logistic function can be used to model restrained population growth. From the left to right, the graph of a logistic function (with b . 0) starts out looking very much like an increasing exponential function. It is asymptotic to y 5 0 and is increasing and concave up. At the point of infl ection the rate of growth is the greatest. Aft er the point of infl ection the rate of growth slows, the function becomes concave down, approaching a horizontal asymptote. Th e maximum value the function approaches is called the carrying capacity, or the maximum sustainable population. Th e y-coordinate of the point of infl ection is equal to one-half of c, the carrying capacity. Th e logistic function is symmetric about the point of infl ection.

A logistic function is the only type of function in this chapter that changes concavity. At the point of infl ection, the rate of growth of the function is a maximum.

Exploration 2-7 asks students to fi t a logistic growth model to some population data. In the problem the maximum sustainable population is 43,000 people, so c 5 43. Th e values of a and b are calculated by using the fi rst and last data points.

Example 1 provides another opportunity to fi nd a logistic model for a set of data and to examine the role of the infl ection point. When you discuss this example, be sure to emphasize the real-world importance of the point of infl ection.

Th e properties box aft er the example summarizes the features of logistic functions. Notice that the box includes the case where b , 0, in which the logistic function is decreasing. Problem 6 involves


Suppose that this table lists the population of a small community, in thousands of people. � e � gure shows a scatter plot of the data.

x (years)y (thousands

of people)

1 22 33 5

95 136 197 278 329 36

10 39

1020304050

5 10 15 20

y (thousands of people)

x (years)

1.

2. At � rst the population seems to be increasing exponentially with time. On a copy of the given graph, sketch the graph of an exponential function that would � t the � rst six data points reasonably well.

3. Toward the end of the 10-yr period, the function seems to be leveling o� . A function that models such population growth is the logistic function. Its general equation is

y c ________ 1 a e bx

where x and y are the variables, e is the base of the natural logarithm, and a, b, and c stand for constants. � e community has room

c Calculate a and b using the � rst and the tenth points. Write the particular equation, and plot it on the same screen as the data. Sketch the result.

4. What does the logistic function indicate the population was at time x 0 yr?

5. What graphical evidence do you have that the maximum population in the community

6. logistic in a dictionary, and � nd the origin of the word.


Suppose that this table lists the population of a 3. Toward the end of the 10-yr period, the

E X P L O R AT I O N 2-7: T h e L o g i s t i c Fu n c t i o n f o r P o p u l a t i o n G r o w t h

Figure 2-7c shows the graphs of

f (x) 2 x and g (x) 2 x ______ 2 x 1

Function f is an exponential function, and function g is a logistic function. For large positive values of x, the graph of g levels o� to y 1. � is is because, for large values of x, 2 x is large compared to the 1 in the denominator. So the denominator is not much di� erent from the 2 x in the numerator, and the fraction representing g (x) approaches 1.

g (x) 2 x __ 2 x 1

For large negative values of x, the 2 x in the denominator is close to zero. So the denominator is close to 1. � us the fraction representing g (x) approaches 2 x .

g (x) 2 x _____ 0 1 2 x

y

x

gf

55

1

Figure 2-7c


Logistic Functions for Restrained GrowthSuppose that the population of a new subdivision is growing rapidly. � is table shows monthly population � gures.

x (mo) y (houses)

2 103117

6 132

8 10 167

Figure 2-7a shows the plot of points and a smooth (dashed) curve that goes through them. You can tell that it is increasing, is concave up, and has a positive y-intercept, suggesting that an exponential function � ts the points. Using the � rst and last points gives the function y 91.2782… (1.0622… ) x , the curve shown in the � gure, which � ts the points almost exactly. Suppose that there are only 1000 lots in the subdivision. � e actual number of houses will level o� , approaching 1000 gradually, as shown in Figure 2-7b.

10 155

250 Concave up

200

150

100

50

y

x

Figure 2-7a

y

x

Restrained growthFigure 2-7b

In this section you will learn about logistic functions that are useful as mathematical models of restrained growth.

In this exploration, you’ll � t the logistic function to restricted population growth.

Logistic Functions for Restrained Growth

2-7

Objective

111

2. Sample equation: y 5 1.1972... 1.6229... x

y

x5 10 15 20

5040302010

3. b 5 0.5886... a 5 36.9312... y 5 43 ___________________

1 1 36.931 2...e 20.5886...x

y

x5 10 15 20

5040302010

4. There were approximately 1134 people.

5. The graph seems to be leveling off at about 43,000.

6. From Greek logos, meaning “word” or “calculation”


Differentiating Instruction• Help ELL students with Problem 13

and let them answer Problem 14 in their primary language.

• Some languages have more than one word for grow, depending on what is growing; this can make discussion of growth harder for many ELL students to understand. Visualizations will help.

• Have students write the logistic equation and its properties in their journals.

• Visually explain “increasing at an increasing rate” and “increasing at a decreasing rate.” Have students put this in their journals.

• Have ELL students do Problem 14 in pairs.

a real-world situation that can be modeled with a decreasing logistic function.

Exploration Notes

Exploration 2-7 demonstrates how to fit a logistic function to restricted population growth from two points and the upper horizontal asymptote. A blackline master for Problem 2 is available in the Instructor’s Resource Book. Allow approximately 15 minutes for this activity.

1. y

x5 10 15 20

5040302010

Section 2-7: Logistic Functions for Restrained Growth


c. Make a table showing that the logistic function � ts all the points closely.

d. Use the logistic function to predict the number of houses that will be occupied at the value of x corresponding to 2 yr. Which process do you use, extrapolation or interpolation?

e. Find the value of x at the point of in� ection. What is the real-world meaning of the fact that the graph is concave up for times before the point of in� ection and concave down therea� er?

Start with the second form of the logistic function.

a. y 1000 ________ 1 a b x � e vertical dilation is 1000.

103 1000 ________ 1 ab 2

167 1000 _________ 1 a b 10

Substititute points (2, 103) and (10, 167)

103 103a b 2 1000 167 167a b 10 1000

Divide to eliminate a and simplify.

b 833 ___ 897 103 ___ 167 1/8

1.0721… Store without rounding.

103a (1.0721… ) 2 897 Substitute for b.

a 10.0106… Start without rounding

y 1000 ______________________ 1 10.0106… (1.0721…) x Write the particular equation.

b. Figure 2-7b on page 110 shows the graph.

c. x (months) y (houses) Logistic Function

2 103 103 (exact)117 116.60… (close)

6 132 131.73… (close)8 (close)

10 167 167 (exact)

d. Trace the function to x

y

Start with the second form of the logistic function. SOLUTION

103a b 2 897 167a b 10 833

167a b 10 ________ 103a b 2 833 ___ 897

b 8 833 ___ 897 103 ___ 167


As you can see in Figure 2-7c, the logistic function is almost indistinguishable from the exponential function for large negative values of x. But for large positive values, the logistic function levels o� , as did the number of occupied houses represented by Figure 2-7b. You can � t logistic functions to data sets by using the same dilations and translations you have used for other types of functions. You’ll

General Logistic FunctionYou can transform the equation of function g in Figure 2-7c so that only one exponential term appears.

g (x) 2 x ______ 2 x 1

2 x ______ 2 x 1 2 x ___ 2 x Multiply by a clever form of 1.

1 _______ 1 2 x

To get a general function of this form, replace the 1 in the numerator with a constant, c, to give the function a vertical dilation by a factor of c. Replace the exponential term 2 x with ab x or with a e bx if you want to use the natural exponential function. � e result is shown in the box.

DEFINITION: Logistic Function General Equationf (x) c ________ 1 ae bx or f (x) c ________ 1 a b x

where a, b, and c are constants and the domain is all real numbers

Use the information on the occupied houses from the beginning of the section.

x (months) y (houses)

2 103117

6 132

810 167

a. Given that there are 1000 lots in the subdivision, use the points for 2 mo and 10 mo to � nd the particular equation of the logistic function that satis� es these constraints.

b. the result.

Use the information on the occupied houses from the beginning of the section.EXAMPLE 1 ➤


Technology Notes

Presentation Sketch: Logistic Present.gsp at www.keypress.com/keyonline demonstrates the different logistic growth models as the parameters a, b, and c change. Additional pages of the document provide a brief introduction to the iterative approach to logistic growth.

Exploration 2-7 asks students to fit a logistic growth model to some population data. The data can be plotted and the function can be graphed in Fathom. You might encourage students to use sliders in order to find the best a and b to fit the data and then to compare their values to those they find algebraically, as the exploration suggests.


c. Make a table showing that the logistic function � ts all the points closely.

d. Use the logistic function to predict the number of houses that will be occupied at the value of x corresponding to 2 yr. Which process do you use, extrapolation or interpolation?

e. Find the value of x at the point of in� ection. What is the real-world meaning of the fact that the graph is concave up for times before the point of in� ection and concave down therea� er?

Start with the second form of the logistic function.

a. y 1000 ________ 1 a b x � e vertical dilation is 1000.

103 1000 ________ 1 ab 2

167 1000 _________ 1 a b 10

Substititute points (2, 103) and (10, 167)

103 103a b 2 1000 167 167a b 10 1000

Divide to eliminate a and simplify.

b 833 ___ 897 103 ___ 167 1/8

1.0721… Store without rounding.

103a (1.0721… ) 2 897 Substitute for b.

a 10.0106… Start without rounding

y 1000 ______________________ 1 10.0106… (1.0721…) x Write the particular equation.

b. Figure 2-7b on page 110 shows the graph.

c. x (months) y (houses) Logistic Function

2 103 103 (exact)117 116.60… (close)

6 132 131.73… (close)8 (close)

10 167 167 (exact)

d. Trace the function to x

y

Start with the second form of the logistic function. SOLUTION

103a b 2 897 167a b 10 833

167a b 10 ________ 103a b 2 833 ___ 897

b 8 833 ___ 897 103 ___ 167


As you can see in Figure 2-7c, the logistic function is almost indistinguishable from the exponential function for large negative values of x. But for large positive values, the logistic function levels o� , as did the number of occupied houses represented by Figure 2-7b. You can � t logistic functions to data sets by using the same dilations and translations you have used for other types of functions. You’ll

General Logistic FunctionYou can transform the equation of function g in Figure 2-7c so that only one exponential term appears.

g (x) 2 x ______ 2 x 1

2 x ______ 2 x 1 2 x ___ 2 x Multiply by a clever form of 1.

1 _______ 1 2 x

To get a general function of this form, replace the 1 in the numerator with a constant, c, to give the function a vertical dilation by a factor of c. Replace the exponential term 2 x with ab x or with a e bx if you want to use the natural exponential function. � e result is shown in the box.

DEFINITION: Logistic Function General Equationf (x) c ________ 1 ae bx or f (x) c ________ 1 a b x

where a, b, and c are constants and the domain is all real numbers

Use the information on the occupied houses from the beginning of the section.

x (months) y (houses)

2 103117

6 132

810 167

a. Given that there are 1000 lots in the subdivision, use the points for 2 mo and 10 mo to � nd the particular equation of the logistic function that satis� es these constraints.

b. the result.

Use the information on the occupied houses from the beginning of the section.EXAMPLE 1 ➤

113

CAS Suggestions

Students can use a CAS to produce logistic functions in diff erent forms, gathering useful information about the curve from each form.



Q4. �e function in ? ? property.

Q5. �e expression ln x is a logarithm with the number ? as its base.

Q6. Write in exponential form: h lo g p m Q7. Write in logarithmic form: c 5 j Q8. If an object rotates at 100 revolutions per

minute, how many degrees per second is this? Q9. Write the general equation of a quadratic

function. Q10. �e function g (x) 3 f 5 x 6 is a

horizontal translation of function f by A. 3 B. C. 5 D. 6 E. 6

1. Given the exponential function f (x) 2. 2 x and the logistic function g (x) 2. 2 x ______ 2. 2 x 1 ,

a. domain 10 x 10. Sketch the result.

b. How do the two graphs compare for large positive values of x? How do they compare for large negative values of x?

c. Find graphically the approximate x-value of the point of in�ection for function g. For what values of x is the graph of function g concave up? Concave down?

d. has a horizontal asymptote at y 1.

e. Transform the equation of the logistic function so that an exponential term appears only once. Show numerically that the resulting equation is equivalent to g (x) as given.

2. Figure 2-7d shows the graph of the logistic function

f (x) 3 e 0.2x _______ e 0.2x

a. horizontal asymptote at y 3.

b. Read the point of in�ection from the graph. Find the x-coordinate algebraically.

c. For what values of x is the graph concave up? Concave down?

d. Transform the equation so that there is only one exponential term. Con�rm by graphing that the resulting equation is equivalent to f (x) as given.

3. Spreading the News Problem: You arrive at school and meet your mathematics teacher, who tells you today’s test has been canceled! You and your friend spread the good news. �e table shows the number of students, y, who have heard the news a�er time x, in minutes, has passed since you and your friend heard the news.

x (min) y (students)

0 210 520 1330 35

90

a. points. Is the graph of this function concave up, concave down, or both?

b. �ere are 1220 students in the school. Use the numbers of students at time 0 min and at time

function that meets these constraints. c.

�rst 3 hours. d. Based on the logistic model, how many

students have heard the news at 9:00 a.m. if you heard it at 8:00 a.m.? How long will it be until all but ten students have heard the news?

x

f(x)3

302010Figure 2-7d


� e process is extrapolation because x given points.

e. � e point of in� ection is halfway between the x-axis and the asymptote at y 1000. Trace the function to a value that is close to y 500. Use the intersect feature to � nd x occurs at about 33 mo. Before 33 mo, the number of houses is increasing at an increasing rate. A� er 33 mo, the number is still increasing but at a decreasing rate.

Note that if a, b, and c are all positive, the logistic function will have two horizontal asymptotes, one at the x-axis and one at the line y c. � e point of in� ection occurs halfway between these two asymptotes.

PROPERTIES: Logistic Functions � e logistic function is y c _______ 1 a e bx where a, b, and c are constants such thata 0, b 0, c 0. � e domain is all real numbers. � e logistic function has

y 0 and another at y c

y c _ 2

If b 0 If b 0

y c

xAsymptote at y 0Logistic function

Asymptote at y c

Note: a 0, c 01 ae–bx

Point of in�ectionat y c

2

x

Note: a 0, c 0y c1 ae–bx

Asymptote at y c


2Asymptoteat y 0

Logistic function

➤

Reading AnalysisFrom what you have read in this section, what do you consider to be the main idea? What is the main di� erence between the graph of a logistic function and the graph of an exponential function? For what kind of real-world situations are logistic functions reasonable mathematical models?

Quick Review

Q1. An exponential function has the ? ? property.

Q2. A power function has the ? ? property.

Q3. � e equation y 3 5 ln x de� nes a ? function.

5min

Problem Set 2-7


PRO B LE M N OTES

Supplementary problems for this section are available at www.keypress.com/keyonline.

Q1. Add–multiply Q2. Multiply–multiply Q3. LogarithmicQ4. Multiply–addQ5. eQ6. p h 5 m Q7. j 5 lo g 5 cQ8. 600 deg/sQ9. y 5 ax 2 1 bx 1 c, a 0Q10. D

Problems 1 and 2 are similar to the example in the text. You may want to remind students that the equation in Problem 1 can be rewritten to look like the form of the logistic function on page 112.1a.

1b. Th e graphs are almost the same for large negative values of x, but widely diff erent for large positive values of x.1c. x 5 0; concave up for x , 0 and concave down for x . 01d. As x grows very large, the 1 in the denominator becomes insignifi cant in comparison to the 2.2 x , so g (x) 5 2.2 x _______ 2.2 x 1 1 2.2 x ____ 2.2 x 5 1

1e. g (x) 5 1 ________ 1 1 2.2 2x . A table of values shows that the expressions are equivalent.

5

10

x

f(x)

–5 5

g(x)

y


Q4. �e function in ? ? property.

Q5. �e expression ln x is a logarithm with the number ? as its base.

Q6. Write in exponential form: h lo g p m Q7. Write in logarithmic form: c 5 j Q8. If an object rotates at 100 revolutions per

minute, how many degrees per second is this? Q9. Write the general equation of a quadratic

function. Q10. �e function g (x) 3 f 5 x 6 is a

horizontal translation of function f by A. 3 B. C. 5 D. 6 E. 6

1. Given the exponential function f (x) 2. 2 x and the logistic function g (x) 2. 2 x ______ 2. 2 x 1 ,

a. domain 10 x 10. Sketch the result.

b. How do the two graphs compare for large positive values of x? How do they compare for large negative values of x?

c. Find graphically the approximate x-value of the point of in�ection for function g. For what values of x is the graph of function g concave up? Concave down?

d. has a horizontal asymptote at y 1.

e. Transform the equation of the logistic function so that an exponential term appears only once. Show numerically that the resulting equation is equivalent to g (x) as given.

2. Figure 2-7d shows the graph of the logistic function

f (x) 3 e 0.2x _______ e 0.2x

a. horizontal asymptote at y 3.

b. Read the point of in�ection from the graph. Find the x-coordinate algebraically.

c. For what values of x is the graph concave up? Concave down?

d. Transform the equation so that there is only one exponential term. Con�rm by graphing that the resulting equation is equivalent to f (x) as given.

3. Spreading the News Problem: You arrive at school and meet your mathematics teacher, who tells you today’s test has been canceled! You and your friend spread the good news. �e table shows the number of students, y, who have heard the news a�er time x, in minutes, has passed since you and your friend heard the news.

x (min) y (students)

0 210 520 1330 35

90

a. points. Is the graph of this function concave up, concave down, or both?

b. �ere are 1220 students in the school. Use the numbers of students at time 0 min and at time

function that meets these constraints. c.

�rst 3 hours. d. Based on the logistic model, how many

students have heard the news at 9:00 a.m. if you heard it at 8:00 a.m.? How long will it be until all but ten students have heard the news?

x

f(x)3

302010Figure 2-7d


� e process is extrapolation because x given points.

e. � e point of in� ection is halfway between the x-axis and the asymptote at y 1000. Trace the function to a value that is close to y 500. Use the intersect feature to � nd x occurs at about 33 mo. Before 33 mo, the number of houses is increasing at an increasing rate. A� er 33 mo, the number is still increasing but at a decreasing rate.

Note that if a, b, and c are all positive, the logistic function will have two horizontal asymptotes, one at the x-axis and one at the line y c. � e point of in� ection occurs halfway between these two asymptotes.

PROPERTIES: Logistic Functions � e logistic function is y c _______ 1 a e bx where a, b, and c are constants such thata 0, b 0, c 0. � e domain is all real numbers. � e logistic function has

y 0 and another at y c

y c _ 2

If b 0 If b 0

y c

xAsymptote at y 0Logistic function

Asymptote at y c

Note: a 0, c 01 ae–bx


2

x

Note: a 0, c 0y c1 ae–bx

Asymptote at y c


2Asymptoteat y 0

Logistic function

➤

Reading AnalysisFrom what you have read in this section, what do you consider to be the main idea? What is the main di� erence between the graph of a logistic function and the graph of an exponential function? For what kind of real-world situations are logistic functions reasonable mathematical models?

Quick Review

Q1. An exponential function has the ? ? property.

Q2. A power function has the ? ? property.

Q3. � e equation y 3 5 ln x de� nes a ? function.

5min

Problem Set 2-7

115

Problem 2b can be addressed using a Solve command on a CAS.

2a. As x grows very large, the 4 in the denominator becomes insignifi cant in comparison to e 0.2x so

f(x) 5 3 e 0.2x _______ e 0.2x 1 4 3 e 0.2x ____ e 0.2x 5 3

2b. Point of infl ection at x 5 6.9314… . 2c. f is concave up for x , 6.9314… and concave down for x . 6.9314…2d. f(x) 5 3 _________ 1 1 4 e 20.2x . Th e graphs coincide.

Problems 3 and 5 are real-world applications of logistic modeling. 3a. Concave up

x

y

60 120 180

50

100

150

3b. y 5 1220 ___________________ 1 1 (609) (1.1019…) 2x

3c.

3d. 435 students; 115.4930… min

60 120 180

400

800

1200y

x



6. Rabbit Overpopulation Problem: Figure 2-7e shows two logistic functions represented by the equation

y 1000 ________ 1 ae x

Both functions represent the population of rabbits in a particular woods as a function of time x, in years. �e value of the constant a is to be determined under two di�erent initial conditions.

2000

1000

1 2 3 4 5

y

g

fx

Figure 2-7e

a. For y f (x) in Figure 2-7e, 100 rabbits were introduced into the woods at time x 0. Find the value of the constant a under this condition. Show that your answer is correct by plotting the graph of f on your grapher.

b. How do you interpret this mathematical model with regard to what happens to the rabbit population under the condition in part a?

c. For y g (x) in Figure 2-7e, 2000 rabbits were introduced into the woods at time x 0. Find the value of a under this condition. Show that the graph agrees with Figure 2-7e. How does the sign of a represent a generalization of the de�nition of logistic function?

d. How do you interpret the mathematical model under the condition of part c? What seems to be the implication of trying to stock a region with a greater number of a particular species than the region can support?

7. Given the logistic function

f (x) c _________ 1 a e x

a. a graphs of f for c 1, 2, and 3. Use as a domain 10 x 10. Sketch the results. True or false? “�e constant c is a vertical dilation factor.”

b. Figure 2-7f shows the graph of f with c 2 and with a 0.2, 1, and 5. Which graph is which? How does the value of a transform the graph?

2

10

f(x)

x5510

Figure 2-7f

c. g (x) c ___________ 1 a e x 3) .

What transformation applied to f does this represent? Con�rm that your answer is correct by plotting f and g on the same screen using c 2 and a 1.

d. What value of a in the equation of f (x) would produce the same transformation as in part c?


4. Spreading the News Simulation Experiment: In this experiment you will simulate the spread of

and then selects two people at random to “tell” the news to. Do this by selecting two random integers between 1 and the number of students in your class, inclusive. (It is not actually necessary to tell any news!) �e random number generator on one student’s calculator will help make the random selection. �e two people with the chosen numbers stand. �us, a�er the �rst iteration, there will probably be three students standing (unless a duplicate random number

two more people to “tell” the news to by selecting a total of six (or four) more random integers. Do this for a total of ten iterations or until the entire class is standing. At each iteration, record the number of iterations and the total number of people who have heard the news. Describe the results of the experiment. Include things such as

�e plot of the data points. A function that �ts the data, and a graph of

chose the function you did. A statement of how well the logistic model �ts the data. �e iteration number at which the good news was spreading most rapidly.

5. Ebola Outbreak Epidemic Problem: In the

out in the Gulu district of Uganda. �e table shows the total number of people infected from

infections. �e �nal number of people who were infected during this epidemic wasa virus that causes internal bleeding and is fatal in most cases.)

x (days) y (total infections)

1 71

10 182

15 239

21 281

30 321

50 370

A Red Cross medical o�cer instructs villagers about the Ebola

virus in Kabede Opong, Uganda.

a. �ts the data. Is the graph of this function concave up or concave down?

b. Use the second and last data points to �nd the particular equation of a logistic function that �ts the data.

c. the logistic function from part b. Sketch the results.

d. Where does the point of in�ection occur in the logistic model? What is the real-world meaning of this point?

e. Based on the logistic model, how many people

f. Find data about other epidemics. Give your source. Try to model the spread of the epidemic for which you found data.


Problem Notes (continued)

Problem 4 is an interesting class simulation experiment. If you have the time, it is worthwhile. If the same number is generated a second time, ignore it and generate another random number.4. Simulations will vary. 5a. Concave down

50

200

400y

x

5b. y 5 396 _______________________ 1 1 (2.7532…)(1.0888… ) 2x

5c.

5d. The point of inflection occurs at (11.9037…, 198). Before approximately 12 days passed, the rate of new infection was increasing; after that, the rate was decreasing.5e. Approximately 363 people were infected.5f. Answers will vary.

Problem 6 presents a real-world situation that is modeled by a decreasing logistic function. In part c, there are too many rabbits to be supported by the environment, so they begin to die off faster than they are born, thereby reaching the maximum sustainable population. Note that when the y-coordinate of the initial value is greater than 1000, the logistic function decreases.

50

200

400

y

x

6a. a 5 9, so f(x) 5 1000 ______ 1 1 9 e 2x . The graph is correct.6b. The natural ceiling on the number of rabbits is 1000. If the population is less than this, it will grow toward this limit.6c. a 5 2 1 _ 2 , so g (x) 5 1000 ______

1 2 1 _ 2 e 2x . The graph

is correct. The sign of a is negative, whereas the definition of logistic function states that a . 0, so this is a generalization of the definition.

6d. If the population is greater than the number the region can support, it will decrease toward that limit.


6. Rabbit Overpopulation Problem: Figure 2-7e shows two logistic functions represented by the equation

y 1000 ________ 1 ae x

Both functions represent the population of rabbits in a particular woods as a function of time x, in years. �e value of the constant a is to be determined under two di�erent initial conditions.

2000

1000

1 2 3 4 5

y

g

fx

Figure 2-7e

a. For y f (x) in Figure 2-7e, 100 rabbits were introduced into the woods at time x 0. Find the value of the constant a under this condition. Show that your answer is correct by plotting the graph of f on your grapher.

b. How do you interpret this mathematical model with regard to what happens to the rabbit population under the condition in part a?

c. For y g (x) in Figure 2-7e, 2000 rabbits were introduced into the woods at time x 0. Find the value of a under this condition. Show that the graph agrees with Figure 2-7e. How does the sign of a represent a generalization of the de�nition of logistic function?

d. How do you interpret the mathematical model under the condition of part c? What seems to be the implication of trying to stock a region with a greater number of a particular species than the region can support?

7. Given the logistic function

f (x) c _________ 1 a e x

a. a graphs of f for c 1, 2, and 3. Use as a domain 10 x 10. Sketch the results. True or false? “�e constant c is a vertical dilation factor.”

b. Figure 2-7f shows the graph of f with c 2 and with a 0.2, 1, and 5. Which graph is which? How does the value of a transform the graph?

2

10

f(x)

x5510

Figure 2-7f

c. g (x) c ___________ 1 a e x 3) .

What transformation applied to f does this represent? Con�rm that your answer is correct by plotting f and g on the same screen using c 2 and a 1.

d. What value of a in the equation of f (x) would produce the same transformation as in part c?


4. Spreading the News Simulation Experiment: In this experiment you will simulate the spread of

and then selects two people at random to “tell” the news to. Do this by selecting two random integers between 1 and the number of students in your class, inclusive. (It is not actually necessary to tell any news!) �e random number generator on one student’s calculator will help make the random selection. �e two people with the chosen numbers stand. �us, a�er the �rst iteration, there will probably be three students standing (unless a duplicate random number

two more people to “tell” the news to by selecting a total of six (or four) more random integers. Do this for a total of ten iterations or until the entire class is standing. At each iteration, record the number of iterations and the total number of people who have heard the news. Describe the results of the experiment. Include things such as

�e plot of the data points. A function that �ts the data, and a graph of

chose the function you did. A statement of how well the logistic model �ts the data. �e iteration number at which the good news was spreading most rapidly.

5. Ebola Outbreak Epidemic Problem: In the

out in the Gulu district of Uganda. �e table shows the total number of people infected from

infections. �e �nal number of people who were infected during this epidemic wasa virus that causes internal bleeding and is fatal in most cases.)

x (days) y (total infections)

1 71

10 182

15 239

21 281

30 321

50 370

A Red Cross medical o�cer instructs villagers about the Ebola

virus in Kabede Opong, Uganda.

a. �ts the data. Is the graph of this function concave up or concave down?

b. Use the second and last data points to �nd the particular equation of a logistic function that �ts the data.

c. the logistic function from part b. Sketch the results.

d. Where does the point of in�ection occur in the logistic model? What is the real-world meaning of this point?

e. Based on the logistic model, how many people

f. Find data about other epidemics. Give your source. Try to model the spread of the epidemic for which you found data.

117

7b. Changing a seems to translate the graph horizontally.

5

2

x

f(x)

a � 0.2

a � 1 a � 5

�5 7c. Horizontal translation by 3

5

2

x

y

f(x)

g(x)

�5 7d. a 5 e 1.2 5 3.3201....


1. A logistic function contains the points (21, 6), (0, 3), and (1, 1.2). Determine an equation for the function. What are the coordinates of the infl ection point of the function?

2. What are the coordinates of the infl ection point of a logistic function in the form y 5 c __________ (1 1 a b x ) ?

3. Two common forms of logistic-function equations are y 5 c __________ (1 1 a b x ) and y 5 c ________ 1 1 b (x2d) .

What is the graphical eff ect of each parameter: a, b, c, and d? Which two parameters control horizontal translations? Under what algebraic conditions are these two equations equivalent? How does this confi rm the identifi cation of the parameters controlling horizontal translations?Problem 7 deals with dilations and

translations of the parent logistic function and reviews material from Chapter 1.

7a. True.

5

2

x

f(x)

c � 1

c � 2

c � 3

�5


See page 983 for answers to CAS Problems 1–3.

119Section 2-8: Chapter Review and Test

R2. a. Find the particular equation of a linear function containing the points (7, 9) and (10, 11). Give an example in the real world that this function could model.

b. Sketch two graphs showing a decreasing exponential function and an inverse variation power function. Give two ways in which the graphs are alike. Give one way in which they are di� erent.

c. How do you tell that the function graphed in Figure 2-8a is an exponential function, not a power function? Find the particular equation of the exponential function. Give an example in the real world that this exponential function could model.

y

x

(5, 16)

(2, 6)2

20

10

5 Figure 2-8a

d. Find the particular equation of the quadratic function graphed in Figure 2-8b. How does the equation you � nd show that the graph is concave down? Give an example in the real world that this function could model.

y

x

(2, 15.2)

(4, 18.8)(6, 12.8)

2 4 6 8

20

10

Figure 2-8b

e. A quadratic function has the equation y 3 2(x 5 ) 2 . Where is the vertex of the graph? What is the y-intercept?

R3. For each table of values, tell from the pattern whether the function that � ts the points is linear, quadratic, exponential, or power.

a. x f (x)

3 6 12 9 6

12 3

b. x g (x)

3 6 12 9 8

12 6

c. x h (x)

3 6 30 9 36

12

d. x q (x)

3 6 12 9 18

12

e. Suppose that f (3) 90 and f (6) 120. Find f (12) if the function is i. An exponential function ii. A power function iii. A linear function

f. of exponential functions is true for the function f (x) 53 1. 3 x by showing algebraically that adding the constant c to x multiplies the corresponding f (x)-value by a constant.

R4. a. � e most important thing to remember about logarithms is that a logarithm is ? .

b. Write in logarithmic form: z 1 0 p c. What does it mean to say that

log 30 d. Give numerical examples to illustrate these

logarithmic properties: i. log(xy) log x log y ii. log x __ y log x log y iii. log x y y log x

e. log 5 log ?


Chapter Review and TestIn this chapter you have learned graphical and numerical patterns for various types of functions:

� ese patterns allow you to tell which type of function might � t a given real-world situation. Once you have selected a function that has appropriate

the particular equation by calculating values of the constants. You can check your work by seeing whether the function � ts other given points. Once you have the correct equation, you can use it to interpolate between given values or extrapolate beyond given values to calculate y when you know x, or to calculate x when you know y.

Chapter Review and TestIn this chapter you have learned graphical and numerical patterns for various

2- 8

R0. Update your journal with what you have learned in this chapter. Include things such as the de� nitions, properties, and graphs of the functions listed. Show typical graphs of the various functions, give their domains, and make connections between, for example,

the logarithmic functions. Show how you can use logarithms and their properties to solve for unknowns in exponential or logarithmic equations, and explain how these equations arise in � nding the constants in the particular equation of certain functions. Tell what you have learned about the constant e and where it is used.

R1. � is problem concerns these � ve function values:

x f (x)

2 1.2

6 10.88 19.2

10 30.0

a. On the same screen, plot the data points and the graph of f (x) 0.3 x 2 .

b. Is the function increasing or decreasing? Is the graph concave up or concave down?

c. Name the function in part a. Give an example in the real world that this function might model. Is the y-intercept of f reasonable for this real-world example?

R0. Update your journal with what you have R1. � is problem concerns these � ve function

Review Problems



Class Time 2 days (including 1 day for testing)

Homework AssignmentDay 1: R0–R7, T1–T28Day 2: Problem Set 3-1

(aft er Chapter 2 Test)

Teaching ResourcesExploration 2-8a:

Rehearsal for Chapter 2 TestBlackline Master

Problem C3Test 6, Chapter 2, Forms A and B

TE ACH I N G

Section Notes

Section 2-8 contains a set of review problems, a set of concept problems, and a chapter test. Th e review problems include one problem for each section in the chapter. You may wish to use the chapter test as an additional set of review problems.

Encourage students to practice the no-calculator problems without a calculator so that they are prepared for the test problems for which they cannot use a calculator.

Diff erentiating Instruction• Go over the review problems in class,

perhaps by having students present their solutions. You might assign students to write up their solutions before class starts.

• Have students write Problem C1 in their journals. Consider allowing ELL students to skip Problem C2.

• ELL students may need help with the language in Problem C3.

• Clarify the concept of slope fi eld in Problem C3 for ELL students.

• Because many cultures’ norms highly value helping peers, ELL students oft en help each other on tests. You can limit this tendency by making multiple versions of the test.

• Consider giving a group test the day before the individual test, so that students can learn from each other as they review, and they can identify what they don’t know prior to the individual test. Give a copy of the test to each group member,

have them work together, and then randomly choose one paper from the group to grade. Grade the test on the spot, so students know what they need to review further. Make this test worth 1 _ 3 the value of the individual test, or less.

• ELL students may need more time to take the test.

• ELL students will benefi t from having access to their bilingual dictionaries while taking the test.


R2. a. Find the particular equation of a linear function containing the points (7, 9) and (10, 11). Give an example in the real world that this function could model.

b. Sketch two graphs showing a decreasing exponential function and an inverse variation power function. Give two ways in which the graphs are alike. Give one way in which they are di� erent.

c. How do you tell that the function graphed in Figure 2-8a is an exponential function, not a power function? Find the particular equation of the exponential function. Give an example in the real world that this exponential function could model.

y

x

(5, 16)

(2, 6)2

20

10

5 Figure 2-8a

d. Find the particular equation of the quadratic function graphed in Figure 2-8b. How does the equation you � nd show that the graph is concave down? Give an example in the real world that this function could model.

y

x

(2, 15.2)

(4, 18.8)(6, 12.8)

2 4 6 8

20

10

Figure 2-8b

e. A quadratic function has the equation y 3 2(x 5 ) 2 . Where is the vertex of the graph? What is the y-intercept?

R3. For each table of values, tell from the pattern whether the function that � ts the points is linear, quadratic, exponential, or power.

a. x f (x)

3 6 12 9 6

12 3

b. x g (x)

3 6 12 9 8

12 6

c. x h (x)

3 6 30 9 36

12

d. x q (x)

3 6 12 9 18

12

e. Suppose that f (3) 90 and f (6) 120. Find f (12) if the function is i. An exponential function ii. A power function iii. A linear function

f. of exponential functions is true for the function f (x) 53 1. 3 x by showing algebraically that adding the constant c to x multiplies the corresponding f (x)-value by a constant.

R4. a. � e most important thing to remember about logarithms is that a logarithm is ? .

b. Write in logarithmic form: z 1 0 p c. What does it mean to say that

log 30 d. Give numerical examples to illustrate these

logarithmic properties: i. log(xy) log x log y ii. log x __ y log x log y iii. log x y y log x

e. log 5 log ?


Chapter Review and TestIn this chapter you have learned graphical and numerical patterns for various types of functions:

� ese patterns allow you to tell which type of function might � t a given real-world situation. Once you have selected a function that has appropriate

the particular equation by calculating values of the constants. You can check your work by seeing whether the function � ts other given points. Once you have the correct equation, you can use it to interpolate between given values or extrapolate beyond given values to calculate y when you know x, or to calculate x when you know y.

Chapter Review and TestIn this chapter you have learned graphical and numerical patterns for various

2- 8

R0. Update your journal with what you have learned in this chapter. Include things such as the de� nitions, properties, and graphs of the functions listed. Show typical graphs of the various functions, give their domains, and make connections between, for example,

the logarithmic functions. Show how you can use logarithms and their properties to solve for unknowns in exponential or logarithmic equations, and explain how these equations arise in � nding the constants in the particular equation of certain functions. Tell what you have learned about the constant e and where it is used.

R1. � is problem concerns these � ve function values:

x f (x)

2 1.2

6 10.88 19.2

10 30.0

a. On the same screen, plot the data points and the graph of f (x) 0.3 x 2 .

b. Is the function increasing or decreasing? Is the graph concave up or concave down?

c. Name the function in part a. Give an example in the real world that this function might model. Is the y-intercept of f reasonable for this real-world example?

R0. Update your journal with what you have R1. � is problem concerns these � ve function

Review Problems

119

PRO B LE M N OTES

R1c. Quadratic power function. Real-world interpretations may vary.

Problems R2a and R2c could be solved using systems on a CAS.R2a. y 5 2 _ 3 x 1 13 __ 3 . Real-world interpretations may vary.R2b.

y

x

y

x

Both are decreasing and have the x-axis as an asymptote. Th e exponential function crosses the y-axis, whereas the inverse function has no y-intercept (and has the y-axis as an asymptote). See the graphs above.R2c. Th e y-intercept is nonzero. y 5 (3.1201…)(1.3867… ) x . Real-world interpretations may vary.R2d. y 5 1.2x 2 1 9x 1 2; the coeffi cient of x 2 is negative, which indicates the graph is concave down. Real-world interpretations may vary.R2e. Vertex (5, 3); y-intercept: 53 R3a. ExponentialR3b. Power (inverse variation)R3c. Linear R3d. QuadraticR3e. i. f (12) 5 213 1 _ 3 R3e. ii. f (12) 5 160 R3e. iii. f (12) 5 180 R3f. f (x 1 c) 5 53 1.3 x1c 5 53 1.3 x 1.3 c 5 1.3 c f (x)

Problems R4e and R5c–R5e can be solved directly on a CAS without any knowledge of the underlying logarithm laws.

Exploration Notes

Exploration 2-8a may be used as an in-class “rehearsal” for the chapter test, or as a review assignment for homework.

R0. Journal entries will vary.R1a.

R1b. Increasing for x . 0, decreasing for x , 0, concave up

2 4 6 8 10

10

20

30f(x)

x

Section 2-8: Chapter Review and Test

See page 984 for answers to Problem R4.


C1. Rise and Run Property of Quadratic Functions Problem: � e sum of consecutive odd counting numbers is always a perfect square. For instance,

1 1 2

1 3 2 2

1 3 5 9 3 2

1 3 5 7 16 2

� is fact can be used to sketch the graph of a quadratic function by a “rise-run” technique similar to that used for linear functions. Figure 2-8c shows that for y x 2 , you can start at the vertex and use the pattern “over 1, up 1; over 1, up 3; over 1, up 5; . . . .”

y

x+1

+1+1

+1

+3

+5

Figure 2-8c

a. On graph paper, plot the graph of y x 2 by using this rise-run technique. Use integer values of x pattern for values of x from 0 to

b. � e graph of y 5 (x 2 ) 2 is a translation of the graph of y x 2 the vertex, and then plot the graph on graph paper using the rise-run pattern.

c. � e graph of y 5 0.3(x 2) 2 is a vertical dilation of the graph in part b. Use the rise-run technique for this function, and then plot its graph on the same axes as in part b.

C2. Log-log and Semilog Graph Paper Problem: f (x) 1000 0 0.6 5 x be the number of

bacteria remaining in a culture over time x, in g (x) 0.09 x 2 be the area of skin, in

square centimeters, on a snake of length x, in centimeters. Figure 2-8d shows the graph of the exponential function f plotted on semilog graph paper. Figure 2-8e on the next page shows on the graph of the power function g plot ted on log-log graph paper. On these graphs, one or both axes have scales proportional to the logarithm of the variable’s value. � us the scales are compressed so that a wide range of values can � t on the same sheet of graph paper. For these two functions, the graphs are straight lines.

f(x) = 1000 0.65x

x

1000

100

10

10 5 10 15

C1. Rise and Run Property of Quadratic Functions C2. Log-log and Semilog Graph Paper Problem:

Concept Problems


R5. a. Write in exponential form: p lo g c m b. Find lo g 7 30. c. ln 7 2 ln 3 ln ? d. Solve the equation:

log(x 1) log(x 2) 1 e. Solve the equation: 3 2x 1 7 x

R6. a. On the same screen, plot the graphs of f 1 (x) ln x and f 2 (x) e x . Use the same scale on both axes. Sketch the results. How are the two graphs related to each other and to the line y x?

b. For the natural exponential function f (x) 5 e x , write the equation in the form f (x) ab x . For the exponential function g (x) x , write the equation as a natural exponential function.

Sunlight Under the Water Problem (R6c–R6e): �e intensity of sunlight underwater decreases with depth. �e table shows the depth, y, in feet, below the surface of the ocean you must go to reduce the intensity of light to the given percentage, x, of what it is at the surface.

x (%) Depth y (ft)

100 0

50 13

25 26

12.5 39

c. What numerical pattern tells you that a logarithmic function �ts the data? Find the particular equation of the function.

d. On the same screen, plot the data and the logarithmic function. Sketch the result.

e. Based on this mathematical model, how deep do you have to go for the light to be reduced to 1% of its intensity at the surface? Do you �nd this by interpolation or by extrapolation?

R7. a.same screen and sketch the results.

f (x) 10 2 x _______ 2 x 10

g (x) 2 x

b. f (x) is very close to g (x) when xwhen x is a large positive number, f (x) is close to 10 and g (x) is very large.

c. Transform the equation of f (x) in part a so that it has only one exponential term.

d. Transform the equation of g (x) in part a so that it is expressed in the form g (x) e kx .

e. Population Problem: A small community is built on an island in the Gulf of Mexico. �e population grows steadily, as shown in the table.

x (months) y (people)

6 7512 15318 260

355

be a reasonable mathematical model for population as a function of time. If the

�nd the particular equation of the logistic function that contains the points for 6 mo

approximately the correct solutions for

sketch the result. When is the population predicted to reach 95% of the capacity?


Problem Notes (continued)R5a. c p 5 mR5b. lo g 7 30 5 1.7478…R5c. 63R5d. x 5 4 , the equation is undefi ned if x 5 23.R5e. x 5 1 _______

2 2 log 7 ____ log 3 5 4.3714…

R6a. f 1 (x) and f 2 (x) are refl ections of each other across the line y 5 x .

x

y

f1

5

5(x)

f2(x)

R6b. f(x) 5 5 0.6703 … x ;g (x) 5 4.3 e 2.0014…x R6c. Multiply–add property; y 5 213 lo g 2

x ___ 100 R6d.

R6e. 86.3701… ft deep (by extrapolation).R7a.

R7b. When x is a large negative number, the denominator of f(x) is essentially equal to 10, so for large negative x, f(x) 5 10 2 x _____ 2 x 1 10 10 2 x _____ 10 5 2 x 5 g (x). But for large positive x, the 10 in the denominator of f(x) is negligible compared with the 2 x ; so f(x) 5 10 2 x _____ 2 x 1 10 10 2 x _____ 2 x 5 10.

25 50 75 100

20

40

60

x

y

x

y

f(x)

g(x)

5 10 15 20�5

5

10

15

Problem R7c appears to be surprisingly immune to the TI-Nspire CAS. Th e desired output formatting of the CAS prevents the user from getting a clean result when dividing the numerator and denominator by 2 x . You could apply the Expand o r Propfrac commands to the logistic function, but this approach requires some quite sophisticated transformations to obtain the desired results. Alternatively,

you could deal with the numerator and denominator individually, but the eff ort involved would be far greater than simply solving the problem without a CAS.R7c. f(x) 5 10 __________ 1 1 10 2 2x R7d. g (x) 5 e (ln 2)x


C1. Rise and Run Property of Quadratic Functions Problem: � e sum of consecutive odd counting numbers is always a perfect square. For instance,

1 1 2

1 3 2 2

1 3 5 9 3 2

1 3 5 7 16 2

� is fact can be used to sketch the graph of a quadratic function by a “rise-run” technique similar to that used for linear functions. Figure 2-8c shows that for y x 2 , you can start at the vertex and use the pattern “over 1, up 1; over 1, up 3; over 1, up 5; . . . .”

y

x+1

+1+1

+1

+3

+5

Figure 2-8c

a. On graph paper, plot the graph of y x 2 by using this rise-run technique. Use integer values of x pattern for values of x from 0 to

b. � e graph of y 5 (x 2 ) 2 is a translation of the graph of y x 2 the vertex, and then plot the graph on graph paper using the rise-run pattern.

c. � e graph of y 5 0.3(x 2) 2 is a vertical dilation of the graph in part b. Use the rise-run technique for this function, and then plot its graph on the same axes as in part b.

C2. Log-log and Semilog Graph Paper Problem: f (x) 1000 0 0.6 5 x be the number of

bacteria remaining in a culture over time x, in g (x) 0.09 x 2 be the area of skin, in

square centimeters, on a snake of length x, in centimeters. Figure 2-8d shows the graph of the exponential function f plotted on semilog graph paper. Figure 2-8e on the next page shows on the graph of the power function g plot ted on log-log graph paper. On these graphs, one or both axes have scales proportional to the logarithm of the variable’s value. � us the scales are compressed so that a wide range of values can � t on the same sheet of graph paper. For these two functions, the graphs are straight lines.

f(x) = 1000 0.65x

x

1000

100

10

10 5 10 15

C1. Rise and Run Property of Quadratic Functions C2. Log-log and Semilog Graph Paper Problem:

Concept Problems


R5. a. Write in exponential form: p lo g c m b. Find lo g 7 30. c. ln 7 2 ln 3 ln ? d. Solve the equation:

log(x 1) log(x 2) 1 e. Solve the equation: 3 2x 1 7 x

R6. a. On the same screen, plot the graphs of f 1 (x) ln x and f 2 (x) e x . Use the same scale on both axes. Sketch the results. How are the two graphs related to each other and to the line y x?

b. For the natural exponential function f (x) 5 e x , write the equation in the form f (x) ab x . For the exponential function g (x) x , write the equation as a natural exponential function.

Sunlight Under the Water Problem (R6c–R6e): �e intensity of sunlight underwater decreases with depth. �e table shows the depth, y, in feet, below the surface of the ocean you must go to reduce the intensity of light to the given percentage, x, of what it is at the surface.

x (%) Depth y (ft)

100 0

50 13

25 26

12.5 39

c. What numerical pattern tells you that a logarithmic function �ts the data? Find the particular equation of the function.

d. On the same screen, plot the data and the logarithmic function. Sketch the result.

e. Based on this mathematical model, how deep do you have to go for the light to be reduced to 1% of its intensity at the surface? Do you �nd this by interpolation or by extrapolation?

R7. a.same screen and sketch the results.

f (x) 10 2 x _______ 2 x 10

g (x) 2 x

b. f (x) is very close to g (x) when xwhen x is a large positive number, f (x) is close to 10 and g (x) is very large.

c. Transform the equation of f (x) in part a so that it has only one exponential term.

d. Transform the equation of g (x) in part a so that it is expressed in the form g (x) e kx .

e. Population Problem: A small community is built on an island in the Gulf of Mexico. �e population grows steadily, as shown in the table.

x (months) y (people)

6 7512 15318 260

355

be a reasonable mathematical model for population as a function of time. If the

�nd the particular equation of the logistic function that contains the points for 6 mo

approximately the correct solutions for

sketch the result. When is the population predicted to reach 95% of the capacity?

121

C1a.

C1b. Vertex at (2, 25)

x

�7

�1

�5

�1

�3

�1

�1�1

�1�1

�1

�3�1

�5

�1

�7

y

4

10

�7

�1

�5

�1

�3

�1

�1�1

�1�1

�1

�3

�1

�5

�1

�7

x

y

�4

R7e. The size of the population would be limited by the capacity of the island. f(x) 5 460 ________________________ 1 1 (13.2906…)(1.1718… ) 2x

f(12) 5 154.2335… f(18) 5 260.5072… 20 40

200

400

x

y

34.8878… months


See page 984 for answers to Problem C1c.


d. Describe the behavior of the tree population for each of the three initial conditions in part a. In particular, explain what happens if too few trees are planted and also what happens if too many trees are planted.

e. Without doing any more computations, sketch on the slope � eld the graph of the tree population if, at time x 0, i. 500 trees had been planted. ii. 1500 trees had been planted. iii. 200 trees had been planted.

f. How does the slope field allow you to analyze graphically the behavior of many related logistic functions without doing any computations?

y

x50

0 10

1500

1000

500

Figure 2-8f

Part 1: No calculators allowed (T1–T9) T1. Write the general equation of

a. A linear function b. A quadratic function c. A power function d. An exponential function e. A logarithmic function f. A logistic function

T2. What type of function could have the graph shown? a. y

x

b. y

x

c. y

x

d. y

x

e. y

x

f. y

x

T3. What numerical pattern do regularly spaced data have for a. A linear function b. A quadratic function c. An untranslated power function d. An untranslated exponential function e. An untranslated logarithmic function

Part 1: No calculators allowed (T1–T9) c. y d. y

Chapter Test


a. Read the values of f (9) and g (60) from the graphs. �en calculate these numbers algebraically using the given equations. If your graphical answers are di�erent from your calculated answers, explain what mistakes you made in reading the graphs.

b. You’ll need a sheet of semilog graph paper and a sheet of log-log graph paper for graphing. On the semilog paper, plot the function h(x) 2 1. 5 x using several values of x in the domain [0, 15]. On the log-log paper, plot the function p(x) 700 x 1.3 using several values of x in the domain [1, 100]. What do the graphs of the functions look like?

c. Take the logarithm of both sides of the equation f (x) 10000 0.65 x . Use the properties of logarithms to show that log f (x) is a linear function of xhow this fact is connected to the shape of the graph.

d. Take the logarithm of both sides of the equation g (x) 0.09 x 2 . Use the properties of logarithms to show that log g (x) is a linear function of log x. How does this fact relate to the graph in Figure 2-8e?

C3. Slope Field Logistic Function Problem: �e logistic functions you have studied in this chapter model populations that start at a relatively low value and then rise asymptotically to a maximum sustainable population. �ere may also be a minimum sustainable population. Suppose that a new variety of tree is planted on a relatively small island. Research indicates that the minimum sustainable population is 300 trees and that the maximum sustainable population is 1000 trees. A logistic function modeling this situation is

y 300C 1000 e 0.7x ______________ C e 0.7x

where y is the number of trees alive at time x, in decades a�er the trees were planted. �e coe�cients 300 and 1000 are the minimum and maximum sustainable populations, respectively, and C is a constant determined by the initial condition, the number of trees planted at time x 0. a. Determine the value of C and write the

particular equation if, at time x 0, i.

ii. 1300 trees are planted. iii. 299 trees are planted.

b. a window with 0 x 10 and suitable y-values. What are the major di�erences among the three graphs?

c. Figure 2-8f shows a slope �eld representing functions with the given equation. �e line segment through each grid point indicates the slope the graph would have if it passed through that point. On a copy of Figure 2-8f, plot the three equations from part a. How are the graphs related to the line segments on the slope �eld?

g(x) = 0.09x2

x

1000

100

10

11 10 100

Figure 2-8e


Problem Notes (continued)

Problems C2 and C3 are good problems for a research project or for extra credit. A blackline master for Problem C3 is available in the Instructor’s Resource Book.C2a. f(9) 5 20.7119…; g (60) 5 324C2b. The graphs look linear.C2c. log f(x) 5 log 1000 1 x log 0.65; y-intercept is log 1000; slope is log 0.65. The graph is linear.C2d. log g (x) 5 log 0.09 1 2 log x; y-intercept is log 0.09; slope is 2

C3a. i. y 5 1800 1 1000 e 0.7x ______________ 6 1 e 0.7x

C3a. ii. y 5 290 1 1000 e 0.7x _____________ 20.3 1 e 0.7x

C3a. iii. y 5 2210,300 1 1000 e 0.7x __________________ 2701 1 e 0.7x

C3b.

C3c. The graphs follow the direction of the line segments.

x

y

5

400

1000

1300

x

y

c � 6

c � �0.3

c � �701

C3d. If 400 trees are planted, the population increases at first and then levels off at 1000. If 1300 (too many) trees are planted, the population decreases to level off at 1000. If 299 (too few) trees are planted, the population dwindles until all trees are dead.


d. Describe the behavior of the tree population for each of the three initial conditions in part a. In particular, explain what happens if too few trees are planted and also what happens if too many trees are planted.

e. Without doing any more computations, sketch on the slope � eld the graph of the tree population if, at time x 0, i. 500 trees had been planted. ii. 1500 trees had been planted. iii. 200 trees had been planted.

f. How does the slope field allow you to analyze graphically the behavior of many related logistic functions without doing any computations?

y

x50

0 10

1500

1000

500

Figure 2-8f

Part 1: No calculators allowed (T1–T9) T1. Write the general equation of

a. A linear function b. A quadratic function c. A power function d. An exponential function e. A logarithmic function f. A logistic function

T2. What type of function could have the graph shown? a. y

x

b. y

x

c. y

x

d. y

x

e. y

x

f. y

x

T3. What numerical pattern do regularly spaced data have for a. A linear function b. A quadratic function c. An untranslated power function d. An untranslated exponential function e. An untranslated logarithmic function

Part 1: No calculators allowed (T1–T9) c. y d. y

Chapter Test


a. Read the values of f (9) and g (60) from the graphs. �en calculate these numbers algebraically using the given equations. If your graphical answers are di�erent from your calculated answers, explain what mistakes you made in reading the graphs.

b. You’ll need a sheet of semilog graph paper and a sheet of log-log graph paper for graphing. On the semilog paper, plot the function h(x) 2 1. 5 x using several values of x in the domain [0, 15]. On the log-log paper, plot the function p(x) 700 x 1.3 using several values of x in the domain [1, 100]. What do the graphs of the functions look like?

c. Take the logarithm of both sides of the equation f (x) 10000 0.65 x . Use the properties of logarithms to show that log f (x) is a linear function of xhow this fact is connected to the shape of the graph.

d. Take the logarithm of both sides of the equation g (x) 0.09 x 2 . Use the properties of logarithms to show that log g (x) is a linear function of log x. How does this fact relate to the graph in Figure 2-8e?

C3. Slope Field Logistic Function Problem: �e logistic functions you have studied in this chapter model populations that start at a relatively low value and then rise asymptotically to a maximum sustainable population. �ere may also be a minimum sustainable population. Suppose that a new variety of tree is planted on a relatively small island. Research indicates that the minimum sustainable population is 300 trees and that the maximum sustainable population is 1000 trees. A logistic function modeling this situation is

y 300C 1000 e 0.7x ______________ C e 0.7x

where y is the number of trees alive at time x, in decades a�er the trees were planted. �e coe�cients 300 and 1000 are the minimum and maximum sustainable populations, respectively, and C is a constant determined by the initial condition, the number of trees planted at time x 0. a. Determine the value of C and write the

particular equation if, at time x 0, i.

ii. 1300 trees are planted. iii. 299 trees are planted.

b. a window with 0 x 10 and suitable y-values. What are the major di�erences among the three graphs?

c. Figure 2-8f shows a slope �eld representing functions with the given equation. �e line segment through each grid point indicates the slope the graph would have if it passed through that point. On a copy of Figure 2-8f, plot the three equations from part a. How are the graphs related to the line segments on the slope �eld?

g(x) = 0.09x2

x

1000

100

10

11 10 100

Figure 2-8e

123

C3e.

C3f. You can draw the graph following the direction of the line segments to get an idea of what happens at different initial conditions.T1a. y 5 ax 1 bT1b. y 5 ax 2 1 bx 1 c, a 0T1c. y 5 ax b , a 0T1d. y 5 ae bx or y 5 ab x , a, b 0, b > 0 and b 1 in the case of y 5 ab x T1e. y 5 a 1 b log c x, b 0 and c . 0, c 1T1f. y 5 c ________ 1 1 ae 2bx or y 5 c ________ 1 1 ab 2x , a, b, c 0, b > 0 and b 1 in the case of y 5 c ________ 1 1 ab 2x

T2a. LogarithmicT2b. ExponentialT2c. LogisticT2d. QuadraticT2e. PowerT2f. LinearT3a. Add–addT3b. Constant-second differencesT3c. Multiply–multiplyT3d. Add–multiplyT3e. Multiply–add

x

y



Model Rocket Problem: A precalculus class launches a model rocket out on the football �eld. �e rocket

measures the rocket’s height, �nding the values in

t (s) h (ft)

2 1663 216

5 2206

T20. to the data. Is the graph of this function concave up or concave down? What kind of function would be a reasonable mathematical model for this function?

T21. Show numerically that a quadratic function would �t the data by showing that the second di�erences in the height data are constant.

T22. Use any three of the data points to �nd the particular equation of the quadratic function that �ts the points. Show that the equation gives the correct values for the other two points.

T23. Logarithmic Function Problem: A logarithmic function f has f (2) f (6)

functions to �nd two more values of f (x). Use the given points to �nd the particular equation in the form f (x) a b ln x.

Population Problem: a new subdivision that opens in a small town. �e population of the subdivision increases as new families move in. �e table lists the population of the subdivision various numbers of months a�er its opening.

Months People

2 36357 579

11 830

T24. Find the particular equation of the (untranslated) exponential function f that �ts the �rst and last data points. Show that the values of f (5) and f (7) are fairly close to those in the table.

T25. Show that the logistic function g gives values for the population that are also fairly close to the values in the table.

g (x) 3500 ____________ 1 10.8 e 0.11x

T26. On the same screen, plot the four given points, the graph of f, and the graph of g. Use a window with 0 x 70 and 0 y 5000. Sketch the result.

T27. g gives more reasonable values for the population than the exponential function f when you extrapolate to large numbers of months.

T28. What did you learn from taking this test that you did not know before?


T4. Write the equation log a b c in exponential form.

T5. Show how to use the logarithm of a power property to simplify log 5 x .

T6. ln 80 ln 2 ln 20 ln ?

T7. log 5 2 log 3 log ? T8. Solve the equation: x 3 2 x 0 T9. Solve the equation:

lo g 2 (x l og 2 (x 3) 8

Part 2: Graphing calculators allowed (T10–T28)

Shark Problem: Suppose that from great white sharks caught in the past, �shermen �nd these weights and

x (ft) f (x) (lb)

5 7510 60015 2025

20

T10. Show that the data set in the table has the

T11. Write the general equation of a power function. �en use the points (5, 75) and (10, 600) to calculate algebraically the two constants in the equation. Store these values without rounding. Write the particular equation.

T12. correct by showing that it gives the other two data points in the table.

T13. From fossilized shark teeth, naturalists think there were once great white sharks 100 ft long. Based on your mathematical model, how heavy would such a shark be? Is this surprising?

T14. A newspaper report describes a great white shark that weighed 3000 lb. Based on your mathematical model, about how long was the shark? Show the method you use.

Co�ee Cup Problem: You pour a cup of co�ee. �ree °F

above room temperature. You record its temperature every 2 minutes therea�er, creating this table of

x (ft)g (x)

(°F above room temperature)

35 76.87 62.29

11

T15. whether the graph of the function you can �t to the points is concave up or concave down.

reasonable for this function but a linear or a power function would not.

T16. Find the particular equation of the exponential function that �ts the points at x 3 and x 11. Show that the equation gives approximately the correct values for the other three times.

T17. to estimate the temperature of the co�ee when it was poured.

T18. Use your equation to predict the temperature of the co�ee a half-hour a�er it was poured.

T19. �e Add–Multiply Property Proof Problem: y 7(1 3 x ), then log y is a linear

function of x.


Problem Notes (continued)T4. a c 5 bT5. log 5 x 5 x log 5T6. 8T7. 45 T8. x 5 2T9. No solutions.

T10. f(10)

____ f(5) 5 600 ___ 75 5 8 5 4800 ____ 600 5 f(20)

____ f(10)

T11. f(x) 5 ax b ; f(x) 5 0.6 x 3 T12. 0.6(1 5) 3 5 2025, 0.6(2 0) 3 5 4800; the function is correct. T13. f(100) 5 600,000 lb 5 300 tonsT14. 17.0997… ft T15.

Graph will be concave up. Th e function appears to start at a positive number, decrease rapidly, and then level off as x grows large. A linear function cannot work, because the graph appears to be concave. An inverse variation power function cannot work, because it appears that the graph will intersect the vertical axis.

Problems T16–T18 work particularly well on a CAS. Th e point of these problems is not the manipulation, so students could use a system solver for Problem T16, use a Solve command for Problem T17, and evaluate the function directly for Problem T18.T16. f(x) 5 (130.0510…)(0.8999… ) x ;f(5) 5 76.7840…Ff(7) 5 62.1919…Ff(9) 5 50.3729…FT17. 130.0510… F above room temperature.T18. 5.5088… F above room temperature.

10 20 30

50

100

x

g (x)

A quick solution for Problem T19 is shown at the right for students who know how to read the CAS output. In this case, knowing how to ask the question and interpret the answer has signifi cant value.

T19. log y 5 log 7 1 (log 13) x


Model Rocket Problem: A precalculus class launches a model rocket out on the football �eld. �e rocket

measures the rocket’s height, �nding the values in

t (s) h (ft)

2 1663 216

5 2206

T20. to the data. Is the graph of this function concave up or concave down? What kind of function would be a reasonable mathematical model for this function?

T21. Show numerically that a quadratic function would �t the data by showing that the second di�erences in the height data are constant.

T22. Use any three of the data points to �nd the particular equation of the quadratic function that �ts the points. Show that the equation gives the correct values for the other two points.

T23. Logarithmic Function Problem: A logarithmic function f has f (2) f (6)

functions to �nd two more values of f (x). Use the given points to �nd the particular equation in the form f (x) a b ln x.

Population Problem: a new subdivision that opens in a small town. �e population of the subdivision increases as new families move in. �e table lists the population of the subdivision various numbers of months a�er its opening.

Months People

2 36357 579

11 830

T24. Find the particular equation of the (untranslated) exponential function f that �ts the �rst and last data points. Show that the values of f (5) and f (7) are fairly close to those in the table.

T25. Show that the logistic function g gives values for the population that are also fairly close to the values in the table.

g (x) 3500 ____________ 1 10.8 e 0.11x

T26. On the same screen, plot the four given points, the graph of f, and the graph of g. Use a window with 0 x 70 and 0 y 5000. Sketch the result.

T27. g gives more reasonable values for the population than the exponential function f when you extrapolate to large numbers of months.

T28. What did you learn from taking this test that you did not know before?


T4. Write the equation log a b c in exponential form.

T5. Show how to use the logarithm of a power property to simplify log 5 x .

T6. ln 80 ln 2 ln 20 ln ?

T7. log 5 2 log 3 log ? T8. Solve the equation: x 3 2 x 0 T9. Solve the equation:

lo g 2 (x l og 2 (x 3) 8

Part 2: Graphing calculators allowed (T10–T28)

Shark Problem: Suppose that from great white sharks caught in the past, �shermen �nd these weights and

x (ft) f (x) (lb)

5 7510 60015 2025

20

T10. Show that the data set in the table has the

T11. Write the general equation of a power function. �en use the points (5, 75) and (10, 600) to calculate algebraically the two constants in the equation. Store these values without rounding. Write the particular equation.

T12. correct by showing that it gives the other two data points in the table.

T13. From fossilized shark teeth, naturalists think there were once great white sharks 100 ft long. Based on your mathematical model, how heavy would such a shark be? Is this surprising?

T14. A newspaper report describes a great white shark that weighed 3000 lb. Based on your mathematical model, about how long was the shark? Show the method you use.

Co�ee Cup Problem: You pour a cup of co�ee. �ree °F

above room temperature. You record its temperature every 2 minutes therea�er, creating this table of

x (ft)g (x)

(°F above room temperature)

35 76.87 62.29

11

T15. whether the graph of the function you can �t to the points is concave up or concave down.

reasonable for this function but a linear or a power function would not.

T16. Find the particular equation of the exponential function that �ts the points at x 3 and x 11. Show that the equation gives approximately the correct values for the other three times.

T17. to estimate the temperature of the co�ee when it was poured.

T18. Use your equation to predict the temperature of the co�ee a half-hour a�er it was poured.

T19. �e Add–Multiply Property Proof Problem: y 7(1 3 x ), then log y is a linear

function of x.

125

T20. Th e graph will be concave down. A quadratic function might fi t the data.

1 2 3 4 5 6 7�1

100

200

t

h

T21. Th e second diff erences are all 232.

Problems T22–T24 can be solved immediately using systems on a CAS. T22. h(t) 5 216 t 2 1 130t 2 30h(5) 5 216( 5 2 ) 1 130(5) 2 30 5 220, which agrees.h(6) 5 216( 6 2 ) 1 130(6) 2 30 5 174, which agrees.T23. f(18) 5 5.5, f(54) 5 6.2; y 5 3.6583… 1 0.6371… ln xT24. f(x) 5 (302.0582…)(1.0962… ) x ; f(5) 5 478.2229…; f(7) 5 574.7067…T25. g (2) 5 362.0488…; g (5) 5 484.0232…; g (7) 5 583.2807…;g (11) 5 829.2796… T26.

T27. Th e town can hold only a limited number of people. T28. Answers will vary.

50

2000

y

x

f

g


Properties of Elementary Functions · CAS Activity 2-5a: Dilations of Logarithmic Functions Chapter...

Documents

Transcript of Properties of Elementary Functions · CAS Activity 2-5a: Dilations of Logarithmic Functions Chapter...