Handouts

Marjorie Fernandez Karwowski Valencia Community College East Campus Mathematics Department

MFK/Fall 08

Calculator Help and Basic Function Guide

Calculator Check: Before using your calculator first do the following

Choose the “mode” key, make sure everything on the left hand side is highlighted Choose the “y=” key, make sure all the “plots” are not highlighted. If they are then use

your “up arrow” key, place your cursor on the ones that are highlighted then hit “enter”. Under the “y=” key use your “down arrow” key to access any equations and press “clear”

to erase them. On the TI-83 you are able to enter 10 equations. Choose “zoom”, then choose “#6, that is z-standard”. This will give you a window that is -10 to 10 on the x and y axis with a scale of 1, or that is [-10,10,1] by [-10,10,1].

Basic Functions

1.) Difference between a negative sign and a subtraction symbol For example, if you want to compute - 9 – 10.8 you must first choose the “( - )” key

(left of the enter key) then choose –10.8. *Please note that the negative sign will appear smaller than the subtraction symbol. Remember the negative sign must always be used when an equation, number or computation begins with a negative.

2.) The n root The “^” raises a number to a power or that is exponent.

For example if you want to find the , you want to use the “ ” key or type in 78.6^(1 ) or 78.6^0.5.

If you want to find the it’s easiest to type in 78.6^( ). You can also use the “math” key option #4.

3.) Changing decimals to fractions and vica versa For example, to convert .236 to a reduced fraction, type in .236 then “Math”,

choose “Frac” then hit enter. Do the same to change a fraction to a decimal by choosing “Dec” under the Math key.

4.) Basic viewing windows under the “zoom” key There are 2 basic built in windows that we use in the class, they are found under

the “zoom” key. The standard window, as stated, above gives you a [-10,10,1] by [-10,10,1] window. The z-decimal window will give you a close up view of the graph in comparison to the z-standard window. The z-decimal window gives you a

[-4.7,4.7,1] by [-3.1,3.1,1] window. You should always hit the “window” key to see the window that you are currently using.

5.) Graphing using the “y=” key

1

After entering your equation(s) under the “y=” key, always hit “trace” afterwards to explore the coordinates. You may enter up to 10 equations at a time.

If you want to save an equation under the “y=” key but not display its graph, use your left arrow key, go on top of the “=” sign then hit enter. When the equal sign is not highlighted the graph is so called “turned off”. To turn the graph back on just simply go over the equal sign and hit enter.

The “trace” key allows you to see the points on the graph by displaying the coordinates on the bottom of your screen.

6.) Evaluating at a particular x-value You may evaluate a graph (remember hit trace first) at x-values by simply typing

in the x value on the keypad then hit enter. *Please note that your x-values must be no smaller than your x-min and no larger than your x-max *.

For example let’s evaluate at x = 6. First type in under the

“y=” key, hit trace, type in 6 then hit enter. The calculator will give you the y-value and the cursor will be located at the point (6, -.25). Note: Notice that we used parenthesis for (x + 2). If you type – 2 x + 2, the calculator will only divide – 2 by x and not by the whole expression (x + 2).

You will need to be in the z-standard window for the example above because using z-decimal will result in an error due to 6 being larger than x-max which is 3.1.

7.) Using tables, always set up the table first*There are 2 choices, you may have the calculator complete the table for you or you may enter various x-values then have the calculator give you the corresponding y-values*

Having the calculator complete the table for you. To have the calculator complete the table for you first go to “Tblset”. Enter the x-value that you want to start with, then enter the increment value or that is the “ΔTbl” value. Choose “auto” for both the Indpnt and Depend options (the Indpnt represents the x, and the Depend represents the y). Now hit “2nd” then “Graph” and your table is now complete. You may use your up and down arrow keys to look at other ordered pairs.

Entering various x-values on your own. To complete a table by entering various x-values once again go to “Tblset” first. Once you are there just simply choose the “Ask” option for the Indpnt , now hit “2nd” then “Graph” and your table should appear blank. You may now enter your x-values then hit enter to obtain the corresponding y-value.

Note: Fractions are most important in this class, they will be used throughout the entire course. You must be able to add, subtract, multiply and divide fractions by

2

hand without using a calculator. We will also be evaluating expressions throughout the course that involves fractions.

Fraction review

Adding and subtracting: A common denominator must be found. (The least common denominator, L.C.D.)

Multiplying: Always simplify before you multiply. Dividing: Keep change and flip, then proceed per instructions with

multiplication. Squaring a fraction: Square the numerator and denominator. Remember that a

negative square is always positive.

Add or Subtract

1.) 2.) 3.)

4.) -1 +

Multiply or divide

5.) 6.) 7.) 8.) - 24

Simplify (remember PEMDAS)

9.) 10.) 11.) - 9

– 12.)

13.) 19 14.) 0

3

1105 Chapter 1

Functions and Their GraphsSec 1.1 Linear Models

The general (or standard) form of a liner equation is denoted by ax + by = c, where a and b cannot both be = 0

To graph: find the x-intercept (set y = 0), find the y-intercept (set x = 0) then a line connecting the points.

To find the slope: (shortcut for slope m = then change the sign)

1. Graph the following by hand and state the intercepts and slope.

a. 3x – 9y = 9 y-int_____, x-int______ b. - 20x – 10y = 40 y-int_____, x-int____

m = ___________ m = _______

c. - 2x + = 4 y-int_____, x-int______ d. y-int______, x-int________

m = ___________ m = _______

4

2. Given: 2x – 4y = - 48 a. x int. _______ b. y-int ________c. Solve for y _____________

d. State a window in which both intercepts are visible [ , ] by [ , ] then graph.

3. Given: -2x + = 10 a. x int. _______ b. y-int ________c. Solve for y _____________

d. State a window in which both intercepts are visible [ , ] by [ , ] then graph.

Slope is discussed in section 1.4 but we will discuss here.

The slope intercept form is denoted by: y = mx + b, where m is the slope and b is the y –intercept.

4. Write an equation in slope-intercept form for the lines below.

Slope: In the context of applications is the rate of change. For example someone’s salary changing over time, the rate at which tree is growing over time, the rate at which a car’s value

5

depreciates over time, the rate at which a drug leaves the body etc. We’re assuming the rate is constant.

See page 14 #1 – 3 for applications using the slope-intercept form.

See page 17 # 25 – 28 for applications using the standard/general form.

5. The temperature in the Kent, Ohio at 5 a.m. was 52 F. The temperature rose 4 degrees every hour until is reached it’s maximum temperature at about 5 p.m. Complete the table of values for the temperature, T, at h hours after 5 a.m.

h 0 3 6 8 10

T

a. Write an equation (in slope intercept form) for the temperature, T, in terms of h.

b. Graph the equation.

c. What is the temperature at 10 a.m.? At noon?

d. When was the temperature 88 ?

6. Tony has $450 in his savings account for school expenses. He withdraws $50 every month for miscellaneous items per month. Complete the table of values for the amount of money, A, left in his account after m months.

m 0 4 6 8 9

A

a. Write an equation that expresses the amount of money, A, in his account in terms of the number of months, m.

b. Graph the equation.

6

c. How much did his money decrease between the third month and the sixth month? Show this on the graph.

d. When will his account contain more than $200? Show this on the graph.

7. Juanita plans to diversify her retirement savings in 2 different accounts. The IRA CD account pays 3.5% interest per year, the Traditional CD account pays 4.0% interest per year. Juanita plans to earn $600 per year on her retirement savings.

a. How much interest will she earn in the IRA account? Use the variable “x”.

b. How much interest will she earn in the Traditional account? Use the variable “y”.

c. Write an equation in general form that describes her retirement savings account if she earns $600 per year.

d. Find the intercepts and describe their meaning. Round to the nearest dollar.

Sec. 1.1 starting on page 14Homework: 1, 3, 14 – 17 all, 25a-d, 29a-c, 33, 45, 46, 49

Sec. 1.2 Functions A function is a relationship between 2 variables such that for each x value (input) there exists a unique y value (output).

1. The following table denotes the Daytona 500 winners’ average speed (in miles per hour) where x = 0 corresponds to the year 2000.

x 0 1 2 3 4

r 155.669 161.783 142.971 133.870 156.345

a.) Is r a function of x? _______

b.) From the table find r(3). _____________

c.) Find x such that r(x) = 161.783. ___________________

7

d.) Interpret your result from part b. _________________________________________

2. The graph below describes the population of a town where P is a function of t. P denotes the population (in thousands) and t denotes the time in years since 1995.

a. In what year was the population 10,000?

b. Approximate the population in 2000.

c. How long did it take the population in part b to double?

d. During what years was the population between 60,000 and 110,000?

3. Evaluate the following, simplify if possible. Do by hand then verify with your calculator.

a. f(x) = - , find f(-2) b. f(x) = find f( - 2/3)

c. g(x) = , g(12) then find g(4) d. h(x) = , h( 5) then find h(2)

Sec. 1.2 starting on page 30 homework: 15, 17, 19, 27, 31a-b, 35a-b, 37a-c, 41a-d, 43a-d

Sec. 1.3 Graphs of Functions

Vertical Line Test A graph represents a function if and only if when you draw a line through the graph it crosses the graph atmost once.

1. See page 50 #9 of your textbook.

2. Complete the table for g(x) =

3. Complete the table for f(x) = 3.5x + x

x - 5 -3 -1 1y

8

4. Complete the table for f(x) = 23.5x x22

Note: It’s often helpful to rewrite function

notation, that is set f(x) = y, P(r) = y etc.

6. Given the graph find the following: a. Find f( - 1) b. For what value(s) of x is f(x) = 1?

c. the value(s) of x where f(x) = - 2 d. Find where f(x) = 0

e. Find the minimum value of f(x). f. Find the intervals of increasing and decreasing.

g. Find where f(x) < 1 h. Find where f(x) > - 2

7. Use the graph to find: a. Estimate R(- 1) b. Estimate the value(s) where R(s) = 0.6

c. Find the maximum and minimum value(s) of R(s).

d. For what value(s) of s does R have a maximum and minimum values?

e. Intervals of increasing and decreasing.

x 3 7 12 27

x 4 7 10 13

9

8. The graph below is y = 3/2x – 1, use the graph to find the following:

a. y = - 2 b. y > - 2 c. y < - 1 d. solve 3/2x – 1 = 5

Sec. 1.3 starting on page 48 homework: 1a-f, 2a-f, 5a-d, 9, 13, 17a-c19a-c, 21a-c, 29a-b, 33a, 37

Sec. 1.4 Slope and Rate of Change

1. Find the slope for each line a. b.

Recall that given an x and y intercept the slope can be found using the shortcut

m = then change the sign

2. Graph the following using the intercept (cover method) method and find the slope for each.

a. – 2x + 4y = - 8 m = _______ b. – 8y + 6x = 48 m = _______

10

The slope between two points ( ) and ( ) is m =

The slope “m” or that is the “constant rate of change” is also described by

3. Find the slope between the points ( 3, - 4) and ( - 5, 7)

4. Find the slope from the graph.

Again, the slope can be described as a “rate of change” when working with applications.

5. The graph below denotes cost, C (in dollars) of renting a laptop computer in terms of h hours.

a. Find the slope

b. Explain the meaning of the slope with respect to the context of the problem.

11

Sec. 1.4 starting on page 67 homework: 5-7 all, 9, 11, 13, 21a-d, 23, 33, 35 Sec. 1.5 Linear Functions

Recall that the slope-intercept form of a linear equation is given by y = mx + b, Where m is the slope and b is the y-intercept. To graph using our calculator it is necessary to write the equation in this form in order to enter it under the y = key.

1. Write the equations in the slope intercept form, then state the slope and y-intercept.

a. - 3x + 7y = 9 b. 2x – 3y = 1

2 . i. Sketch by hand the graph of the line with given slope and y-intercept. ii. Write the equation of the line in slope intercept form. iii. Find the x-intercept. a. m = - 1/2 and b = - 2 b. m = 3/4 and b = 3

equation _________________ equation _________________

x-int _______ x-int _______

12

Note that you can start at any pt. that lies on the graph not just the y-intercept when sketching a graph. When given a point (other than “b”) and a slope we can use the “point slope form”, then from here we can write it in slope intercept form.

The point slope form of a linear equation that passes through the pt. ( ) with slope m

is y = m(x – x ) + y , distribute the m and simplify to express in slope intercept form.

3. i. Sketch by hand the graph. ii. Write equation in point slope form iii. Write equation in slope intercept form.

a. ( -1, 2); m = -2/3 b. (3, - 2); m = 2

Interpreting the slope: Often we are interested as to how a situation changes over time. 13

How much does our car depreciate each year, how does our salary change each year, is the price of gas increasing or decreasing each day?

4. The prenatal growth of a fetus more than 12 weeks old can be approximated by the formula L = 1.53t – 6.7, where L is the length in cm and t is the age in weeks. Prenatal length can be determined by X rays. a. What is the slope? _________

b. Interpret the slope as a rate of change.

c. Estimate the age of a fetus that is 12 cm long.

5. The value in dollars of a copy machine is denoted by the function f(x) = -370x + 5100, where x is years after 1998.

a. Interpret the slope as a rate of change.

b. Interpret the y-intercept. (vertical intercept)

c. What was the value of the copy machine in 2001?

6. The graph below denotes the value, V, (in thousands) of a tool making machine after t years.

a. Find the slope from the graph and explain it’s meaning with context of the problem. b. Find a formula for the function that describes the graph. (in slope intercept form)

c. Explain the meaning of the vertical intercept. (the y-intercept)

7. Match the equation with it’s graph. The scales on the graphs are all the same.

14

a. y = b. y = c. y = - 2x + 3 d. y = - 4x + 3

Sec. 1.5 starting on page 85 homework: 1, 2, 11, 13,23, 33, 35,53a-d

Chapter 8 Linear Systems

Sec. 8.1 Systems of Linear Equations in 2 variables

The solution(s) given any 2 graphs is at the point of intersection.

1. Find the solution from the graph below.

Solving a system of linear equations using your calculator:

Solve for y for both equations and place under the y1 and y2 key. Hit graph, then trace.

15

2nd trace, choose #5 then hit enter 3 times.

2. Solve the following using your grapher.

a. y = .68x – 2.1 and y = 4.98 – 4.23x b. 3.1x + 2.3y = -1.1 and y = -2.3x – 5.6

We will be discussing 3 different ways in which to solve a system of 2 equations; graphing by hand or with our calculator, using “linear combinations, that is using substitution or elimination. Before doing so let’s discuss the different types of systems.

In this section we discuss consistent, inconsistent, independent and dependent systems. A “consistent” system will have at least one solution. An “inconsistent” system will have no solutions. (parallel lines) An “independent” system for 2 linear equations will have 0 or 1 solutions. A “dependent” system will have infinite solutions (coinciding lines).

3 Case Scenarios when solving a system of linear equations.

From a graph, the solution(s) if they exist, are at the point(s) of intersection

I. Consistent and independent: One solution exists, one example would be perpendicular lines.

the solution is at (2.5, -0.5)

**When solving by hand you will get a value for x and a value for y. For example, x = 2.5, y = -0.5

II. Consistent and dependent: Infinite solutions, the lines are the same.

For example: The system of equations x – y = 2 and 2x – 2y = 4 are the same equations, one is just a multiple of the other, the graphs are the same.

16

**When solving by hand you will get a “true result” 0 = 0, or – 5 = -5 etc. III. Inconsistent and independent: No solutions the lines are paralell.

For example, the system y = 2x + 3 and y = 2x -1, the slopes are equal therefore parallel lines.

**When solving by hand you will get a “false result” 7 =3, 0 = -9 etc.3. Solve by graphing, identify as consistent or inconsistent, then dependent or independent.

a. y = 3x – 4 b. 3y – 6x = 12 x + 2y = 6 y = 2x + 1

Solving using linear combinations, that is using substitution or elimination

For the following problems, solve by hand (symbolically) using substitution then elimination. State whether the system is consistent or inconsistent and whether the system is dependent or independent.

4. x + y = 2

17

3x + y = 4

5. 16x – 6y = - 4 - 8x + 3y = 2

6. x + y = 7

x + y = 8

7. 3x – y = 6 - 4x + 2y = - 8

Note that Revenue = price quantity and cost = fixed cost(s) + variable cost(s)

Write equations for the following, do not solve.

8. Ron and Javier are ticket sellers for their band to raise money for charity, Ron handling student tickets that sell for $2.00 each and Javier selling adult tickets for $4.50 each. If their total income for 364 tickets was $1,175.50, how many did Ron sell and how many did Javier sell? Use R for Ron and J for Javier.

a. Write an equation for the number of tickets sold.

b. Write an equation for the revenue from the tickets sold.

9. Lauri Anne’s beaded jewelry company sells there bracelets for $30 each. When she first started the company her initial fixed cost was $350 and $6.00 per bracelet.

a. Express Lauri Anne’s cost C in terms of the number x of bracelets produced.

b. Express her revenue R in terms of the number x of bracelets sold.

Sec. 8.1starting on page 651 homework: 3, 5, 7, 10, 11, 17, 18, 25a-b, 27b-c

Sec. 8.4 Linear Inequalities

18

1. Graph each inequality a. - 2x + 3y > -6 b. y > 3

c. – 3 < x 1 d. y + 2x > 4, y ≥ 1 – x

e. - x – 2y ≥ - 4 , 4x – 6y < 12 f. y < - , x ≥ 0, y ≥ 0

Sec. 8.4starting on page 689 homework: 1, 3, 9, 10, 15, 21, 23, 28

Chapter 6 Quadratic Functions

Sec. 6.1 Factors and x-intercepts

Solving quadratics using factoring: Move all terms to the left, right hand side = 0

19

If the left is not factored and there are parenthesis, remove the parenthesis then factor Once the left is factored then set each quantity = 0 and solve for the variable.

1. Solve by factoring. a. (x – 2)(x + 9) = 0 b. 9y + 12y – 5 = 0

c. 5x + 27x = - 10 d. 3(2x - 1) = 7x e. (x + 1)(x = 3) = 5

Recall 1a from above, notice how if “x – 2” is a factor then x = 2 is a solution? Therefore the reverse is true, if x = 2 is a solution then (x - 2) is a factor, if x = -9 is a solution then (x + 9)is a factor etc.

2. Write a quadratic equation (in standard form with integer coefficients) whose solutions are given.

a. x = 8, x = - 2 b. x = 1/3, x = - 7

3. A model rocket is launched upward at an initial velocity of 128 ft. per second. It’s height “h” above ground can be modeled by h(t) = - 16t + 128t.

a.) Complete the table starting at x = 2 and incrementing by 1

b.) Determine when the rocket hits the ground

c.) Determine when the rocket is 240 ft. above the ground on the way up

d.) Determine the maximum height of the rocket

4. Nancy stands at the top of an 282 ft cliff and throws her College Algebra book directly upward with an initial velocity of 25 feet per second. The height of her book above the ground t seconds later is given by h(t) = - 16t + 25t + 282, where h is in feet.

a. Use your grapher to make a table of values for the height equation starting at 0 and incrementing by 0.5 seconds.

t H(t)

20

b. Graph the equation and use the table of values to help you choose the window.

c. What’s the highest altitude Nancy’s book reaches? Use the table and the trace key.

d. How long does it take for her book to reach the maximum height? Use the table and the trace key.

e. When does Nancy’s book pass her on it’s way down? Remember she’s standing on a 282 ft cliff.

f. How long will it take her book to reach the ground?

Sec. 6.1starting on page 485 homework: 1a-e, 3, 7, 11, 13, 15, 17, 2329, 33

Sec. 6.2 Solving Quadratic Equations

In this section we will solve quadratic equations using the quadratic formula. The quadratic formula can be used when factoring is not possible.

When using the quadratic formula make sure your equation is in standard form with all terms on the left hand side and the right hand side equal to zero. That is, ax + bx + c = 0 where the quadratic formula is defined by

x =

1. Solve the following using the quadratic formula, write answers in exact form and then round to 3 decimal places where possible.

a. 3x + 9x = -5 b. 2x = 4x + 3

t H(t)

21

2. An object is propelled upward from a height of 80 feet at an initial velocity of 80 ft per second, then it’s height after t seconds is given by the equation h = -16t + 80t +80 where h is in feet.

a. Construct a table of values showing the object’s height in 0.5 second intervals after it is propelled.

b. What is the height of the object in 2 seconds on the way up?

c. How long will it take the object to reach the ground? Solve algebraically.

Sec. 6.2 starting on page 499 homework: 27, 31, 39a-c

Sec. 6.3 Graphing Parabolas

A quadratic equation is nothing more than a polynomial that has degree 2. We will begin discussing quadratic equations that are in “vertex form”. The nice thing about them written in “vertex form” is that they are much easier to graph by hand, and we can easily identify many of it’s characteristics like the vertex, line of symmetry and the x and y intercepts.

*Note that when we talk about “characteristics” we are referring to a comparison to the standard parabola which is defined by f(x) = x . This graph opens upward with the vertex at the origin and is symmetric about the y-axis.

Given f(x) = a x What does the “a” value describe?

t h

22

“a” describes the concavity, that is how wide or narrow it opens, if “a” is positive then the parabola opens upward (concave up) if “a” is negative then it opens downward (concave down) Concavity: If “a” is greater then 1 the graph opens narrower If “a” is between 0 and 1 then the graph opens wider

1. State whether the graph opens downward or upward. State whether the graph opens narrower or wider.

a. y = 2.5 b. f(x) = - 2.5 c. f(x) = 0.24 d. y = - 0.24

The vertex form of a quadratic equation is defined byf(x) = a(x – h) + k

What does the (h,k) describe?

The (h,k) is called the vertex, it is where the minimum or maximum occurs on the parabola. If you draw a vertical line through this point then you will see that the parabola is symmetric about this line. To find the line of symmetry which is denoted by “x = h”, simply set x – h = 0 than solve for x. The vertex also describes the shifting of the parabola with respect to y = . For example if the vertex is (- 2, 3) then we know that the graph was shifted left 2 then up 3. Sometimes it’s easier to first determine the vertex then describe the shifting.

2. i. Determine if the parabola is concave up or down. ii. State the vertex. iii. State the line of symmetry. iv. State the shifting v. opens wider or narrower a. f(x) = (x – 9) b. y = x - 12

c. y = (x – 1) + 2 d. (x + 3) - 1

e. f(x) = 4(x – 8) + 9 f. y = (x + 9) - 11

Finding the vertex from standard form y = a x + bx + c

The “h” value or that is the x-coordinate is x =

23

To find the “k” value or that is the y-coordinate, simply plug the x-value that you found from step 1 into the equation to find the y. (Remember when we evaluated equations earlier in the semester.)

For example, let’s say y = 3x - 6x + 1 the x-coordinate will be = 1 Now when we plug

1 into the equation we get y = 3(1) - 6(1) + 1 = - 2. Therefore the vertex for the equation is (1, - 2)

3. Do the following by hand, show all work, do not use decimals.

o Find the vertex using the formula above, do by hand. (Remember to write the vertex in ordered pair form)

o State whether there is a minimum or a maximum point on the graph. o Find the coordinates of the intercepts.

a. y = - 3x - 12x + 4 b. f(x) = 2x + 4x – 2

Determining the nature of solutions from the discriminant

Recall the quadratic formula, x = , is called the discriminant

There are 3 case scenarios that determine the type of solutions

I. Disc. is negative , that is < 0, there are 0 real solutions (Quantity under the radical is negative, there are 0 real solutions)

When the graph does not cross the x-axis, then there will be no “real” solutions

II. Disc. = 0 that is = 0, there is 1 real s olution (Quantity under the radical is 0, there’s 1 real solution)

When the graph just touches the x-axis there will be 1 real solution at the x-intercept

24

III. Disc. is positive, that is > 0, there are 2 real solutions

(Quantity under the radical is positive, there’s 2 real solutions)

When the graph crosses the x-axis twice, then there will be 2 real solutions at those x-

intercepts

* Remember solutions exist at the x-intercept(s) Disc. # of real solutions Positive 2 Zero 1 Negative 0

4. Use the discriminat to determine the nature of the solutions.

a. x - 2x + 3 = 0 b. 3x + 5x = 9 c. 2x - 4x + 2 = 0

Sec. 6.3 starting on page 510 homework: 1a-d, 3a-d, 15a, 15c, 17, 21, 29, 30, 43a-c, 44c

Sec. 6.4 Problem Solving

* Remember earlier that we discussed how minimums and maximums occur at the vertex with quadratic equations. In business, we are most concerned with minimizing costs and maximizing revenue and profit.

1. A projectile is thrown upward where it’s distance above the ground after t seconds is given by h(t) = - 16t + 532t. Use your calculator. Find it’s maximum height and the time it takes to reach it’s maximum height.

25

2. Bob owns a watch repair shop and has found that the cost of operating his shop is given by C(x) = 4x - 344x + 61 where x is the number of watches repaired.

How many watches must he repair to have the lowest cost and what will the total cost be?

Break-Even: Break-even is when a company has zero loss or zero profit. The profit equation is: profit = revenue – cost. Therefore break-even is when, profit = 0 or revenue – cost = 0Remember to write the break-even points in ordered pair form.Remember when asked to find a minimum or maximum value use x = -b/2a!

There are 2 different ways you can solve these word problems. You may choose either way for the test.

i. Find the profit equation, that is revenue – cost = 0, solve for x by using the quadratic formula. Plug the x-value into either the revenue equation or the cost equation to get the y value. Plugging into the revenue equation is always easier.

ii. Put the profit equation under y1, the revenue equation under y2, find the point of intersection using your grapher. You will be given the window on your test. This method is the easiest. 3. Juan’s Ladder company produces ladders and it costs his company C = .065x + x + 1500 to make x number of ladders. His company receives R = 60x dollars in revenue from the sale of each ladder. Round all answers to the nearest whole number.

a. Write a profit equation for his company.

b. Use the window [- 100, 1000, 40] by [- 100, 80000, 1000] to find the break-even points on your grapher.

c. Now use the quadratic formula to find the break-even points.

d. Find the value of x for which profit is a maximum.

Sec. 6.4 starting on page 524 homework: 5, 7, 17a-c, 21a-b, 22a-b, 49 use the window [ - 10, 2000] by [ - 10, 22000], 51use the window [ - 10, 500] by [ - 10, 270000]

Sec. 6.5 Graphing Inequalities

Recall in chapter 1 when we solve equations graphically then also solve where f(x) > 0 etc. Remember solutions are at the x-intercepts, for inequalities you want to start off by finding the x-

26

intercepts.

1. Using the graph below find where: a. (x + 5)(x – 2) = 0

b. (x + 5)(x – 2) > 0 c. (x + 5)(x – 2) < 0

2. Use your grapher to solve x + 4.9x – 10.56 > 6, with the window [ - 9.4, 9.4] by [ - 50, 50]

Recall the zero-product property in which if ab = 0 then a = 0 or b = 0

For example, (x – 3)(x + 2) = 0, set x – 3 = 0 solve for x then set x + 2 = 0 and solve for x. 3. Solve (x – 3)(x + 2) > 0, first solve for x then use test points to find the intervals in which the solutions lie.

4. Solve, write answer in interval notation 2x - 7x < 4

5. A rocket is fired from the ground. It’s height in feet after t seconds is denoted by h = - 16t + 297t.

a. Write an inequality for the following. In what time interval will the rocket’s height be higher than 1331 feet?

b. Use your grapher to find the time interval in which the rocket is higher than 1331 feet. Use the window [- 10, 20, 1] by [1200, 1500, 50]

27

6. A company sells personalized pen at 480 – 20p pen each month in which they charge p dollars Per pen. The company wants their revenue from the pens to be over $2160 per month.

a. Write an inequality that denotes their revenue. (Remember revenue = price quantity)

b. In what range should they keep the price of their personalized pens? Use [ - 2, 30 ] by [ - 10, 3000]

7. The cost in dollars of producing x number of items is given by C = - 0.2x + 13x + 1500 where 0 x 680. How many items can be produced if the total cost is to be under $2500? Write answer in interval notation. Use [ - 100, 900, 100 ] by [- 10, 5000, 200 ]

Sec. 6.5 starting on page 535 homework: 5a-b, 7a-b, 19, 21, 31, 35, 51, 53, 55

Chapter 2 Modeling with Functions

Sec. 2.1 Non-linear models

In this section we will be solving by extracting roots. It’s important to remember that the opposite of squaring is taking the square root and vica versa.

Extracting roots: ax + c = 0, isolate the x term, then take the of the other side.

Example 1, solve x - 11 = 0, first put the 11 on the right, x = 11 now take , x =

Example 2, solve -3(x + 2) + 15 = 0, first put the 15 on the right, - 3(x + 2) = - 15

now divide both sides by – 3, (x + 2) = 5 now take the x + 2 and set equal to , x + 2 = now solve for x, x = - 2

Example 3, solve 2(x – 9) – 32 = 0, 2(x – 9) = 32, (x – 9) = 16, x – 9 = x – 9 = 4 x – 9 = 4 x – 9 = - 4 x = 4 + 9 x = - 4 + 9 x = 13 and x = 5

28

Solve the following, give exact answers then round to 3 decimal places where needed.

1. a. 5x = 30 b. = 12

c. (2x – 5) = 4 d. (5t – 8) = 17 e. 7(x + 8) – 21 = 0

2. Solve for the specified variable. D = , solve for x

Compound Interest: B = P(1 + r) , A is the amount of the money in an account compounded annually for n years at an interest rate r and P is the principal.

3. Alba wants to invest $5500 in a savings account that pays interest compounded annually.

a. Write a formula for the balance B in her account after 3 years as a function of the interest rate r .

b. If Alba wants to have $6750 in her account after 3 years, what would the interest rate r have to be?

Again recall that when solving an x equation we take the square root of both sides.Therefore when solve a square root equation we square both sides.

3. Solve the following, don’t forget to check your answers. Isolate the term then solve.

a. = 3 b. + 2 = 13 c. 2 = 6

Sec. 2.1 starting on page 133 homework: 1, 3, 5, 7, 9, 13, 31, 33, 35, 36, 51, 52, 55a-b

29

Sec. 2.2 Some Basic Functions

This section goes over the 8 basic functions that are often used. There will be matching of these basic functions and shifting of them on the final exam. In this section you will be required to graph by hand shifting and or transformations of only y = and y =

Turn to page 145 to see the 8 basic graphs that are shown below. Note that the y = graph is incorrect in the book, the correct graph below.

1. y = x 2. y = x 3. y =

4. y = 5. y = 6. y = x

7. y = 8. y =

First lets go over some basic absolute value problems. A few of these will be on the test.

Simplify 1. a. - b. 2 - c. 3 - 7

In section 2.3 shifting/transformations is discussed, we’ll go ahead and discuss it now.

30

Given y = f(x) and y = f(x) + k, the graph will be shifted up “k” units. Given y = f(x) and y = f(x) – k, the graph will be shifted down “k” units.

For example, y = and y = - 2 is shown below, state the shifting.

Given y = f(x + k) is the graph of f(x) shifted left k units. Given y = f(x – k) is the graph of f(x) shifted right k units.

The graph of y = and y = is shown below, state the shifting.

Recall in the earlier chapters how we solved equalities and inequalities from a graphical perspective. Assume there are arrows on the graphs.

2. The graph of f(x) = x - 6 is given below. a. Decsribe the shifting

b. Solve x - 6 = 21 c. Solve x - 6 < 21 d. Solve x - 6 > - 6

31

3. The graph of f(x) = is given below. a. Decsribe the shifting

b. Solve = 0 c. Solve < 5 d. Solve > 3

Piecewise-defined function: Piecewise-defined functions are functions represented by more than one equation. Sometimes an application can be modeled by a linear equation then it changes to a quadratic model.

4. Graph the piecewise-defined functions by hand a. y = { if x and y = { – x + 2 if x > 1

b. y = { - x + 3 if x 2 and y = { x - 2 if x < 2

32

Sec. 2.2 starting on page 148 homework: 1a-b, 5, 19a-c, 23a-c, 25, 26, 27a, 27c, 28a, 28c, 41, 43

Sec. 2.3 Transformations of Graphs

Recall from sec. 2.2

Given y = f(x) and y = f(x) + k, the graph will be shifted up “k” units. Given y = f(x) and y = f(x) – k, the graph will be shifted down “k” units.

Given y = f(x + k) is the graph of f(x) shifted left k units.Given y = f(x – k) is the graph of f(x) shifted right k units.

Identify each graph as a translation of a basic function, and write the formula for the graph.

1. 2. 3.

4. a. State the basic graph. b. Describe the shifting

c. Sketch the basic graph and the function that is given, label 3 points on each for each function.

i. g(x) = ii. y = + 2

33

iii. f(x) =

Scale Factors and Reflections

Recall from chapter 6 the effect the “a” value had on the standard parabola y = , recall the

difference between y = , y = 3 and y = . Recall y = - a will result in the same graph

but reflected across the x-axis. In this section we call this “a” value the “scale factor”, if > 1 then the graph will be stretched vertically (narrower), if 0 < < 1 the graph is compressed vertically, it will open wider .

5. a. Identify the scale factor and describe how it affects the graph of it’s basic function b. Sketch the basic graph and then the given function. Label 3 points on the graphs.

i. y = ii. y = 2 and iii. y = -2

6. Given the graphs below find the following: a. State the graph as a transformation of the basic. b. State the equation for the graph.

34

Sec. 2.3 starting on page 163 homework: 1, 2, 5, 6, 7, 8, 10, 12, 14, 15, 21, 22, 23, 24, 26, 33, 51a, 53a, 65, 68, 69

Sec. 2.5 The Absolute Value Function

There are 3 types of absolute value functions that we will be solving.

The rules are as follows:

I. For = a and a > 0, set x = a then x = -a and solve for x. (There’s no solution to = -

For example: Solve the following: = 7

II. For < a set –a < x < a, then solve for x. Do the same for a

For example: Solve the following: < 8

III. For > a set x > a solve for x, then set x < - a solve for x. Do the same for For a.

For example: Solve the following: 3

Solve the following: (first isolate the absolute value expression then use one of the rules above) 1. a. - 2 < 8 b. + 10 15

35

c. 4 = 8 d. 2 + 1 11 e. = - 8

Sec. 2.5 starting on page 201 homework: 13, 18, 21, 23, 25

Sec. 2.6 Domain and Range

Finding the domain from a graph . Because the domain is related to the x-values, find the “leftmost or the smallest x-value, then find the “rightmost” or the largest x-value then write answer in interval notation. We will assume all the graphs have endpoints, that is there are no arrows.

Finding the range from a graph. Because the range is related to the y-values, find the “bottom most” or the smallest y-value, then find the “top most” or the largest y-value then write answer in interval notation. Again, we will assume all the graphs have endpoints, that is there are no arrows 1. From the graphs below, find the domain and range using interval notation. a.) Domain ______________ b.) Domain _____________

Range _______________ Range ______________

c.) Domain ______________ d.) Domain _____________

Range _______________ Range ______________

36

Finding the domain and range given a function:

When given an equation that contains a denominator, simply set the denominator 0, then solve for x to find the domain. Graph the function to assist in finding the range.

2. State the domain only for the following functions.

a. f(x) = b. f(x) =

When given an equation that contains a , set the radicand 0 then solve for x. Graph the function to assist in finding the range.

3. State the domain and then graph with your grapher to find the range.

a. f(x) = b. f(x) = - 3

Sec. 2.6 starting on page 211 homework: 1, 5, 26a-b, 29a-b

Sec. 3.1 Variation

37

An example of variation is the area of a circle. A = r states that the area of a circle with radius “r” is directly proportional to the square of it’s radius, the constant of proportionality is . Notice how the dependent variable “A” increases as the radius “r” increases.

Direct Variation with a Power: If y varies directly with x to the power n, then y = kx , k and n are positive exponents with “k” being the “constant of proportionality” or the constant of variation”.

1. Write an equation for the following then find the constant of proportionality.

a. L varies directly to V. L = 10 and V = 2

b. G varies directly to the square of h. G = 26 and h = 3

If y varies inversely with x to the power n, then y = .

2. Write an equation for the following then find the constant of proportionality.

a. y is inversely proportional to x squared. y = 12 and x = 2

b. t varies inversely to m. t = 9 and m = 5

3. a. Use the table of values to find the constant of variation b. Write y as a function of x. That is find the equation of variation. c. Fill in the rest of the table with the correct values.

y varies directly with x y varies directly with the square of x

y varies inversely with x y varies inversely with the cube of x

x y4

7 17.5

11

32.5

x y10.8

- 1

1.2

5 30

x y- 1

- .7

14 -.25

20

x y- 10 -.002

2

.016

11

38

See page 250 # 15, 17, 19

Solving applications: i. Write the equation of variation. ii. Use 2 data points to find “k”. iii. Plug “k” into your equation. iv. Use this equation containing to answer any following questions.

Sec. 3.1 starting on page 249 homework: 13a-c, 15a-c, 17a-c, 19a-c, 21, 23, 25

This is the formula sheet that you will be able to use on the Chapter 4 and Chapter 5 tests. Please have it with you in class also.

Formula Sheet for Chapter 4 and 5

P(t) = P b P = P (1 + r) P = P (1 - r) A(t) = P(1 + )

y = log x then x = b y = ln(x) then x = e

1. log b = 1 2. log 1 = 0

3. log b = x (basically log b = something) 4. b = x b = something

Base 10 and base “e” logarithmic properties

5. log10 = 1 ln e = 1

6. log 1 = 0 ln 1 = 0

7. log10 = x ln e = x log10 = something

and the ln e = something

8. 10 = x e = x e = something

9. product property: log (xy) = log x + log y

39

10. quotient property: log = log x - log y

11. power property: log x = k log x

12. log x = or log x = .

13. i. x ∙ x = x ii. (x ) = x iii. = iv. .

Chapter 4 Exponential Functions

Sec. 4.1 Exponential Growth and Decay

Compare the graphs. One is denoting exponential growth, the other linear growth. f(x) = 3 f(x)= 3x both graphs

y1 = 3x y2 = 3

From the table above notice how for each unit increase in x 3 is added to the value of y1. However, for each unit increase in x, 3 is multiplied to the value of y2. This is one example of exponential growth.

Now lets compare y = 3 to y = 2 3 (this is the darker graph)

y = 3 y1 = 3 and y2 = 2 3

40

1. From the table above determine the initial value for when x = 0 for both graphs.

2. From the table and graphs, state the growth for each.

The Exponential Growth function is given by P(t) = b for b > 1, denotes the initial value, b denotes the growth factor and t is for time.

3. A rancher started with 65 alpacas and finds that the number of alpacas increases by a factor of 2 every year.

a. Write the exponential growth function. b. Complete the table starting at 0 and incrementing by 1.

t 0 1 2 3 4 5P(t)

c. Observing your table of values give an appropriate graphing window [ , ] by [ , ]

d. Graph the function using your calculator.

e. How many alpacas does the rancher have after 9 years?

Note: If the problem states that there is an increase by a factor of “b” every month, week

year etc. then the equation is simply P(t) = b . If it states “every 2 weeks, 6 months, 3 years etc. then the equation will be P(t) = b , P(t) = b , P(t) = b etc.

***Hence the words “every n amount of time t” will be t/n.

4. A less experienced rancher starts off with 65 alpacas and finds that his number increases by a factor of 2 every 2 years.

41

a. Write the exponential growth function. b. Complete the table starting at 0 and incrementing by 1.

t 0 1 2 3 4 5P(t)

c. How many alpacas does this rancher have after 9 years?

d. Comment on the differences in the tables above.

5. A colony of bacteria starts with 450 organisms and triples every 3 weeks.

a. Write a formula for the population of the bacteria colony after t weeks.

b. How many bacteria will there be in 2 weeks?

c. How many bacteria will there be in 7 weeks?

Recall the compounded annually interest formula from chapter 2 A = P(1 + r) , where P is the amount invested, r is the interest rate, t is time in years and A is the amount in the account after t years.

In this chapter we call the (1 + r) the “growth factor” where “r” is the percent increase.

6. Since the opening of a manufacturing facility, the population of a certain city has grown according to the model f(t) = 210,000(1.015) , where t is the number of years since 1999.

a. State the initial value. b. What does 210,000 represent? c. What is the growth factor?

d. What does t = 0 represent? e. What is the percent increase? f. What was the population in 2005?

7. Reggie invests $6,500 into a retirement certificate of deposit account that pays 4.5%, interest compounded annually. a. Write an equation that denotes exponential growth.

b. How much will he have in his account after 5 years?

8. Lakisha bought a house in Alameda California for $400,500. Since 1995 the prices of homes in that area rose on average 7% per year.

42

a. State the initial value of the home.

b. What does t = 0 represent?

c. What is the percent increase?

d. Write an equation that denotes exponential growth.

e. How much was her house worth in 2000?

The Exponential Decay function is given by P(t) = b for 0 < b < 1, denotes the initial value, b denotes the decay factor and t is for time.

y1 = 3 y2 = y1 = 3 and y2 =

For percent decrease word problems, b = 1 – r is the decay factor and “r” represents the percent

decrease.

9. Vic’s boat cost $14,000 in 2000 and has depreciated by 8% every 2 years.

a. State the decay factor and the percent decrease.

b. Write a function that describes the exponential decay.

c. How much was his boat worth in 2005?

10. The number of bass in Lake Campbell has declined to one-third of it’s previous number due to pollution every 4 years since 1990 when the number of bass was estimated at 5,000.

a. State the . b. Write a function that describes the exponential decay.

c. How many bass were there in 1995?

Note: Recall that when solving an equation we take the “n the root” of the other side.

43

11. Solve for the variable, give exact answer then round estimate to 3 decimal places, assume all positive answers.

a. 1875 = 3b b. 151,875 = 20,000b c. 10.56 = 12.4(1 – r)

12. A new Toyota Corolla cost $18,000 in 2005. The value of course decpreciates exponentially over time. A 2 year old Toyota Corolla cost $14,600.

a. Write an equation that denotes the value of the Toyota Corolla.

b. Find the decay factor and the percent rate of depreciation.

c. How much would a 5 year old Toyota Corolla?

13. Given the table below find the following.

a. Determine whether the table represents exponential growth or decay.

b. State the initial value.

c. Write an equation that denotes exponential growth or decay.

d. Find the growth or decay factor.

e. Find the percent rate or percent decrease.

f. What is the q-intercept?

g. Write a formula that represents the data in the table.

h. Complete the table.

14. Given the graph below find the following.

a. Find the initial value.

b. Find the growth factor b. (no decimals) c. Write a formula for the function.

p 0 1 `2 3 4q 84 30.24 18.144

44

For #15 and 16, assume that each population grows exponentially with constant annual percent increase, r. 15. a. The population of Ohio was 11,353,140 in 2000. Write a formula in terms of r for the population of Ohio t years later.

b. In 2006 the population of Ohio was 11,478,006. Write an equation and solve for r.

c. What was the annual percent increase to the nearest hundredth of a percent?

16. The population of Tiffin was 2,500 in 1995 and tripled in 4 years. a. Write an equation that denotes Tiffin’s population growth.

b. Find the growth rate r.

c. Find the annual percent of increase to the nearest hundredth of a percent?

45

Sec. 4.1 starting on page 334 homework: 5a-b, 7a-b, 9, 11, 13, 15, 17, 33, 37, 41a-c, 45, 49, 55, 57, 63a-b, 65a

Sec. 4.2 Exponential Functions

Recall in sec. 4.1 that an exponential growth function was denoted by P(t) = b for b > 1, and the exponential decay function was denoted by P(t) = b for 0 < b < 1. We were working with all applications in sec. 4.1. Both functions are exponential, it’s just that with applications we use the words growth and decay and also we use the above mathematical notation.

In section 4.2 we also work with exponential functions but not within applications, therefore we will be using slightly different notation. Instead of using the words growth and decay we will be indentifying whether or not the functions are increasing or decreasing.

Also in this section we will be working with shifting, reflection, domain, range, x and y intercepts as we did in previous chapters.

An exponential function that is increasing is denoted by: f(x) = a(b )

For y = a(b ) , b > 1 and a > 0 no x-intercept y-int (0,a) (recall b = 1) domain: all real numbers or that is ( - , ) range (0, ) horizontal asymptote: y = 0 increases from left to right

The graphs will have the basic shape below. The graphs are increasing from left to right. The larger the base “b” and the larger the initial value “a” the faster the graph will increase. Notice also how the graphs do not cross the x-axis, therefore there exists a horizontal “asymptote” at y = 0.

y = 10 y = 2 y = 3(2 )

Note: You will need to memorize the basic shape and their properties.

Note the same type of shifting applies with exponential functions as they did with quadratics. Withf(x) k there is a vertical shift up or down and with f(x h) there is a horizontal left or right.

y = 3 y = 3 + 4 y = 3

46

An exponential function that is decreasing is denoted by: f(x) = a(b )

(the properties below are the same for the increasing exponential) For y = a(b ) where 0 < b < 1 and a > 0

no x-intercept y-int (0,a) (recall b = 1) domain: all real numbers that is ( - , ) range (0, ) horizontal asymptote: y = 0 decreases from left to right

** Be Careful: Recall that x = , so x = etc., therefore 2 = =

Therefore exponential functions that are decreasing can also be denoted by

y = a(b ) where b > 1 and a > 0 because y = a(b ) can be re-written as y = a

I would advise for any negative exponents, rewrite with positive exponents then decide if the function is increasing or decreasing.

y = 2 y = 2 that is y = graphed together

1. Find the y-intercept (x = 0) and determine whether the functions are increasing or decreasing.

a. f(x) = 21(1.015) b. y = 1.3(3) c. y = d. g(x) =

When we replace x with –x that is given y = f(x) the y = f(-x) will be reflected across the “y-axis”. The y graphs are darkened. That is

y = 2 and y = 6 y = 2 and y = 6

47

Recall earlier the difference between y = and y = - , the y = - is the same except it is reflected across the “x-axis”.

2. Observing the graphs below comment on their differences.

y = 2 y = 2 that is y = y = - 2

3. Compare the following with y = 4 . State whether the function is increasing or decreasing and describe any shifting and reflection.

a. y = 4 - 2 b. y = 4 c. h(x) = - 4 d. y = 4 e. y = 4

4. See page 348 # 3a-b. Make a table of values and graph each function.

x - 2 - 1 0 1 2 3 4y = 2

y =

48

Lastly in this section we will start to solve exponential functions. Here in this section, the key is to rewrite one or both of the bases “the b” so they are the same, then set the exponents equal to eachother then solve for the variable.

For example, if 8 = 8 , what would x have to be in order to make the statement true?

What if instead we had 2 = 8, what would x have to be? Couldn’t we rewrite 8 as 2 , therefore we now have 2 = 2 , now it’s more obvious that x = 3.

Recall the following exponent rules to assist you in rewriting one or both of the bases in order to make them

i. x ∙ x = x ii. (x ) = x iii. = remember that 27 = 3 , 9 = 3 , = 2 etc.

5. Solve each by hand, no calculator.

a. 5 = 5 b. 3 = 9 c. 9 = 81

d. 8 = 64 e. 4 = f. 49 = 7

g. 27 = 9 h. 2 ∙ 4 = 8

49

Sec. 4.2starting on page 348 homework: 1a-d, 3a-b, 5a-b, 7a-d, 27, 31, 33, 35, 36

Sec. 4.3 Logarithms Recall in section 4.1 the following word problem: Since the opening of a manufacturing facility, the population of a certain city has grown according to the model f(t) = 210,000(1.015) , where t is the number of years since 1999. We were then asked to find the population in 2005. We simply plugged 6 into t.

What if we wanted to know the year in which the population reached 250,000? We would have to solve for the “t”. In order to do so we have to learn the properties of logarithms.

Let’s say we have y = 2 and we want to solve for x, we can do so using the property of logarithms. We will learn that if y = 2 then x = log y.

The logarithmic function is denoted by y = log x where b > 0, x > 0 and b # 1.and can be written as x = b

The logarithm y = log(x) is called the “common logarithm”, the base is 10 though it’s not written.

Recall the exponential function from sec. 4.2 then observe the graph of y = log(x). y = 10 y = log(x) both graphs along with y = x

Notice on your calculator how the 10 key and the log(x) are right behind eachother? Likewise with the x and the . The opposite of squaring is taking the square root, hence the opposite of y = 10 is y = log(x), we say they are “inverses of eachother”. If 2 functions are inverses of eachother that means that when you graph them they are reflections on one another about the line y = x. This is shown in the last graph above. We will be working with inverses in chapter 5.

In this section we will mainly be rewriting from one form to another, evaluating without a calculator and will be solving just a few word problems.

50

1. Find each logarithm without using a calculator. (Look at the rule/formula sheet. You may need to rewrite using the fact that if y = log x then x = b )

a. log (25) b. log ( ) c. log (8)

d. log (7) e. log (6) f. log (1)

g. log(10) h. log i. log(1)

2. Evaluate with a calculator to 3 decimal places. What happens as the x value increases? Comment on the amount of increase.

a. log(12) b. log(120) c. log (1200)

3. Rewrite each equation in logarithmic form. (Remember that if then x = b y = log x )

a. 8 = 2 b. x = 7 c. 5 = q

d. 4 = y e. 5 = 25 f. m = 4.5

Solving exponential equations

Isolate the term that contains the exponent Rewrite the exponential equation to logarithmic form, remember if x = b

then y = log x. Solve for the variable.

Note that 2(10 ) # 20 . Also # log7

51

For example: Solve and round to 3 decimal places.

2(10 ) = 50 Isolate the term with the exponent by dividing both sides by 2

10 = 25 Rewrite to log form: 3x = log(25) Solve for x: x = (exact solution)

Note that you cannot divide the 25 by the 3, first find the log of 25 then divide by 3

x = .466 is the approximate solution.

4. Solve the following, give exact answers where possible then round to 3 decimal places if needed.

a. 36 = 10 b. 1120 = 40(10 ) c. 5(10 ) = 365

d. 2 + 4(10 ) = 10 e. 24(10 ) – 23.5 = 364

5. The population of a certain city increased during the years 1990 to 2000 according to the formula P(t) = 2,456,000(10) . Where t is the number of years since 1990.

a. What was the population in 1998?

b. When will the population reach 5,350,000? Round to the nearest year.

Sec. 4.3 starting on page 363 homework: 1 – 7 odd, 11, 15, 17, 23a, 31a-d, 35-41 odd, 45, 49a, 49c

Sec. 4.4 Properties of Logarithms

In this section we will mainly be simplifying, rewriting and solving. There will be of course some applications.

On page 46 from the formula/rule sheet we will be using the following properties and rules:

product property: log (xy) = log x + log y

52

quotient property: log = log x - log y

power property: log x = k log x

i. x ∙ x = x ii. (x ) = x iii. = iv.

1. Using the properties above, combine into one logarithm and simplify.

a. log x + log (8) b. 3log(x) + log(y) c. log x - log y

d. log(3a) + log(b) e. log (2x – 7) – log b

f. 7 log (x) + 8 log(y) g. log(x - 25) - log(x + 5)

2. Use properties to expand in terms of simpler logs.

a. log(7x) b. log c. log[x (y )]

d. log [x (4x + 3) ] e. log

3. If log 2 = 0.43 and log 3 = 0.68, evaluate the following.

a. log 6 b. log 9 c. log

2 methods in solving exponential equations that contain a base other than 10.

53

Method 1: i. Isolate the term that contains the exponent. ii. Take the log of both sides.iii. Use the property log(b) = xlog(b) then solve for x.

For example: Solve 2 = 6, log(2) = log(6), xlog(2) = log(6),

x = is the exact answer, 2.585 is the estimate. Note: log(2)!

Method 2: i. Isolate the term that contains the exponent. ii. Rewrite using the fact that if y = b then x = log y. iii. Use the change of base formula to solve for x.

Change of base formula: log x =

4. Solve , give exact answer then round to 3 decimal places.

a. 4 = 9 b. 4 = 7 c. 2.13 = 8.1

5. In 2003 the Soccer for Charity organization had 2,575 members with an annual growth rate of 4.5%.

a. Write a formula for the membership in the Soccer for Charity as a function of time, assuming that the organization continued to grow at the same rate.

b. How many members did they have in 2006?

c. When will the organization have 4,500 members? Round to the nearest year.

The formula for compound interest is as follows: A(t) = P(1 + )

A(t) is the amount in the account after P dollars is invested at an interest rate “r” after “t” years and “n” is the number of times compounded per year. If the money is compounded monthly n = 12, quarterly then n = 4, semi-annually n = 2.

6. What rate of interest is required so that $750 will yield $975 after 3 years if the interest is

54

a. Compounded quarterly?

b. Compounded monthly?

Sec. 4.4 starting on page 373 homework:

Chapter 5 Logarithmic Functions

Sec. 5.1 Inverse Functions

If 2 functions are inverses. then their graphs are reflections of eachother about the line y = x.

Recall from sec. 4.3 the following.

y = 10 y = log(x) both graphs along with y = x

Observe the table of values:

Note: If 2 functions are inverses of eachother then their domain and range values are interchanged.

x y0 11 102 1003 1000

x y1 010 1100 21000 3

55

The inverse of f(x) is denoted by f (x). Note that f (x)

1. Let f(4) = 7, f(5) = 10, f(6) = 13 and f(7) = 16.

a. Make a table of values for f(x) and another table for it’s inverse f (x) .

Steps to finding the inverse given a function

I. Replace f(x) with y II. Solve for x. III. Replace x with f (x) and y with x.

For example: f(x) = -3x – 1, y = -3x – 1, 3x = -y – 1, x = , therefore f (x) =

2. Find a formula for the inverse then graph using your calculator to verify.

a. y = 2x + 4 b. f(x) =

3. Find the inverse then make a table of values for f(x) = 4x - 8

4. Find the function for the following then graph it’s inverse.

x f(x)

x f (x)

x f(x)

x f (x)

56

5. Use the graph below to draw it’s inverse beside it. Label all the ticks and label 2 points on the graph.

One-to-One Function and Horizontal Line Test

A function is one-to-one if for each output there exists only one input.

Note that example below is not one-to-one. This is a quadratic function; if we draw a horizontal line the line would intersect the graph in more than one place. This is called the horizontal line test. The graph represents a function but not a one-to-one function. Hence the function does not have an inverse.

6. Determine whether the graphs and functions below have inverses.

57

a. b. c.

d. y = - 4.9x + 7 e. f(x) = 3 - f. y = 4x

Sec. 5.1 starting on page 415 homework: 1a-c, 23, 27, 28, 43a

Sec. 5.2 Logarithmic Functions

Recall from sec. 5.1 how to find an inverse, recall that if y = log x then x = b

1. Given y = 3

a. Find it’s inverse b. Make a table of values c. graph both functions

xf (x)

- 1 0 1 2x - 1 0 1 2f(x)

58

2. Compare the following with y = log(x) to describe the shifting or the reflection.

a. y = -log(x) b. f(x) = log(x – 3) c. y = log(x) – 4 d. f(x) = log(-x)

Note the 2 properties: log10 = x and 10 = x

3. Simplify: a. log10 b. 10 c. 10

4. Given f(x) = log x Solve for x a. f(x) = 3.45 b. f(x) = -1.76

5. Convert each logarithm equation to exponential form.

a. y = log x b. log (4) = w c. log(5) = p d. log (g) = d

Note: When asked to solve a logarithm convert to exponential, when asked to solve an exponential convert to a logarithm.

Note: Evaluate log(-10) Recall that for log (x), x > 0, therefore check your answers when solving to make sure the x-values are not where the log is undefined.

59

6. Solve for the unknown value. Leave answer in exact form where needed. Remember to combine logarithms into one by using the previously learned properties.

a. log 25 = 5 b. log 49 = c. log (x) = 12

d. 3log ( x) = 4 e. log x = 2 f. log(x) = 2

g. 4log (x) = 12 h. log (4x - 1) = 2 i. log x = -2

j. 5 + 2logx = 13 k. log x + log(x – 3) = 1

l. log (5x – 1) - log (x) = 2 m. log (x + 8) + log (x + 2) = 2

Sec. 5.2 starting on page 430 homework: 1, 3, 19a-c, 20a-c, 25a-d, 29a-c, 31, 33, 37-53 odd

Sec. 5.3 The Natural Base

The number “e”

The natural exponential function is the function y = e . Many real world applications within the biology and finance involves the irrational number e. This specific exponential function is defined by y = e . Because “e” is approximately equal to 2.718, it’s graph lies between y = 2 and y = 3 .

Graph all 3 functions under the “z-decimal window”.

As with the other exponential functions, this specific function can also describe growth or decay. Notice how they are symmetric about the y-axis, therefore they are “even functions”.

60

y = e . y = e .

The “natural logarithm”

The logarithm function with base “e” , that is y = log x which can be rewritten as x = e is called the “natural logarithm. The natural logarithm function is written as y = ln(x). RecallThat y = log (x) is written y = log(x). The natural logarithm y = log x is written as y = ln(x).

Notice how the graphs of y = ln(x) and y = e are inverses of each other. Just as y = 10 and y = log(x) were.

y = lnx y = e graphed together

Keep in mind that the natural and common logarithmic functions are merely log functions with unique bases that are often used in science and business applications.

Converting log functions to exponential and vica versa

To solve a natural exponential function we convert it to the natural logarithm, to solve a natural logarithm, we convert it to the natural exponential function. When solving we do the same as with the other exponential and logarithmic functions.

y = ln(x) is rewritten as x = e y = e is rewritten as x = ln(y)

1. Solve for x. Give exact answers where possible, then round to 3 decimal places.

a. 5.9 = e b. 192 = 16e c. 2 + 4(10 ) = 10

61

d. 4 - e = 2 e. ln(x) = 3.5 f. 4.5 = 4e + 3.3

Recall the following properties given in chapter 4.

i. ln e = x that is ln e = something

ii. e = x that is e = something

iii. ln e = 1 iv. ln 1 = 0

2. Use the properties above to simplify.

a. e b. ln e c. e d. ln( ) e. ln

3. The growth of a colony of bacteria is given by Q(t) = 450e . There are initially 450 bacteria present and t is given in hours.

a. Does this model indicate that the population is increasing or decreasing? Explain.

b. How many bacteria are there after 11 hours?

c. Graph using [0, 10] by [450, 3160]. Determine the bacteria present after 6 hours.

d. Using the table function on your grapher, determine the number of hours that will pass in which the amount of bacteria reach approximately 2600.

4. The number of milligrams of a drug in a persons system after t hours is given by the function D = 30e .

62

a. Does the model indicate that the amount of drug will increase or decrease as time goes by? Explain.

b. Find the amount of the drug after 3 hours.

c. When will the amount of drug be 4.959 milligrams? Which method will you use to determine this?

d. Use the table function to interpret D(2).

e. When will the amount of the drug be 0.09 milligrams, that is almost completely gone from the system?

5. The price of an airline ticket to Paris was $850 in 1998. In 2005 the prices rose to $1900.

a. What is P if t = 0 represents 1998?

b. Use the 2005 price find the growth factor e .

c. Find a growth law of the form P(t) = P e for the price of an airline ticket to Paris.

Sec. 5.3 starting on page 446 homework: 1, 3, 5a-d, 7, 9, 11a,b,d, 13b,c, 23-29 odd, 47a-d, 48a-c, 49a-c

Chapter 7 Polynomial and Rational Functions

Sec. 7.1 Polynomial Functions

In this section we go over polynomial functions and the basic characteristics such as the degree and leading coefficient. We will also multiply polynomials and factor the sum or difference of cubes.

1. Multiply: a. (x – 3)(x + 3)(2x – 5) b, (4x – 2)(3x + 7x – 5)

2. Given: 3x + 7x – 5 – 2x Find: The degree and the leading coefficient

63

3. Without performing multiplication, give the degree of the product.

a. (3x + 7x)( 2x – 5) b. (7x – 5)(4x + 6)( 3x – 5)

Factoring the sum or difference of cubes:

x - y = (x – y)(x +xy + y ) x + y = (x + y)(x - xy + y )

4. Factor: a. x + 8 b. 27 x - 8y

Sec. 7.1 starting on page 572 homework: 1, 3, 17a-b, 35, 37, 41

Sec. 7.2 Graphing Polynomial Functions

In chapter 1 we worked with linear functions (degree 1) described by y = mx + b. In chapter 2 we worked with quadratic functions of degree 2 described by y = ax + bx + c. A linear function has atmost 1 zero or solution, a quadratic function has atmost 2 zeros or solutions.

Therefore a polynomial of degree n will have atmost n solutions/zeros/x-intercepts. A polynomial of degree 3 is called a “cubic”, and a polynomial of degree 4 is called a

“quartic”.

Recall that a quadratic had a minimum or a maximum. At this minimum point the graph changes from decreasing to increasing. The graph with a maximum changes from increasing to decreasing. We called the point a vertex. With higher degree polynomials we call these “turning points”.

The graph below has 2 turning points. The window is [-5, 5] by [-10,10] There is a turning point at (-2.5, 10) and another turning point at (2.5, -10)

Characteristics of functions/polynomials64

function degree # of turning points end behavior linear 1 none positive or negative slope

y = 2x – 1 y = - 2x -1

quadratic degree 2 1 turning pt. increases left and right if a >0 decreases left and right if a <0 (positive leading coefficient) (negative leading coefficient)

y = 2x + 3x y = - 2x + 3x

cubic degree 3 0 or 2 turning pts.

decreases left, increase right if a >0 increases left, decreases right if a <0 (positive leading coefficient) (negative leading coefficient)

y = 4x + 2 (0 turning pts.) y = - x + 2x (2 turning pts.)

65

quartic degree 4 1 or 3 turning pts.

increases left and right if a >0 decreases left and right if a <0 (positive leading coefficient) (negative leading coefficient)

y = x + 2x (1 turning pt.) y = -x + 2x (3 turning pts.)

1. Describe the long term behavior. How many intercepts? How many turning points? Graph using your calculator to decide.

a. - x - x + 6x b. y = 0.3x + 1 c. y = - x + 3x + 1

2. Use your calculator to graph to find the following for y = x + 4x - 11x - 30

a. The x-intercepts. Use xmin = -10 and xmax = 10 and adjust the ymin and ymax to get a good graph.

b. Write the polynomial in factored form.

3. Find the zeros of each by factoring, where needed write answer(s) in exact form

a. y = 4x - 28x + 49 b. P(x) = 4x - 28x + 49

c. f(x) = 8x - 4x d. r(x) = x - 11x e. p(x) = x - 25x

4. Find a possible equation for the polynomial whose graph is shown below. Write answer in factored form.

66

Sec. 7.2 starting on page 584 homework: 1, 3, 11, 13, 21a-b, 23a-b, 39a, 41a, 43a, 44a, 45a, 47, 51

Sec. 7.4 Graphing Rational Functions

Definition: A rational function is nothing more than a polynomial divided by a polynomial, denoted by

f(x) = where Q(x) # 0.

We learned earlier about vertical asymptotes, that is imaginary vertical lines in which the graph approaches but does not cross. These vertical asymptotes occur at the x-values that make the denominator = 0, or that is where the function is undefined.

For example, the graph below shows the equation y = , note that when x = 3 the

denominator = 0.

The graphs below show the function under 3 different window, the standard window and the z-decimal.

The standard window shows the equation of the vertical asymptote whereas the z-decimal does not.

Notice how the graph using the z-decimal approaches x = 3 but does not cross the graph at that particular value.

Also note that at x = 3 the function gives you an error using the table, which implies that when x = 3 the function is undefined.

standard window (asymptote shown at x = 3) z-decimal window f(3) is undefined

67

To find a vertical asymptote set the denominator = 0 and solve for x.

1.) Find any vertical asymptote(s) for the following, graph to verify your answers.

a. f(x) = b. y =

c. f(x) = d. f(x) =

2.) Does f(x) = have a vertical asymptote? Why or why not?

Determining horizontal asymptotes

For the rational function f(x) = =

n is the degree of the numerator and is the leading coefficient

m is the degree of the denominator and is the leading coefficient

3 case scenarios

I. If n < m, there’s a horizontal asymptote at y = 0. (the x-axis)

II. If n = m, there’s a horizontal asymptote at y = .

III. If n > m, there is no horizontal asymptote.

3.) Find any horizontal asymptotes (if they exist) for the following.

68

a. f(x) = b. y = c. f(x) = d. y =

4.) Find any horizontal or vertical asymptotes then graph. Denote all asymptotes with dashed lines.

a. y = b. f(x) =

c. y = d. f(x) =

Sec. 7.4 starting on page 615 homework: 13-17 odd parts a and b

69