Slide 1-1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Graphs, Functions,and Models

Chapter 1


1.1 Introduction to Graphing

Plot points. Determine whether an ordered pair is a solution of an equation. Graph equations. Find the distance between two points in the plane and

find the midpoint of a segment. Find an equation of a circle with a given center and

radius, and given an equation of a circle, find the center and the radius.

Graph equations of circles.


Cartesian Coordinate System

(x, y)

I

(+, +)II

(, +)

III

(, )IV

(+, )

(0,0)


Solutions of Equations

Equations in two variables have solutions (x, y) that are ordered pairs.

Example: 4x + 5y = 20

When an ordered pair is substituted into the equation, the result is a true equation.


Examples

Determine whether the ordered pair (4, 2) is a solution of 3x + 4y = 2.

3(4) + 4(2) ? 2

12 + 8 ? 2

4 2

false

(4, 2) is not a solution.

Determine whether the ordered pair (2, 1) is a solution of 3x + 4y = 2.

3(2) + 4(1) ? 2

6 + 4 ? 2

2 = 2

true

(2, 1) is a solution.


Graphs of Equations

To graph an equation is to make a drawing that represents the solutions of that equation.


x-Intercept

The point at which the graph crosses the x-axis.

An x-intercept is a point (a, 0). To find a, let y = 0 and solve for x.

Example: Find the x-intercept of 3x + 5y = 15.

3x + 5(0) = 15

3x = 15

x = 5

The x-intercept is (5, 0).


y-Intercept

The point at which the graph crosses the y-axis.

A y-intercept is a point (0, b). To find b, let x = 0 and solve for y.

Example: Find the y-intercept of 3x + 5y = 15.

3(0) + 5y = 15

5y = 15

y = 3

The y-intercept is (0, 3).


Example

Graph x + 2y = 6. x-intercept:

x + 2(0) = 6 x = 6

(6, 0)

y-intercept:(0) + 2y = 6 2y = 6 y = 3

(0, 3)

(6, 0)

(0, 3)


Example

Graph y = x2 + 2x 3 = 0 Make a table of values.

x y (x, y)

3 0 (3, 0)

2 3 (2, 3)

1 4 (1, 4)

0 3 (0, 3)

1 0 (1, 0)

2 5 (2, 5)

1. Select values for x.

2. Compute values for y.


The Distance Formula

The distance d between any two points

(x1, y1) and (x2, y2) is given by

.

Example: Find the distance between the points (2, 2) and (3, 5).

2 22 1 2 1( ) ( )d x x y y

2 2

2 2

( 3 2) ( 5 2)

( 5) ( 7) 25 49

74 8.6

d

d

d


Midpoint Formula

If the endpoints of a segment are (x1, y1) and (x2, y2), then the coordinates of the midpoint are

.

Example: Find the midpoint of a segment whose endpoints are (5, 6) and (4, 4).

1 2 1 2,2 2

x x y y

5 4 6 4,

2 2

1 2,

2 2

1, 1

2


Circles

A circle is the set of all points in a plane that are a fixed distance r from a center (h, k).

The equation of a circle with center (h, k) and radius r, in standard form, is

(x h)2 + (y k)2 = r2.


Example

Find an equation of a circle having radius 7 and center (4, 2).

Using the standard form:

(x h)2 + (y k)2 = r2

[x 4]2 + [y (2)]2 = 72

(x 4)2 + (y + 2)2 = 49.


Example

Graph the circle

(x + 4)2 + (y 1)2 = 9

Write the equation in standard form.

[x (4)]2 + [y 1]2 = 32

The center is (4, 1) and the radius is 3.

(4, 1)


1.2 Functions and Graphs

Determine whether a correspondence or a relation is a function. Find function values, or outputs, using a formula. Find the domain and the range of a function. Determine whether a graph is that of a function. Solve applied problems using functions.


Function

A function is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to exactly one member of the range.Function Not a Function7 49 2 4 7 5 7 0 0 6 32 4 8 2


Relation

A relation is a correspondence between the first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to at least one member of the range.

Which of the following relations is a function?

{(8, 2), (8, 4), (7, 3)} Not a function

{(6, 4), (1, 4), (7, 4)} Function


Notation for Functions

The inputs (members of the domain) are values of x substituted into the equation. The outputs (members of the range) are the resulting values of y.

f(x) is read “f of x,” or “f at x,” or “the value of f at x.”

Example: Given f(x) = 3x2 4, find f(6).

f(6) = 3(6)2 4 = 3(36) 4 = 104


Graphs of Functions

We select values for x and find the corresponding values of f(x). Then we plot the points and connect them with a smooth curve.

The Vertical-Line Test

If it is possible for a vertical line to cross a graph more than once, then the graph is not the graph of a function.


Example: Does the graph represent a function?

The graph is a function because we cannot find a vertical line that crosses the graph more than once.


Example: Does the graph represent a function?

The graph is not a function. We can find a vertical line that crosses the graph in more than one point.


Finding Domains of Functions

Find the indicated function values and determine whether the given values are in the domain of the function.

f(1) and f(5), for

f(1) =

Since f(1) is defined, 1 is in the domain of f.

f(5) =

Since division by 0 is not defined, the number 5 is not in the domain of f.

1( )

5f x

x

1 1 1

1 5 4 4

1 1

5 5 0


Another Example

Find the domain of the function

Solution: We can substitute any real number in the numerator, but we must avoid inputs that make the denominator 0. Solve x2 3x 28 = 0.

(x 7)(x + 4) = 0x 7 = 0 or x + 4 = 0 x = 7 or x = 4The domain consists of the set of all real numbers except 4 and 7 or {x|x 4 and x 7}.

2

2

3 10 8( )

3 28

x xg x

x x


Visualizing Domain and Range

Keep the following in mind regarding the graph of a function:

Domain = the set of a function’s inputs, found on the horizontal axis.

Range = the set of a function’s outputs, found on the vertical axis.


Example

Graph the function. Then estimate the domain and range.

Domain = [1, )

Range = [0, )

( ) 1f x x ( ) 1f x x


1.3 Linear Functions, Slope,

and Applications Determine the slope of a line given two points on the line.

Solve applied problems involving slope and linear functions.


Linear Functions

A function f is a linear function if it can be written as

f(x) = mx + b,

where m and b are constants.

If m = 0, the function is a constant function f(x) = b.

If m = 1 and b = 0, the function is the identity function

f(x) = x.


Examples

Linear Functions Nonlinear Functions

3 1xy

y = x y = 3y = x2 + 1

2

xy

2 2 4x y


Horizontal and Vertical Lines

Horizontal lines are given by equations of the type y = b or f(x) = b. They are functions.

Vertical lines are given by equations of the type x = a. They are not functions.

y = 2

x = 2


Slope

The slope m of a line containing the points (x1, y1) and (x2, y2) is given by

2 1 1 2

2 1 1 2

the change in

the change in

risem

runy

x

y y y y

x x x x


Example

Graph the function and determine its slope.Calculate two ordered pairs, plot the points, graph the function, and determine its slope.

1( ) 2

2f x x

1( ) ( ) 2 2 0

24 24f

1( ) ( ) 2 0

20 0 22f

Rise 2

Run 4

2 1

2 1

(0 ( 2)) 2 1

4 0 4 2

y ym

x x


Horizontal and Vertical Lines

A horizontal line has a slope of 0.

Graph y = 4 and determine its slope.

Choose any number for x; y must be 4.

x y

4 4

2 4

1 4

A vertical line has an undefined slope because we cannot divide by zero.

Graph x = 2 and determine its slope.

Choose any number for y; x must be 2.

x y

2 4

2 1

2 2


Types of Slopes

Positive—line slants up from left to right

Negative—line slants down from left to right

Zero—horizontal line

Undefined—vertical line

zero

positive

negative

undefined


1.4 Equations of Lines

and Modeling Find the slope and the y–intercept of a line given the equation y = mx + b, or f(x) = mx + b. Graph a linear equation using the slope and the y-intercept. Determine equations of lines. Given the equations of two lines, determine whether their graphs are parallel or whether they are perpendicular. Model a set of data with a linear function. Fit a regression line to a set of data; then use the linear model to make predictions.


Slope-Intercept Equation

The linear function f given by

f(x) = mx + b

has a graph that is a straight line parallel to y = mx. The constant m is called the slope, and the y-intercept is (0, b). (0, 0)

(0, b) y = mx

f(x) = mx + b


Example

Find the slope and y-intercept of the line with equation y = 0.36x + 4.2.

Solution: y = 0.36x + 4.2

Slope = 0.36; y-intercept = (0, 4.2)


Example

Find the slope and y-intercept of the line with equation

4x + 3y 12 = 0.

Solution: We solve for y:

Thus, the slope is and the y-intercept is (0, 4).

1 13 3

4 3 12 0

3 4 12

(3 ) ( 4 12)

44

3

x y

y x

y x

y x

43


Example

A line has slope 3 and contains the point (2, 5). Find an equation of the line.

Solution:

We use the slope-intercept equation, y = mx + b, and substitute for m. y = 3x + b.

Using the point (2, 5), we substitute for x and y and solve for b. 5 = 3(2) + b

5 = 6 + b

11 = b

The equation of the line is y = 3x + 11.


Example

Graph

Solution: The equation is in

slope-intercept form, y = mx + b. The y-intercept is

(0, 2).

12

2y x

change in 1 move 1 unit up

change in 2 move 2 units right

rise ym

run x


Point-Slope Equation

The point-slope equation of the line with slope m passing through

(x1, y1) is y y1 = m(x x1).

Example: Find the equation of the line containing the points (2, 7) and (1, 8).

Solution: First determine the slope:

Using the point-slope equation, substitute 5 for m and either

of the points for (x1, y1):

8 7

1 215

35

m

1 1( )

( )

7 5 10

5

5 2

3

7

y y m x x

y x

y x

y x


Parallel Lines

Vertical lines are parallel. Nonvertical lines are parallel if and only if they have the same slope and different y-intercepts.

y = 3x + 2

y = 3x 4


Perpendicular Lines

Two lines with slopes m1 and m2 are perpendicular if and only if the product of their slopes is 1:

m1m2 = 1.

Lines are also perpendicular if one is vertical (x = a) and the other is horizontal (y = b).

y = 3x 412

3y x


Examples

Determine whether each of the following pairs of lines is parallel, perpendicular, or neither.

a) y 4x = 3, 4y 8 = x (perpendicular)

b) 2x + 3y = 4, 3x + 2y = 8 (neither)

c) 2y = 4x + 12, y 8 = 2x (parallel)


Example

Write equations of the lines (a) parallel and (b) perpendicular to the graph of the line 3y + 4 = 18xand containing the point (1, 2).

Solve the equation for y:

(a) The line parallel to the given line will have the sameslope. We use either the slope-intercept or point-slope equation for the line.

43

3 4 18

3 18 4

6

y x

y x

y x


Example continued

Substitute and solve the equation.

1 1( )

2 6( 1)

2 6 6

6 8

y y m x x

y x

y x

y x


Example continued

(b) For a line perpendicular: m = 1

6

1 1( )

12 ( 1)

61 1

26 6

1 11

6 6

y y m x x

y x

y x

y x


Curve Fitting

In general, we try to find a function that fits, as well as possible, observations (data), theoretical reasoning, and common sense.

Example: Model the data given in the table on foreign travel on the next slide with two different linear functions. Then with each function, predict the number of U.S. travelers to foreign countries in 2005. Of the two models, which appears to be the better fit?


Curve Fitting continued

Model I:

Choose any two points to determine the equation.

Predict the number of travelers:

5.75 5.080.1675

5 1m

0.1675( )

0.1

5.0

675

8 1

4.9125

y x

y x

0.1675 4.9125

0.1675( ) 4.91212 5

6.923

y x

y

y



Model II:

Choose any two points to determine the equation.

Predict the number of travelers:

6.08 4.650.2383

6 0m

0.2383 4.65y x

0.2383 4.65

0.2383( ) 4.65

7.510

12

y x

y

y



Using model I, we predict that there will be about 6.92 million U.S. foreign travelers in 2006, and using model II, we predict about 7.51 million.

Since it appears from the graphs that model II fits the data more closely, we would choose model II over

model I.


1.5 More on Functions

Graph functions, looking for intervals on which the function is increasing, decreasing, or constant, and estimate relative maxima and minima. Given an application, find a function that models the application; find the domain of the function and function values, and then graph the function. Graph functions defined piecewise.


Increasing, Decreasing, and Constant Functions

On a given interval, if the graph of a function rises from left to right, it is said to be increasing on that interval. If the graph drops from left to right, it is said to be decreasing.

If the function values stay the same from left to right, the function is said to be constant.


Definitions

A function f is said to be increasing on an open interval I, if for all a and b in that interval,

a < b implies f(a) < f(b).


Definitions continued

A function f is said to be decreasing on an open interval I, if for all a and b in that interval,

a < b implies f(a) > f(b).


A function f is said to be

constant on an open interval I,

if for all a and b in that interval, f(a) = f(b).

Definitions continued


Relative Maximum and Minimum Values

Suppose that f is a function for which f(c) exists for some c in the domain of f. Then:

f(c) is a relative maximum if there exists an open interval I containing c such that f(c) > f(x), for all x in I where x c; and

f(c) is a relative minimum if there exists an open interval I containing c such that f(c) < f(x), for all x in I where x c.


Functions Defined Piecewise

Graph the function defined as:

a) We graph f(x) = 3 only for inputs x less than or equal to 0.

b) We graph f(x) = 3 + x2 only for inputs x greater than 0 and less than or equal to 2.

c) We graph f(x) = only for

inputs x greater than 2.

2

3 for 0

( ) 3 for 0 2

1 for 22

x

f x x x

xx

f(x) = 3, for x 0

f(x) = 3 + x2, for 0 < x 2

( ) 1for 22

xf x x

12

x


1.6 The Algebra of Functions

Find the sum, the difference, the product, and the quotient of two functions, and determine the domains of the resulting functions.

Find the difference quotient for a function.

Find the composition of two functions and the domain of the composition; decompose a function as a composition of two functions.


Sums, Differences, Products, and Quotients of Functions

If f and g are functions and x is in the domain of each function, then

( )( ) ( ) ( )

( )( ) ( ) ( )

( )( ) ( ) ( )

( / )( ) ( ) / ( ), provided ( ) 0

f g x f x g x

f g x f x g x

fg x f x g x

f g x f x g x g x


Example

Given that f(x) = x + 2 and g(x) = 2x + 5, find each of the following. a) (f + g)(x) b) (f + g)(5) Solution: a)

b) We can find (f + g)(5) provided 5 is in the domain of each function. This is true.f(5) = 5 + 2 = 7 g(5) = 2(5) + 5 = 15(f + g)(5) = f(5) + g(5) = 7 + 15 = 22 or(f + g)(5) = 3(5) + 7 = 22

( )( ) ( ) ( )

2 2 5

3 7

f g x f x g x

x x

x


Another Example

Given that f(x) = x2 + 2 and g(x) = x 3, find each of the following.a) The domain of f + g, f g, fg, and f/gb) (f g)(x)c) (f/g)(x)

Solution: a) The domain of f is the set of all real numbers. The

domain of g is also the set of all real numbers. The domains of f + g, f g, and fg are the set of numbers in the intersection of the domains—that is, the set of numbers in both domains, or all real numbers.For f/g, we must exclude 3, since g(3) = 0.


Another Example continued

b) (f g)(x) = f(x) g(x) = (x2 + 2) (x 3)= x2 x + 5

c) (f/g)(x) =

Remember to add the stipulation that x 3, since 3 is not in the domain of (f/g)(x).

2

( )( / )( )

( )

2

3

f xf g x

g x

x

x


Difference Quotients

Example: For the function f given by f(x) = x2 + 2x 3,

find the difference quotient

.

Solution: We first find f(x + h):

( ) ( )f x h f x

h

2

2 2

( ) ( ) 2( ) 3

2 2 2 3

f x h x h x h

x xh h x h


Difference Quotients continued

2 2 2

2 2 2

2

( ) ( )

2 2 2 3 ( 2 3)

2 2 2 3 2 3

2 2

(2 2)2 2

f x h f x

h

x xh h x h x x

h

x xh h x h x x

h

xh h h

hh x h

x hh


Composition of Functions

Definition:

The , the of

and , is defined as

( )( ) ( ( )),

where is in the d

composit

omain of

e

and ( ) is in the

domai

function composition

n of .

f g

f g

f g x f g x

x g g x

f


Example

Given that f(x) = 3x 1 and g(x) = x2 + x 3, find:

a)

b)

( )( ) f g x

2

2

2

2

( )( ) ( ( )) ( )

3( ) 1

3

3 3 9 1

3 10

3

3

x x

x

f g x f g x f

x x

x x

x

2

2

2

( )( ) ( ( )) ( )

( ) (

3 1

3 1 3 1) 3

9 6 1 3 1 3

9 3 3

g f x g f x g

x x

x

x

x

x

x

x

( )( )g f x


Decomposing a Function as a Composition

In calculus, one needs to recognize how a function can be expressed as the composition of two functions.

Example: If h(x) = (3x 1)4, find f(x) and g(x) such that

Solution: The function h(x) raises (3x 1) to the fourth power. Two functions that can be used for the composition are:

f(x) = x4 and g(x) = 3x 1.

( ) ( )( ).h x f g x


1.7 Symmetry and Transformations

Determine whether a graph is symmetric with respect to the x-axis, the y-axis, and the origin. Determine whether a function is even, odd, or neither even nor odd. Given the graph of a function, graph its transformation under translations, reflections, stretchings, and shrinkings.


Symmetry

Algebraic Tests of Symmetryx-axis: If replacing y with y produces an equivalent

equation, then the graph is symmetric with respect to the x-axis.

y-axis: If replacing x with x produces an equivalent equation, then the graph is symmetric with respect to the y-axis.

Origin: If replacing x with x and y with y produces an equivalent equation, then the graph is symmetric with respect to the origin.


Example

Test x = y2 + 2 for symmetry with respect to the x-axis, the y-axis, and the origin.

x-axis: We replace y with y:

The resulting equation is equivalent to the original so the graph is symmetric with respect to the x-axis.

y-axis: We replace x with x:

The resulting equation is not equivalent to the original equation, so the graph is not symmetric with respect to the y-axis.

2

2

( ) 2

2

x y

x y

2

2

2

2

x y

x y


Example continued

Origin: We replace x with x and y with y:

The resulting equation is not equivalent to the original equation, so the graph is not symmetric with respect to the origin.

2

2

2

2

( ) 2

2

x y

x y

x y


Even and Odd Functions

If the graph of a function f is symmetric with respect to the y-axis, we say that it is an even function. That is, for each x in the domain of f, f(x) = f(x).

If the graph of a function f is symmetric with respect to the origin, we say that it is an odd function. That is, for each x in the domain of f, f(x) = f(x).


Example

Determine whether the function is even, odd, or neither.

1.

We see that h(x) h(x). Thus, h is not even.

2.

We see that h(x) h(x). Thus, h is not odd.

6 3( ) 4 2 3h x x x x

6 3

6 3

( ) 4( ) 2( ) 3( )

4 2 3

x x xh x

x x x

6 3

6 3

( ) (4 2 3 )

4 2 3

h x x x x

x x x


Vertical Translation

Vertical Translation

For b > 0,

the graph of y = f(x) + b is the graph of y = f(x) shifted up b units;

the graph of y = f(x) b is the graph of y = f(x) shifted down b units.

y = 3x2


Horizontal Translation

Horizontal Translation

For d > 0,

the graph of y = f(x d) is the graph of y = f(x) shifted right d units;

the graph of y = f(x + d) is the graph of y = f(x) shifted left d units.

y = 3x2


Reflections

The graph of y = f(x) is the reflection of the graph of y = f(x) across the x-axis.

The graph of y = f(x) is the reflection of the graph of y = f(x) across the y-axis.

If a point (x, y) is on the graph of y = f(x), then (x, y) is on the graph of y = f(x), and (x, y) is on the graph of y = f(x).


Example

Reflection of the graph y = 3x3 4x2 across the x-axis.

y = 3x3 4x2y = 3x3 + 4x2


Vertical Stretching and Shrinking

The graph of y = af(x) can be obtained from the graph of y = f(x) by

stretching vertically for |a| > 1, orshrinking vertically for 0 < |a| < 1.

For a < 0, the graph is also reflected across the x-axis.

(The y-coordinates of the graph of y = af(x) can be obtained by multiplying the y-coordinates of y = f(x) by a.)


Examples


Horizontal Stretching or Shrinking

The graph of y = f(cx) can be obtained from the graph of y = f(x) by

shrinking horizontally for |c| > 1, orstretching horizontally for 0 < |c| < 1.

For c < 0, the graph is also reflected across the y-axis.

(The x-coordinates of the graph of y = f(cx) can be obtained by dividing the x-coordinates of the graph of y = f(x) by c.)


Examples

Slide 1-1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

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Transcript of Slide 1-1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.