COLLEGE ALGEBRA

Math 1314

Turnell

Updated 1/18/2018

[Type the document subtitle]

Review of Quadratic Formula

The quadratic formula is derived from completing the square on the general equation: 2 0ax bx c

You MUST memorize the formula: 2 4

2

b b acx

a

Process: 1. Write the equation in standard form: 2 0ax bx c 2. Identify , , and a b c . 3. Substitute numbers into formula. 4. Carefully do the arithmetic under the square root sign. 5. If possible, simplify the radical. 6. If possible, reduce the fraction.

1. 23 2 4 0x x 2. 28 5( 1)x x 3. 4 ( 1) 5 0x x

Page 1

Quadratic Types of Equations

ReviewofFactoring:In previous math classes, you have learned to solve quadratic equations by the factoring

method. 24 8 3 0x x 25 19 4 0x x

QuadraticTypesofEquations:We have equations that look like a quadratic, but have different exponents. Some

examples of these equations are:

4 24 8 3 0x x 2 1

3 35 19 4 0x x 2 16 7 3 0x x Solve by factoring:

1. 4 24 8 3 0x x 2. 2 1

3 35 19 4 0x x

3. 2 16 7 3 0x x 4. 1 1

2 44 9 2 0x x

Page 2

Equations with Fractional Exponents

ReviewofExponents:Remember that a fractional exponent can be written in radical form.

2

3 32x x

52 2

5x x

If you encounter an equation that has a variable raised to a fractional exponent, you solve it by raising both sides to the appropriate power.

2

3 32x x

52 2

5x x

Solve:

5. 3

2 26 7 27x x 6. 2

32 9x

Try these on your own:

43 16x

321 125x

Page 3

Page 4

Quadratics, Quadratic Types of Equations and Rational Exponent Equations

Use the quadratic formula to solve the following. Simplify all answers. 1. 29 2 2 0x x 2. 25 4 4 0x x 3. 24 4 0x x 4. 23 10 1 0x x 5. 25 2(2 5)x x 6. (7 3)( 2) 7x x x Find all solutions of the equation.

7. 4 213 40 0x x

8. 4 25 4 0x x

9. 6 32 3 0x x

10. 6 37 8 0x x

11. 2 1

3 33 2 5x x

12. 4 2

3 35 6 0x x

13. 1 1

2 42 1 0x x

14. 1 1

2 44 4 0x x

15. 1 1

2 44 9 2 0x x

16. 1 1

2 43 2 0x x

17. 2 1

3 32 5 3 0x x

18. 2 1

3 33 5 2 0x x

19. 4 2

3 34 65 16 0x x

20. 2 110 24 0x x

21. 2 13 7 6 0x x

22. 2 12 7 4 0x x

23. 2 17 19 6x x

24. 2 15 43 18x x

25. 2 16 2x x

26. 4 29 35 4 0x x

27. 3

2 24 5 64x x

28. 3

2 22 5 6 8x x

29. 237 4x

30. 232 5 9x

31. 251 4x

Page 5

Answers

1. 1 19

9

2. 2 4

5

i

3. 1 3 7

8

i

4. 5 2 7

3

5. 2 3 6

5

6. 5 3 2

7

7. 2 2, 5x

8. 2, 1x

9. 3 3, 1x

10. 2,1x

11. 125

,127

x

12. 2 2, 3 3 x

13. 1x

14. 16x

15. 1

,16256

x

16. 16,1x

17. 1

,278

x

18. 1

, 827

x

19. 1

, 648

x

20. 2 5

,3 8

x

21. 3 1

,2 3

x

22. 1

2,4

x

23. 7 1

,2 3

x

24. 5 1

,2 9

x

25. 3

2,2

x

26. 1

3 ,2

x i

27. 7,3

28. 1,2

2

29. 1,15

30. 16,11

31. 31,33

Page 6

Functions

A function is a set of ordered pairs where for every x-value there is a unique y-value. The x-values are called the domain (left to right). The y-values are called the range (bottom to top). The vertical line test can determine if a graph is a function or not. If a vertical line only crosses the graph once, then the graph is a function.

Determine whether the following is a function. Give the domain and range of each relation.

{(10,8), (6, 4), (2,0), ( 2, 4)}I {(3, 4), (3, 2), (8,9), (1,0)}K

Properties of Graphs 1. Specific Values 2. Domain and Range 3. Intercepts-where the graph crosses the axes. Find the following: Use the graph to find:

( 2)f (3)f

For what value(s) of x does ( ) 4f x ? For what value(s) of x does ( ) 0f x ? Find the x-intercept(s):_____________ Find the y-intercept:_____________

Page 7

Determine the domain, range, any intercepts and values.

( 2)f =? ( 1)f =? (2)f =?

Given ( ) 2 5f x x , evaluate each function at the given values and simplify answers.

( 3)f (2)f ( 1)f x

Page 8

Given 2( ) 3 1f x x x , evaluate each function at the given values and simplify answers. ( 3)f ( )f x ( 5)f x

Given 2

1

xh x

x

, evaluate each function at the given values and simplify answers.

1h 4h h x

Page 9

Page 10

Functions-Interpreting Graphs

Use the vertical line test to determine which of the following are graphs of functions.

1. 2. 3.

4. 5. 6. Determine if the given relation is a function. Give the domain and range of each relation. 7. 1,2 , 1,3 , 4,11 , 6,7J

8. 1,2 , 3,2 , 7,11 , 8,4L

9. 4,7 , 5,7 , 6,1 , 8,3M

10. 9,8 , 8,9 , 0,4 , 4,0J

Use the graphs of the functions f and g (given below) to find the indicated function values 11. ( 4)f 12. ( 2)f

13. (0)f 14. (2)f

15. ( 5)g 16. ( 1)g

17. (1)g 18. (0)g

Page 11

Functions-Interpreting Graphs

Determine the domain, range, and intercepts of each of the functions whose graph is given.

19.

-4 -3 -2 -1 1 2 3 4

-5-4-3-2-1

12345

20.

-4 -3 -2 -1 1 2 3 4

-3-2-1

123456

21.

-4 -3 -2 -1 1 2 3 4

-5-4-3-2-1

12345

22.

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

23.

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

24.

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

25.

-4 -3 -2 -1 1 2 3 4

-3

-2

-1

1

2

3

26.

-4 -3 -2 -1 1 2 3 4

-2

-1

1

2

3

4

5

27.

-4 -3 -2 -1 1 2 3 4

-2

-1

1

2

3

4

5

Evaluate each function at the given values and simplify answers. 28. ( ) 3 4f x x a. ( 3)f b. ( 1)f x c. ( )f x

29. 2( ) 4 7f x x x a. ( 3)f b. ( 2)f x c. ( )f x 30. 2( ) 2 1f x x x a. ( 3)f b. (0)f c. ( 1)f x

31 2

2

4 1( )

xf x

x

a. ( 3)f b. (3)f c. ( )f x

Page 12

Functions-Interpreting Graphs--Answers

1. Yes 2. Yes 3. No 4. No 5. Yes 6. Yes 7. No, Domain = {1,4,6}, Range = {2,3,11,7} 8. Yes, Domain = {1,3,7,8}, Range = {2,4,11} 9. Yes, Domain = {4,5,6,8}, Range = {1,3,7} 10. Yes, Domain = {0,4,8,9}, Range = {0,4,8,9} 11. ( 4) 2f 12. ( 2) 0f 13. (0) 3f 14. (2) 4f 15. ( 5) 2g 16. ( 1) 1g 17. (1) 4g 18. (0) 1g

19.

,

4,

Domain

Range

x-int= 2, 2 , y-int= 4

20.

,

,

Domain

Range

x-int= 2 , y-int= 2

21.

,

, 0

Domain

Range

x-int= 0 , y-int= 0

22.

,

,

Domain

Range

x-int= 2, 0, 2 , y-int= 0

23.

,

, 4

Domain

Range

x-int= 2, 1.5, 2.5, 4 , y-int= 3

24.

3,3

1,2

Domain

Range

x-int= 3, 1.5, 3 , y-int= 2

25.

,

1 0,

Domain

Range

x-int= 0 , y-int= 0

26.

,

, 2 4

Domain

Range

x-int= 2 , y-int= 2

27.

,

, 2 2,

Domain

Range

x-int= 0 , y-int= 0

28. ( 3) 5

( 1) 3 1

( ) 3 4

f

f x x

f x x

29. 2

2

( 3) 10

( 2) 8 5

( ) 4 7

f

f x x x

f x x x

30. 2

( 3) 14, (0) 1

( 1) 2 3

f f

f x x x

31. 2

2

35 35( 3) , (3)

9 9

4 1( )

f f

xf x

x

Page 13

Page 14

Properties of Functions

Increasing/DecreasingIntervals: The part of the DOMAIN where y-values are increasing/ decreasing.

Relative Maxima: A point where a function changes from increasing to decreasing is called a relative maximum. Relative Minima: A point where a function changes from decreasing to increasing is called a relative minimum.

With the given graph, 1. Determine the domain: 2. Determine the range:

3. Determine ( 8)f

4. Solve ( ) 10f x

5. Find the intervals where the function is increasing:_____________ 6. Find the intervals where the function is decreasing:_______________ 7. Find the intervals where the function is constant:_______________ 8. Find the numbers at which f has a relative maximum:_____________ 9. Find the relative maxima:_____________ 10. Find the numbers at which f has a relative minimum:_____________ 11. Find the relative minima:_____________ 12. Find all intercepts:______________ 13. Find the values of x for which ( ) 0f x 14. Find the zeros of f

Page 15

With the given graph, 1. Determine the domain:_________ 2. Determine the range:____________ 3. Find the intervals where the function is increasing: _____________ 4. Find the intervals where the function is decreasing: _______________ 5. Find the intervals where the function is constant: _______________

6. Find the numbers at which f has a relative maximum:_____________ 7. Find the relative maxima:_____________ 8. Find all intercepts:____________ 9. Find the values of x for which ( ) 0f x 10. Find the zeros of f

Page 16

EvenandOddFunctionsandSymmetryAn even function is symmetric with respect to the y-axis.

Algebraically, the function f is an even function if ( ) ( )f x f x for all x in the domain of f.

An odd function is symmetric with respect to the origin.

Algebraically, the function f is an odd function if ( ) ( )f x f x for all x in the domain of f.

Determine whether each of the following functions is even, odd, or neither:

3( ) 6f x x x 4 2( ) 2f x x x 2( ) 2 1f x x x

Page 17

Page 18

Piecewise Functions

A piecewise function is a function in which the formula used depends upon the domain the input lies in. We notate this idea like:

formula 1 if domain to use formula 1

( ) formula 2 if domain to use formula 2

formula 3 if domain to use formula 3

f x

A cell phone company uses the function below to determine the cost, C, in dollars for g gigabytes of data transfer.

25 0 2( )

25 10( 2) 2

if gC g

g if g

Find the cost of using 1.5 gigabytes of data, and the cost of using 4 gigabytes

of data. Evaluate each piecewise function at the given values.

2

1 0( )

0

x if xf x

x if x

a) ( 2)f b) (2)f c) (0)f

0( )

1 0

x if xf x

if x

a) ( 2)f b) (2)f c) (0)f

Page 19

The Difference Quotient

The difference quotient is defined by:

( ) ( )

, 0f x h f x

hh

Find the difference quotient of the following functions: 1. ( ) 5 1f x x 2. 2( ) 2 6f x x x 3. 2( ) 2 3 5f x x x

Page 20

Properties of Functions

Find the intervals where the function is increasing, decreasing, and constant, if any.

1. 2. 3.

4. 5. 6.

Use the graph to determine each of the following. 7. a. the domain b. the range c. the x-intercepts d. the y-intercept e. intervals on which the function is increasing f. intervals on which the function is decreasing g. intervals on which the function is constant h. the number at which the function has a relative minimum i. the relative minimum of the function j. ( 3)f 8. a. the domain b. the range c. the x-intercepts d. the y-intercept e. intervals on which the function is increasing f. intervals on which the function is decreasing g. values of x for which ( ) 0f x h. the number at which the function has a relative maximum i. the relative maximum of the function j. ( 3)f k. values of x for which ( ) 0f x

Page 21

Properties of Functions

9. a. the domain b. the range c. the zeros d. (0)f e. intervals on which the function is increasing f. intervals on which the function is decreasing g. values of x for which ( ) 0f x h. the number at which the function has a relative maximum i. the relative maximum of the function j. values of x for which ( ) 3f x

Evaluate each piecewise function at the given values.

10. 3 5 0

( )4 7 0

x if xf x

x if x

a. ( 2)f b. (3)f c. (0)f

11. 3 3

( )( 3) 3

x if xf x

x if x

a. (0)f b. ( 6)f c. ( 3)f

12.

2 9 3

( ) 36 3

xif x

f x xif x

a. (5)f b. (0)f c. (3)f

The difference quotient is defined by:

( ) ( )

, 0f x h f x

hh

Find the difference quotient of the following functions:

13. ( ) 4f x x 14. ( ) 3 7f x x

15. 2( )f x x

16. 2( ) 4 3f x x x

17. 2( ) 2 1f x x x

18. 2( ) 2 4f x x x

19. 2( ) 2 5 7f x x x

20. 2( ) 2 3f x x x

Determine, algebraically, whether each function is even, odd or neither:

21. 3( )f x x x

22. 2( )f x x x

23. 2 4( )f x x x

24. 2 4( ) 1f x x x

25. 3 5( ) 2 6f x x x

Page 22

Properties of Functions-Answers

1. Increasing: 1,

Decreasing: ,1

2. Increasing: 1,

3. Decreasing: 4,5

4. Increasing: , 1

Constant: 1,

5. Increasing: 3, 1 (1,3)

Decreasing: 1,1 (3,5)

6. Increasing: 2, 4

Constant: , 2 (4, )

7. a. ,Domain

b. [ 4, )Range c. (1,0) and (7,0) d. (0,4) e. (4, ) f. (0,4) g. ( ,0) h. 4x i. 4y j. ( 3) 4f 8. a. ,Domain

b. ( , 4]Range c. ( 4,0) and (4,0) d. (0,1) e. , 2 (1,3)

f. 2,1 (3, )

g. ( , 4] [4, ) h. 2,3x i. 4, 2y y j. ( 3) 3f k. 4,4x

9. a. ( ,3]Domain b. ( , 4]Range c. ( 3,0) and (3,0) d. (0) 3f e. ,1

f. (1,3) g. ( , 3] h. 1x i. 4y j. 0,2x 10. a. ( 2) 1f

b. (3) 19f

c. (0) 7f 11. a. (0) 3f

b. ( 6) 3f

c. ( 3) 0f 12. a. (5) 8f

b. (0) 3f

c. (3) 6f 13. 4 14. 3 15. 2x h 16. 2 4x h 17. 4 2 1x h 18. 2 2x h 19. 4 2 5x h 20. 4 2 1x h 21. odd 22. neither 23. even 24. even 25. odd

Page 23

Page 24

FindingSlope

Find the slopes of the lines passing through the following points.

Slope is defined as the ratio of a change in y to a corresponding change in x. For a linear function, slope may be interpreted as the rate of change of the dependent variable per unit change in the independent variable.

Formula(s) for slope: 2 1

2 1

y ym

x x

or

ym

x

or rise

mrun

Find the slopes of the lines passing through the following points. Ex #1: (7,0) and (0,4) Ex #2: ( 2, 5) and (1,9) Ex #3: (3, 5) and ( 1, 5) Ex #4: (7, 2) and (7,5)

Page 25

Equations of Lines

Find the equation of a line given the slope and a point. Find the equation of the line with the given information. Write answers in slope-intercept form, if possible. You will need to know 2 formulas:

1. Slope-intercept formula: y mx b

2. Point-Slope formula. 1 1( )y y m x x

Ex. #1: 5m ; through ( 2,1) Ex. #2: 3

5m ; through ( 4, 2)

Horizontal Vertical

Equation: y number 0m only has a y-intercept

Equation: x number m is undefined only has an x-intercept

Ex. #3: 0m ; through ( 5,3)

Ex. #4: m is undefined; through ( 2, 7)

Page 26

Find the equation of the line passing through the given points.

1. Find the slope first. 2 1

2 1

y ym

x x

2. Pick one point and now use the Point-Slope formula. 1 1( )y y m x x 3. Write answers in slope-intercept form, if possible.

Ex. #5: Passing through the points ( 1,3) and (4,7)

Ex. #6: Passing through the points (3, 4) and ( 5, 1)

Page 27

Ex. #7: 2 and 1x intercept y intercept

Ex. #8: Passing through the points (5, 6) and ( 3, 6)

Ex. #9: Passing through the points ( 7, 4) and ( 7,8)

Page 28

AverageRateofChange

For a non-linear function, the average rate of change between any two points is the slope of the line containing the two points. This line is called a secant line.

2( ) 2 3f x x x 2 1

2 1

( ) ( )y f x f x

x x x

Find the average rate of change from (a) 1 to 1 (b) 2 to 5

Find an equation of the secant line containing ( 1, ( 1))f and (3, (3))f

3 21 1( ) 2

2 2f x x x x

Page 29

Find an equation of the secant line containing ( 1, ( 1))f and (2, (2))f 22 2

( )3 3

f x x x

Page 30

Graphing Lines

Graphing lines using the slope and y-intercept. 1. Solve the equation for y. 2. Identify m and b. 3. Plot b on the y-axis.

4. From b, use the slope rise

run

to get more points.

Ex. #1: 4 1x y Ex. #2: 2 3 9x y

Ex. #3: 7

22

y x Ex. #4: 2y

Ex. #5: 3 12 0x

Page 31

Page 32

Lines Find the slope of the line passing through each pair of points.

1. 4,7 and 8,10 3. 4, 2 and 3, 2 5. 5,3 and 5, 2

2. 2,1 and 2,2 4. 2,4 and 1, 1

Find the average rate of change of the function between the two points.

6. Find the average rate of change of 2( ) 2 3f x x from 0 to 2

7. Find the average rate of change of 2( ) 2 3f x x from 1 to 3

8. Find the average rate of change of 3( ) 2 1f x x x from 3 to 2

9. Find the average rate of change of 3( ) 2 1f x x x from 1 to 1

Use the given conditions to write an equation of a line in slope-intercept form.

10. Slope = 2, passing through 3,5

11. Slope = 3 , passing through 2, 3

12. Slope = 1

2, passing through the origin.

13. Slope = 2

3 , passing through 6, 2

14. Passing through 1,2 and 5,10

15. Passing through 3, 1 and 2,4

16. Passing through 3, 1 and 4, 1

17. Passing through 2,5 and 2, 9

18. Passing through 2,4 with x-intercept = 2

19. x-intercept =1

2 and y-intercept = 4

Secant lines. Write answers in slope-intercept form. 20. Given 2( ) 2f x x x , find an equation of the secant line containing (3, (3))f and (6, (6))f

21. Given 2( ) 2f x x x , find an equation of the secant line containing ( 1, ( 1))f and (3, (3))f Graph the line.

22. 2 1y x 24. 3

75

y x 26. 2y 28. 1y

23. 3

24

y x 25. 1

2y x 27. 3x 29. 3 18 0x

Find the slope and y-intercept of each line.

30. 3 5 0x y 32. 8 4 12 0x y

31. 2 3 18 0x y 33. 3 9 0y

Page 33

Lines-Answers

1. 3 / 4m 2. 1/ 4m 3. 0m 4. 5m 5. m is undefined 6. 4

7. 8 8. 17 9. 1 10. 2 1y x 11. 3 9y x

12. 1

2y x

13. 2

23

y x

14. 2y x 15. 2y x 16. 1y

17. 2x

18. 2y x

19. 8 4y x

20. 7 18y x

21. 3 6y x

22. 2m ; 1b 23. 3

4m ; 2b 24. 35m ; 7b

25. 1

2m ; 0b 26. 27.

28.

29.

30. 3m ; 5b

31. 23m ; 6b

32. 2m ; 3b

33. 0m ; 3b

Page 34

Quadra

Quadra

Vertex (

Axis of s

Look at graph.

Stepsto

F D P F

Graph:

tic Functio

tic Functio

(or turning

symmetry:

these grap

oGraph:

Find the verDetermine tPlot the verFrom the ve

f x x

on in Gener

on in Stand

g around po

_________

phs where

rtex. the value ortex. ertex, go ov

21 2 . Fi

Quad

ral Form:

dard Form:

oint) = ____

____

each verte

of “a”.

ver one uni

ind the axi

draticFun

__________

: _________

_________

x is (0,0).

it (to the ri

is of symm

nctions

__________

__________

Notice wha

ight and le

etry, the d

_____

_____

at the valu

eft) and the

domain, and

ue of “a” do

en up or do

d the range

es to the

wn “a”.

e.

Page 35

Graph: 22 1 3f x x . Find the axis of symmetry, the domain, and the range.

Graph: 212 4

2f x x . Find the axis of symmetry, the domain, and the range.

Intercepts: 

Where the graph intersects each axis.

How to find:

Look at the graph—does not always give exact answer Let 0x and solve for y Let 0y and solve for x.

Find the intercepts:

212 4

2f x x

Page 36

Find the intercepts:

211 5

3f x x

Find the intercepts:

22 1 3f x x

Page 37

QuadraticFunctionsinGeneralForm

General Form: Standard Form:

Need to find the vertex: ( , )h k

Use the formula:

Graph: 2 6 7f x x x . Find the axis of symmetry, the domain, and the range.

Find the intercepts.

2( )f x ax bx c 2( ) ( )f x a x h k

, ( )2

bh k f h

a

Page 38

Graph: 2 4 2f x x x . Find the axis of symmetry, the domain, and the range.

Find the intercepts.

Graph: 22 4 1f x x x . Find the axis of symmetry, the domain, and the range.

Find the intercepts.

Page 39

Page 40

Quadratic Functions-Worksheet

Find the vertex and “a” and then use to sketch the graph of each function. Find the intercepts, axis of symmetry, and range of each function. Remember the domain is ( , )

1. 2

( ) 4 1f x x

2. 2

( ) 3 1f x x

3. 21

( ) 1 22

f x x

4. 21

( ) 3 12

f x x

5. 2

( ) 2 3f x x

6. 2

( ) 2 2 1f x x

7. 2

( ) 1 4f x x

8. 2

( ) 2 2f x x

For Problems 9-15, also write the equation in standard form.

9. 2( ) 2 5f x x x

10. 2( ) 2 8f x x x

11. 2( ) 4 7f x x x

12. 2( ) 2 3f x x x

13. 2( ) 6 3f x x x

14. 2( ) 2 4 3f x x x

15. 2( ) 2 2f x x x

Page 41

Quadratic Functions Worksheet--Answers

1. x-int(s): 3, 5x ,y-int=15 Axis: 4x ; Range: [ 1, )

-3 -2 -1 1 2 3 4 5 6 7 8

-3

-2

-1

1

2

3

4

5

Vertex( 4 , -1 )

2. x-int(s): 2, 4x y-int= 8 Axis: 3x ; Range: ( ,1]

-6 -5 -4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

Vertex( -3 , 1 )

3. x-int(s): NONE y-int= 5

2

Axis: 1x ; Range: [2, )

4. x-int(s): NONE y-int=112

Axis: 3x ; Range: [1, )

5. x-int(s): 2 3x y-int=1 Axis: 2x ; Range: [ 3, )

-6 -5 -4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

5

6

Vertex( -2 , -3 )

6. x-int(s): 1

22

x

y-int=7 Axis: 2x ; Range: [ 1, )

-5 -4 -3 -2 -1 1 2 3

-3

-2

-1

1

2

3

4

5

Vertex( -2 , -1 )

7. x-int(s): 1, 3x y-int=3 Axis: 1x ; Range: ( , 4]

-4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

4

5Vertex( 1 , 4 )

8. x-int(s): 2 2x y-int= 2 Axis: 2x ; Range: ( , 2]

-6 -5 -4 -3 -2 -1 1 2 3

-4

-3

-2

-1

1

2

3

4

5

Vertex( -2 , 2 )

9. x-int(s): 1 6x y-int= 5 Axis: 1x ; Range: [ 6, )

2( ) ( 1) 6f x x

-4 -3 -2 -1 1 2 3 4

-8

-7

-6

-5

-4

-3

-2

-1

1

2

Vertex( -1 , -6 )

Page 42

Quadratic Functions Worksheet--Answers

10. x-int(s): none y-int= 8 2( ) ( 1) 7f x x

Axis: 1x ; Range: ( , 7]

-4 -3 -2 -1 1 2 3 4

-12

-10

-8

-6

-4

-2

2

Vertex( -1 , -7 )

11. x-int(s): 2 11x 2( ) ( 2) 11f x x , y-int= 7

Axis: 2x ; Range: [ 11, )

-6 -5 -4 -3 -2 -1 1 2 3

-12-11

-10

-9-8-7

-6

-5-4

-3

-2-1

1

2

Vertex( -2 , -11 )

12. x-int(s): 1, 3x 2( ) ( 1) 4f x x

Axis: 1x ; Range: ( , 4]

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5Vertex( 1 , 4 )

13. x-int(s): 3 6x

2( ) ( 3) 6f x x , y-int= 3 Axis: 3x ; Range: [ 6, )

-8 -7 -6 -5 -4 -3 -2 -1 1 2 3

-6

-5

-4

-3

-2

-1

1

2

3

4

Vertex( -3 , -6 )

14. x-int(s): 2 10

2x

2( ) 2( 1) 5f x x , y-int= 3 Axis: 1x ; Range: [ 5, )

-5 -4 -3 -2 -1 1 2 3

-6

-5

-4

-3

-2

-1

1

2

3

4

Vertex( -1 , -5 )

15. x-int(s): NONE 2( ) ( 1) 1f x x ,

Axis: 1x ; y-int= 2 Range: ( , 1]

-5 -4 -3 -2 -1 1 2 3 4 5

-7

-6

-5

-4

-3

-2

-1

1

2

3

Vertex( 1 , -1 )

Page 43

Page 44

Quadratic Inequalities

The standard form of a quadratic equation is: 2 0ax bx c A quadratic inequality replaces the equal sign with inequality signs: , , ,

Process: 1. Write the given inequality in standard form. 2. Solve the corresponding equation by factoring or the quadratic formula. 3. Plot the answers on a number line. 4. Use test points or a graph to determine what interval solves the inequality. 5. Write answers in interval notation.

1. 2 3 10 0x x 2. 22 3 5 0x x

3. 26 14 0x x 4. 2 2 1 0x x 5. 2 2 1 0x x

6. 2 3 5 0x x **

Page 45

Page 46

Quadratic Inequalities

Find and graph the solutions of the following inequalities. Express the solutions sets in interval notation. 1. 23 5 2 0x x

2. 2 9 20 0x x

3. 2 26 3 2 3 2x x x x

4. 2 4 13 0x x

5. 2 8 20 0x x

6. 2 6 9 0x x

7. 23 10 8 0x x

8. 22 15x x

9. 24 7 3x x

10. 25 2 3x x

11. 2 4 0x x

12. 22 3 0x x

13. 2 0x x

Page 47

Quadratic Inequalities-Answers

-2 -53

-43

-1 -23

-13

13

23

1

-1 1 2 3 4 5 6 7

-1 -12

12

1

-5 -4 -3 -2 -1 1 2 3 4 5

1. 1

2,3

2. , 4 5,

3. 1

2

4. ,

5. 6.

7. 2

4,3

8. 5

3,2

9. 3

1,4

10. 1

2,3

11. ( ,0] [4, )

12. 3

, (0, )2

13. 0,1

Page 48

AbsoluteValueFunctions

Absolute Function in General/Standard Form:

______________________________

Vertex (or turning around point) = ____________

The slope of right branch is ________________

The slope of left branch is ________________

When we graphed quadratic functions, we followed these steps:

Find the vertex. Determine the value of “a”. Plot the vertex. From the vertex, go over one unit (to the right and left) and then up or down “a”.

Applying this technique, graph the following functions. State the vertex, the domain, and the range.

4f x x

2f x x

1 3f x x

2 1f x x

3 2 1f x x

3 5f x x

Page 49

Intercepts: 

Where the graph intersects each axis.

How to find:

Look at the graph—does not always give exact answer Let 0x and solve for y Let 0y and solve for x.

Find the intercepts:

1 3f x x

Find the intercepts:

3 2 1f x x

Find the intercepts:

2 1 4f x x

Page 50

Absolute Value Inequalities

Review:1. Graphing horizontal lines.2. Graphing absolute value functions.3. Solving quadratic inequalities.

2 1 2f x x 4, 3f x f x 2 3 10 0x x

SolvingAbsoluteValueInequalitiesProcedure:1. Sketch a graph—one for each side of the inequality.2. Solve the corresponding equation to see where the two graphs intersect.3. Plot answers on a number line4. Shade the appropriate region(s).5. Write answer in interval notation.

1. 4x 2. 2 1x

Page 51

3. 2 3 5 11x

Hint: It might be easier if the absolute value expression is isolated on one side.

4. 3 1 2 9x

5. 5 13 7 6x 6. 3 1 5 9x

Page 52

Absolute Value Functions and Inequalities

Graph the following functions. State the vertex, the domain, range and all intercepts. 1. 3f x x

2. 5f x x

3. 2 4f x x

4. 3 4f x x

5. 2 3 1f x x

6. 2 3 3f x x

Find and graph the solutions of the following inequalities. Express the solutions sets in interval notation. 7. 3x

8. 2x 9. 2 6x 10. 9 5x 11. 5x 12. 7x 13. 5 1x 14. 2 5 3x 15. 2 2 2 6x 16. 2 1 1 8x

17. 4 7 2x

18. 3 5 12 6x

19. 4 1 6 3x

20. 4 1 7 13x

21. 3 1 7 10x

Page 53

Absolute Value Functions and Inequalities—Answers 1. Vertex 3,0 D: , R: [0, )

x-int. 3,0 y-int. 3,0

2. Vertex 0, 5 D: , R: [ 5, )

x-ints. 5,0 , 5,0 y-int. 0, 5

3. Vertex 2, 4 D: , R: [4, )

x-ints. none y-int. 0,6

4. Vertex 4,0 D: , R: [0, )

x-int. 4,0 y-int. 0,12

5. Vertex 3, 1 D: , : [ 1, )

x-ints. 5 7,0 , ,0

2 2

y-int. 0,5

6. Vertex 3,3

2

D: , R: 3,

x-ints. none y-int. 0,6

Page 54

Absolute Value Functions and Inequalities—Answers p7. ( 3,3)

8. ( , 2] [2, )

9. [ 4,4]

10. ( , 14) (14, )

11. 12. ( , )

13. (4,6)

14. ( ,1) (4, )

15. ( 5,3)

16. ( , 3] [4, )

17. 5 9,

4 4

18. 19. ( , )

20. ( , 4) (6, )

21. [0, 2]

Page 55

Page 56

Graphs of Polynomial Functions

In order to sketch a graph of a polynomial function, we need to look at the “end behavior” of the graph and the intercepts. The “end behavior” of the graph is determined by the leading term of the polynomial.

4 2( ) 4 3 2 5f x x x x

3 2( ) 5 7 2 5f x x x x

2 2( ) 3(5 1) ( 2)f x x x

x

y

x

y

x

y

Page 57

2( ) ( 4) ( 2)f x x x

Summary: If the leading coefficient has an even power, then the end behavior is the same: both up or both down If the leading coefficient has an odd power, then the end behavior will be opposites: one up and one down

Intercepts:In order to find the y-intercept, set x = 0 and solve for y. In order to find the x-intercept, set y = 0 and solve for x. 2( ) ( 2) ( 1)f x x x x 3 2( ) 3 3f x x x x

x

y

Page 58

“Multiplicity” of the x-intercepts 1. Multiplicity of 1 or single: the graph passes through the x-axis like a line. 2. Multiplicity of 2 or even: the graph passes “bounces” off the x-axis like a parabola. 3. Multiplicity of 3 or odd: the graph “squiggles” through the x-axis like a cubic function.

Graphthefollowing: End Behavior, y-intercept, the multiplicity of the x-intercepts.

2 3( ) ( 2) ( 1)( 3)f x x x x

4 3( ) 3 3f x x x

x

y

x

y

Page 59

4 2( ) 4 3 1f x x x

2 2( ) (3 1) ( 1)f x x x

3( ) 4 12f x x x

x

y

x

y

x

y

Page 60

Graphs of Higher Degree Polynomials

Determinetheendbehavior,findthex‐intercept(s)andtheirmultiplicity,andthey‐intercept.Sketchthegraphofthepolynomialfunction.

1. ( ) ( 4)( 2)( 1)f x x x x  

 

2. 2( ) ( 2)( 1)f x x x x  

 

3. 3 2( ) 2f x x x x  

 

4. 3 2( ) 2 2f x x x x  

 

5. 3 2( ) ( 2) ( 1)f x x x x  

 

6. 4 3( ) 2 2f x x x  

 

7. 3 2( ) 4f x x x  

 

8. 3 2( ) 2f x x x x  

 

9. 4 3 2( ) 4 4f x x x x  

 

10. 4 2( ) 5 4f x x x  

 

11. 2( ) ( 1)(2 1)f x x x  

Page 61

Graphs of Higher Degree Polynomials-Answers

1. ( ) 4 2 1f x x x x

 

a. End behavior: down left / up right b.  

 

 

 

c. Y‐intercept:  ‐8  

 

X‐intercepts:  ‐4  ‐2  1 

Multiplicity: 

 

1  1  1 

2. 2( ) 2 1f x x x x

 

a. End: down both left & right b.  

 

 

 

c. Y‐intercept:  0  

 

X‐intercepts:  0  2  ‐1 

Multiplicity: 

 

1  1  2 

3. 3 2( ) 2f x x x x

 

a. End behavior: down left / up right b.  

 

 

 

c. Y‐intercept:  0  

 

X‐intercepts:  0  ‐2  1 

Multiplicity: 

 

1  1  1 

Page 62

4. 3 2( ) 2 2f x x x x

 

a. End behavior: down left / up right b.  

 

 

 

c. Y‐intercept:  ‐2  

 

X‐intercepts:  ‐2  1  ‐1 

Multiplicity: 

 

1  1  1 

5. 3 2( ) ( 2) ( 1)f x x x x

 

a. End behavior: up both left & right b.  

 

 

 

c. Y‐intercept:  0  

 

X‐intercepts:  ‐2  1  0 

Multiplicity: 

 

2  1  3 

Page 63

6. 4 3( ) 2 2f x x x

 

a. End behavior: down both left & right b.  

 

 

 

c. Y‐intercept:  0   

 

X‐intercepts:  0  1 

Multiplicity: 

 

3  1 

7. 3 2( ) 4f x x x

 

a. End behavior: up left / down right b.  

 

 

 

c. Y‐intercept:  0  

 

X‐intercepts:  ‐4  0 

Multiplicity: 

 

1  2 

Page 64

8. 3 2( ) 2f x x x x

 

a. End behavior: down left / up right  b.  

 

 

 

c. Y‐intercept:  0  

 

X‐intercepts:  ‐1  0 

Multiplicity: 

 

2  1 

9. 4 3 2( ) 4 4f x x x x

 

a. End behavior: up both left & right b.  

 

 

 

c. Y‐intercept:  0  

 

X‐intercepts:  0  2 

Multiplicity: 

 

2  2 

Page 65

10. 4 2( ) 5 4f x x x

 

a. End behavior: up both left & right b.  

 

 

 

 

 

c. Y‐intercept:  4  

 

X‐intercepts:  ‐2  ‐1  1  2 

Multiplicity: 

 

1  1  1  1 

11. 2( ) 1 2 1f x x x

 

a. End behavior: down left / up right b.  

 

 

 

c. Y‐intercept:  1   

 

X‐intercepts:  ‐1  ½ 

Multiplicity: 

 

1  2 

Page 66

Synthetic Division and Remainder Theorem

Synthetic Division is a condensed method of long division. It is quick and easy. Unfortunately, it can only be used when the divisor is in the form of ( )x a Synthetic division:

1.

29 5 1

1

x x

x

2.

3 125

5

x

x

3.

35 2 4

1

x x

x

RemainderTheorem: When dividing a polynomial by ( )x c , then the remainder is ( )f c .

If 3( ) 5 2 4f x x x , find (1)f . (refer to #3) If 3 2( ) 11 7 19f x x x x , find ( 1)f .

Reminders: 1. Write both polynomials in standard form. 2. Fill in all missing terms with a place holder of zero. 3. Write your answer as a polynomial that is one degree less than the dividend (numerator).

Page 67

If 4 3 2( ) 12 6 5f x x x x , find 2

3f

.

FactorTheorem: Let ( )f x be a polynomial. Then if ( ) 0f c , then ( )x c is a factor of ( )f x . And if ( )x c is a factor of ( )f x , then ( ) 0f c . (refer to #2) Solve the equation 3 22 3 11 6 0x x x , given that 3 is a zero (or factor) of the function

3 2( ) 2 3 11 6f x x x x Solve the equation 4 3 23 17 19 21 18 0x x x x , given that 3 is a zero of multiplicity of two of the function 4 3 2( ) 3 17 19 21 18f x x x x x

Page 68

Synthetic Division and Remainder Theorem

Perform the indicated divisions using synthetic division.

1. 2 3 5

2

x x

x

2. 22 9

2

x

x

3. 3 22 7 6

3

x x x

x

4. 3 7 4

2

x x

x

5. 3 23 11 5

4

x x x

x

6. 4 32 9 10 24

4

x x x

x

7. 3 125

5

x

x

8. 6 64

2

x

x

Use the Remainder Theorem to evaluate ( )f c .

9. 3 2( ) 2 6 9 21f x x x x , find ( 1)f 10. 4 3( ) 15 5 7f x x x x , find (3)f

11. 3 2( ) 2 5 4 3f x x x x , find 1

2f

12. 4 3( ) 6 7 11f x x x x , find (0)f 13. 3 2( ) 2 5f x x x x , find (6)f Solve the equation: 14. Solve the equation 3 22 5 2 0x x x given that 2 is a zero of 3 2( ) 2 5 2f x x x x 15. Solve the equation 3 23 8 3 2 0x x x given that 2 is a zero of 3 2( ) 3 8 3 2f x x x x 16. Solve the equation 4 3 22 21 57 5 75 0x x x x given that 5 is a zero of multiplicity two of the function 4 3 2( ) 2 21 57 5 75f x x x x x 17. Solve the equation 4 3 24 9 3 5 3 0x x x x given that 1 is a zero of multiplicity three of the function 4 3 2( ) 4 9 3 5 3f x x x x x

Page 69

Syn. Division and Remainder Thm.-Answers

1.  3

12

xx

 

2.  17

2 42

xx

 

3.   2 62 4

3x x

x

   

4.   2 22 3

2x x

x

 

5.   2 43 1

4x x

x

 

6.   3 22 4 6x x x  

7.   2 5 25x x  

8.   5 4 3 22 4 8 16 32x x x x x  

9. 1 34 ( )f 10. 3 316 ( )f

11. 10

2

f

12. 0 11( )f 13. 6 174( )f

14. 1

2, ,12

x

15. 1

2, ,13

x

16. 3

5(multiplicity of 2), ,12

x

17. 3

1(multiplicity of 3),4

x

Page 70

Zeros of Polynomial Functions

Some polynomials cannot be factored by traditional methods. The Rational Zero or Root Theorem gives us another method to find the x-intercepts or zeros of a polynomial. The theorem states that a list of possible rational zeros of a polynomial can be found by taking the factors of the constant term (p) and dividing them by the factors of the leading coefficient (q).

Make a list of possible rational zeros: 1. 3 2( ) 4 15f x x x 2. 5( ) 10 2 5f x x x

After making the list, we can use it along with synthetic division and the graph of the function to try and factor the polynomials and find the roots (zeros).

3 2( ) 3 5 4 4f x x x x

Page 71

4 3 2( ) 6 11 21 3f x x x x x

4 3 2( ) 2 7 5 13 3f x x x x x

Page 72

4 3 2( ) 3 3 16 12f x x x x x

5 3 2( ) 8 8 1f x x x x

Page 73

Page 74

Finding Zeros of Polynomials In the following exercises, list all the possible rational zero. Use the Rational Zeros Theorem, the given graph, and synthetic division to find all zeros of each polynomial function. State any multiplicities.

1. 3 2( ) 4 7 10f x x x x 2. 3 2( ) 2 2 1f x x x x

3. 3 2( ) 3 4f x x x 4. 4 3 2( ) 6 25 4 4f x x x x x

5. 5 4 3 2( ) 4 8 7 17 3 9f x x x x x x 6. 3 2( ) 2 12 6f x x x x

7. 3 2( ) 4 8f x x x 8. 4 3 2( ) 3 5 7 3 2f x x x x x

Page 75

9. 4 3( ) 2 5 10f x x x x 10. 4 3 2( ) 4 7 2 4 1f x x x x x

11. 3( ) 10 12f x x x 12. 3( ) 4 11 7f x x x

13. 4 3 2( ) 2 3 20 20f x x x x x 14. 4 2( ) 4 12 9f x x x x

15. 4 3 2( ) 4 12 13 12 9f x x x x x 16. 5 4 3 2( ) 3 3 9 4 12f x x x x x x

Page 76

Answers—Finding Zeros of Polynomials 1. 5 2 1 , ,x

2. 1

1 12

, ,x

3. 2 (multiplicity of 2) 1 ,x

4. 1 1

2 22 3

, , ,x

5. 3

1 (multiplicity of 2) 1 (multiplicity of 2)2

, ,x

6. 1

62

,x

7. 1 5 2 ,x

8. 2

1 2 13

, ,x

9. 3 5 2 ,x

10. 1 5 1

12 4

, ,x

11. 2 1 7 ,x

12. 1 2 2

12

,x

13. 5 55

1 24

, ,

ix

14. 3 1 1 2 , ,x i

15. 3

2 ,x i (multiplicity of 2)

16. 2 3 , ,x i

Page 77

Page 78

Domain of a Function

Finding the domain of a function: 1. The implied domain is the set of all real numbers for which the expression is defined. For all polynomial functions the domain is all real numbers or expressed in interval notation: , Ex. #1: 2( ) 5 6f x x x

But what about rational functions? The question one must ask when finding the domain is “Where is this function NOT defined?” 2. Rational Functions: This is a function that is comprised of a ratio of 2 polynomial functions. Thus, there is a denominator involved. We must always remember that

DIVISION BY ZERO IS UNDEFINED!! To determine the domain of a rational function, set the denominator equal to zero and solve. These are the values that are NOT acceptable.

Ex. #2: 2

1( )

3 40f x

x x Ex. #3:

2 2

1 1( )

9 9f x

x x

Ex. #4:

1( )

53

2

f x

x

3. Radical Functions: This is a function that is underneath some type of radical. Remember when taking the EVEN root of a negative number, the answer is imaginary. Imaginary numbers are NOT acceptable for real valued functions. To determine the domain of a radical function, set the radicand greater than or equal to zero and solve. These are the values that ARE acceptable. Ex. #5: ( ) 2f x x Ex. #6: ( ) 8 6f x x

Page 79

Ex. #7: ( ) 3 5f x x x

Ex. #8:

3( )

6x

f xx

Ex. #9:

2 5( )

4x

f xx

Page 80

Domain of a Function

Find the domain of the following functions. Write answers in interval notation, when convenient.

1. 2

2( )

5 6

xf x

x x

2.

2

3 7( )

6 27

xf x

x x

3. 2

4( )

25

xf x

x

4. ( ) 3 5f x x

5. ( ) 12 24f x x 6. ( ) 2 5f x x

7. 1 3

( )7 9

f xx x

8. 2 2

1 1( )

1 1f x

x x

9. 4

( )3

1f x

x

10. 1

( )4

21

f x

x

11. ( ) 5 35f x x 12. ( ) 24 2f x x

13. ( ) 2 3f x x x 14. 2

( )5

xf x

x

Page 81

Domain of a Function-Answers

1. 3, 2x 2. 3,9x

3. 5,5x 4. ,

5. 2, 6.

5,

2

7. 7,9x 8. 1,1x

9. 0,3x 10. 1,3x

11. 7, 12. ( ,12]

13. 2, 14. 2,5 (5, )

Page 82

Graphing Rational Functions

A rational function is the ratio of two polynomial functions. In order to sketch a graph, we must find all the intercepts and all the asymptotes.

Process for sketching a graph of a rational function: 1. Find the y-intercept by setting x = 0. 2. Find the x-intercept(s) by setting y = 0. 3. Find the vertical asymptotes by setting the denominator = 0. (The denominator of the reduced function.) 4. Find the horizontal asymptote (if one exists) by comparing degree of the numerator to the degree of

the denominator. 5. Plot the intercepts and graph the asymptotes. Plot a few additional points to complete the graph.

1. 3

( )2

xf x

x

y-intercept: x-intercept(s): Vertical Asymptote(s): Horizontal Asymptote: If the degree of the numerator is equal to the degree of the denominator, then the horizontal asymptote will be:

coefficient of the leading term of the numerator

coefficient of the leading term of the denominatory

Page 83

2. 2

2 3( )

6

xf x

x x

y-intercept: x-intercept(s): Vertical Asymptote(s): Horizontal Asymptote: If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote will ALWAYS be __________________

3. 2

2

4 1( )

2

xf x

x x

y-intercept: x-intercept(s): Vertical Asymptote(s): Horizontal Asymptote:

Page 84

Rational Functions

Find the domain of each function.

1. 5

( )4

xf x

x

2.

23( )

( 5)( 4)

xf x

x x

3.

2

7( )

49

xf x

x

Find the vertical asymptotes of each function.

4. ( )4

xf x

x

5.

3( )

( 3)

xf x

x x

6.

2

5( )

12

xf x

x x

Find the horizontal asymptotes of each function.

7. 2

12( )

3 4

xf x

x

8.

2

2

12( )

3 4

xf x

x

9.

2 1( )

3 5

xf x

x

Find the domain, all intercepts, and all asymptotes of each function. Then sketch the graph of the function.

10. 4

( )2

xf x

x

11.

2

2( )

4

xf x

x

12.

2

2

2( )

1

xf x

x

13. ( )2

xf x

x

14.

2

1( )

4f x

x

15.

2

2( )

2f x

x x

16. 2

2( )

6

xf x

x x

17.

2

2

12( )

4

x xf x

x

18.

2

2

3 4( )

2 5

x xf x

x x

.

Page 85

Rational Functions-Answers

 

1. 4x 2. 4,5x

3. 7,7x 4. 4x

5. 0, 3x x 6. 3, 4x x 7. 0y

8. 4y

9. 23y

10. Domain: 2x , x-int= 0 , y-int=0 , VA: 2x , HA: 4y  

11. Domain: 2, 2x , x-int= 0 , y-int=0 , VA: 2, 2x , HA: 0y  

12. Domain: 1, 1x , x-int= 0 , y-int=0 , VA: 1, 1x , HA: 2y  

13. Domain: 2x , x-int= 0 , y-int=0 , VA: 2x , HA: 1y  

14. Domain: 2, 2x , x-int= none ,

y-int= 14 , VA: 2, 2x , HA: 0y  

 

15. Domain: 1, 2x , x-int=none , y-int= 1 ,VA: 1, 2x , HA: 0y  

16. Domain: 2, 3x , x-int= 2 ,

y-int= 13

,VA: 2, 3x , HA: 0y  

17. Domain: 2, 2x , x-int= 4,3 y-int=3 ,VA: 2, 2x , HA: 1y  

18. Domain: 50, 2x , x-int= 41, 3 ,

y-int=none , VA: 50, 2x , HA: 32y

Page 86

Polynomial Inequalities

Process: 1. Write the given inequality in standard form: exponents in descending order and zero on the right hand side. 2. Solve the corresponding equation by factoring. 3. Plot the answers on a number line. 4. Use test points OR a graph to determine what interval solves the inequality. 5. Write answers in interval notation.

1. ( 4)( 1)( 7) 0x x x 2. 2( 3)( 4)( 1) 0x x x

3. 2 2( 2) ( 4)( 6) 0x x x 4. 3 218 30 8 0x x x 5. 3 24 4 0x x x 6. 3 25 9 45 0x x x

Page 87

Rational Inequalities

Process: 1. Make one side of the inequality zero. 2. Combine all of the terms on the non-zero side into a single fraction. 3. Set both the numerator and denominator EQUAL to zero and solve these equations. These are the boundary points. 4. Plot these points on a number line. 5. Look at corresponding graph and shade either above or below x-axis. 6. Write answers in interval notation.

1. 3

02

x

x

2. 2 3

14

x

x

Page 88

3. 2 1

1 2

x

x

4. 2 1

5 3x x

Page 89

Page 90

Polynomial and Rational Inequalities Find and graph the solutions of the following inequalities. Express answers in interval notation. 1. 3 2 5 0 ( )( )( )x x x 2. 21 2 3 0 ( )( )( )x x x 3. 2 21 5 3 0 ( )( ) ( )x x x 4. 3 212 24 9 0 x x x 5. 3 25 9 45 0 x x x 6. 3 23 4 12 0 x x x

7. 5

02

xx

8. 1

06

xx

9. 2 1

03

x

x

10. 2 1

15

x

x 11.

21

6

xx

12. 3

22

x

x

13. 14

x

x 14.

3 1

4 2

xx

15. 2 1

4 1

x x

16. 1 1

2 4

x x 17.

3 1

1 4

x x

Page 91

-7 -6 -5 -4 -3 -2 -1 1 2 3 4

-1 1 2 3 4 -2 -1 1 2 3 4

-1 -12

12

1 32

2 52

3 -7 -6 -5 -4 -3 -2 -1 1 2 3 4

-4 -3 -2 -1 1 2 1 2 3 4 5 6

-2 -1 1 2 3 4 5 6 7 -4 -72

-3 -52

-2 -32

-1 -12

12

1

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 -8 -7 -6 -5 -4

-3 -2 -1 1 2 3 4 5 -1 1 2 3 4 5 6

-10 -8 -6 -4 -2 2 4 6 -8-7-6-5-4-3-2-1 1 2 3 4 5 6

-6 -4 -2 2 -8 -7 -6 -5 -4 -3 -2 -1 1 2

Answers—Polynomial and Rational Inequalities 1. 5 3 2 , ( , )

2. 1 2 3 3 , ( , ) ( , ) 3. 1 ,

4. 1 3

02 2

, , 5. 5 3 3 , ,

6. 3 , 7. 2 5,

8. 1 6 , ( , ) 9. 13

2

, ,

10. 6 5 , 11. 6 ,

12. 2 4 , 13. 2 4 , ( , )

14. 10 4 , ( , ) 15. 6 1 4 , ( , )

16. 4 2 , ( , ) 17. 134 1

2

, ,

Page 92

Graph Transformations

Transformations: ( ) ( ) constantg x f x Moves the graph __________________________ ( ) ( ) constantg x f x Moves the graph __________________________ ( ) ( constant)g x f x Moves the graph __________________________ ( ) ( constant)g x f x Moves the graph __________________________ ( ) ( )g x f x Multiplies all the y-values by ___________ ( ) (Constant) ( )g x f x Multiplies all the y-values by ________________ ( ) a ( b) cg x f x a and c affects the y-values. b affects the x-values Use the graph of ( )y f x to obtain the following graphs: 1. ( ) ( ) 2g x f x 2. ( ) ( 2)g x f x

Page 93

3. ( ) ( )g x f x 4. ( ) ( 2) 2g x f x

5. 1

( ) ( )2

g x f x 6. ( ) 2 ( )g x f x

7. 1

( ) ( 1) 22

g x f x

Page 94

8. ( ) 1 3f x x 9. 3( ) 2( 1) 1f x x

10. 31( ) 3 2

2f x x

Page 95

Page 96

Transformations

Use the graph of ( )y f x to obtain the following graphs: 1. ( ) ( ) 1g x f x 2. ( ) ( 1)g x f x

3. ( ) ( )g x f x 4. ( ) ( 1) 1g x f x

5. ( ) 2 ( )g x f x 6. 1

( ) ( )2

g x f x

Page 97

7. ( ) 2 ( 2) 2g x f x 8. ( ) 2 ( 1) 3g x f x

9. ( ) ( 1) 1g x f x 10. ( ) ( 1) 1g x f x

11. ( ) ( 1) 1g x f x 12. ( ) ( 1) 2g x f x

Page 98

13. 1

( ) ( 1)2

g x f x 14. ( ) 2 ( 1)g x f x

Page 99

Page 100

Transformations-Answers 1. ( ) ( ) 1g x f x

2. ( ) ( 1)g x f x

3. ( ) ( )g x f x

4. ( ) ( 1) 1g x f x

5. ( ) 2 ( )g x f x

6. 1

( ) ( )2

g x f x

7. ( ) 2 ( 2) 2g x f x

8. ( ) 2 ( 1) 3g x f x

9. ( ) ( 1) 1g x f x

Page 101

10. ( ) ( 1) 1g x f x

11. ( ) ( 1) 1g x f x

12. ( ) ( 1) 2g x f x

13. 1

( ) ( 1)2

g x f x

14. ( ) 2 ( 1)g x f x

Page 102

Basic Functions Linear

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

Quadratic

4. 2y x 5. 22y x 6. 22y x

Absolute Value

7. y x 8. 2y x 9. 1

2y x

Page 103

Square Root 10. y x 11. 2y x 12. 2y x

13. y x 14. 2y x Cubic

15. 3y x 16. 3 2y x 17. 32 2y x

Cube Root

18. 3y x 19. 3y x 20. 31

2y x

Page 104

Additional Graphing Techniques In problems 1-40 use the techniques of shifting, reflecting, and stretching to sketch the graph of the following functions. 1. 2( ) ( 1) 3f x x 2. 2( ) ( 1) 4f x x

3. ( ) 1 2f x x 4. ( ) 2 1f x x

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

( ) 2 2 1f x x

7. ( ) 4 4f x x 8. ( ) 2 3 3f x x

9. 3( ) 2 3f x x 10. 3( ) 1 2f x x

11. ( ) 2 1 1f x x 12. 2( ) 2 3 2f x x

13. 2( ) 3f x x 14. 21( )

4f x x

15. ( ) 3f x x 16. ( ) 1f x x

17. ( ) 4f x x 18. ( ) 1f x x

19. 1

( ) 3 32

f x x 20. 1

( ) 2 12

f x x

21. ( ) 1 3f x x 22. 1

( ) 1 42

f x x

23. 3( ) 2 1 2f x x 24. ( ) 3f x x

25. 31( ) 2 1

2f x x 26. 3

( ) 3f x x

27. 2( ) 2 3 5f x x 28. 31

( ) 1 42

f x x

29. 3( ) 2 1 2f x x 30. 3( ) 2 3f x x

31. 1

( ) 3 22

f x x 32. ( ) 2 1 1f x x

33. 31( ) 1 3

2f x x 34.

1( ) 4 2

2f x x

35. 3( ) 2 1f x x 36. 31( ) 4 1

2f x x

37. 3( ) 4 1f x x 38. 3( ) 3 1f x x

39. ( ) 2 3 1f x x 40. 3( ) 5 3f x x

Page 105

-4 -3 -2 -1 1 2 3 4

-5-4-3-2-1

12345

-4 -3 -2 -1 1 2 3 4

-5-4-3-2-1

12345

Answers-Graphing Techniques 1. 3. 5.

-2 -1 1 2 3 4 5 6

-5-4-3-2-1

12345

7. 9. 11. 13. 15. 17. 19. 21. 23.

-1 1 2 3 4 5 6 7-2-1

12345678

-6 -5 -4 -3 -2 -1 1 2

-4-3-2-1

1234

-2 -1 1 2 3 4 5 6

-3-2-1

1234567

-4 -3 -2 -1 1 2 3 4

-5-4-3-2-1

12345

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

-4 -3 -2 -1 1 2 3 4

-5-4-3-2-1

12345

Page 106

25. 27. 29. 31. 33. 35. 37. 39.

-8 -7 -6 -5 -4 -3 -2 -1 1 2

-4-3-2-1

1234

-3 -2 -1 1 2 3 4 5 6

-4-3-2-1

1234

-4 -3 -2 -1 1 2 3

-5-4-3-2-1

1234

-4 -3 -2 -1 1 2 3

-5-4-3-2-1

1234

-2 -1 1 2 3 4 5 6

-8-7-6-5-4-3-2-1

1

-2 -1 1 2 3 4 5 6

-5-4-3-2-1

1234

-4 -3 -2 -1 1 2 3

-2-1

1234567

-6 -5 -4 -3 -2 -1 1 2

-4-3-2-1

1234

Page 107

Page 108

Piecewise Functions

GraphingPiecewiseFunctionsTo graph a piecewise defined function, choose several values for each domain including the endpoints of each domain, whether or not that the endpoint is included in the domain.

1. 2 1 0

( )3 0

x if xf x

if x

2. 2 1

( )1 1

x if xf x

x if x

3. 3 2 2

( )3 2

x if xf x

if x

Page 109

4.

3 4 0

( ) 2 0

0

x if x

f x if x

x if x

5. 3

4 4

( ) 4 1

1

x if x

f x x if x

x if x

Page 110

Piecewise Functions

Graph the following:

1.

2 0( )

1 0x if x

f xif x

2. 3 0

( )4 0x if x

f xif x

3. 2 3 1

( )3 2 1

x if xf x

x if x

4. 2

1 0( )

0

x if xf x

x if x

5. 3

2 0

( ) 3 0

0

x if x

f x if x

x if x

6. 3 3 0

( ) 2 0

0

x if xf x if x

x if x

7. 3

3

1 0( )

0

x if xf x

x if x

8. 0

2 1 0

x if xf x

x if x

9. 4 2

2 2 2

x if xf x

x if x

10. 2

1 1

1 1

x if xf x

x if x

11. 1 -1

0 1 1

1 1

x if x

f x if x

x if x

12. 1 0

1 0

x if xf x

if x

13. 1 3

2 8 3

x if xf x

x if x

14. 1

0 1

2 1

x if x

f x if x

x if x

15. 2

2 4 1

4 1

1 1

x if x

f x if x

x if x

16. 0

1 0

x if xf x

if x

17. 1 1

2 1

x if xf x

if x

Page 111

Answers-Piecewise Functions

1. 2 0

( )1 0

x if xf x

if x

2. 3 0

( )4 0x if x

f xif x

3. 2 3 1

( )3 2 1

x if xf x

x if x

4. 2

1 0( )

0

x if xf x

x if x

5.

3

2 0

( ) 3 0

0

x if x

f x if x

x if x

6.

3 3 0( ) 2 0

0

x if xf x if x

x if x

7.

3

3

1 0( )

0

x if xf x

x if x 8.

0

2 1 0

x if xf x

x if x

9. 4 2

2 2 2

x if xf x

x if x

Page 112

10. 2

1 1

1 1

x if xf x

x if x 11.

1 -1

0 1 1

1 1

x if x

f x if x

x if x

12. 1 0

1 0

x if xf x

if x

13. 1 3

2 8 3

x if xf x

x if x 14.

1

0 1

2 1

x if x

f x if x

x if x

15. 2

2 4 1

4 1

1 1

x if x

f x if x

x if x

16. 0

1 0

x if xf x

if x

17. 1 1

2 1

x if xf x

if x

Page 113

Page 114

Composite Functions

( ) 3 1f x x 2( ) 2g x x

x ( )f x

2

5

6

TheCompositionofFunctions The composition of the function f with g is denoted by f g and is defined by the

equation: ( ) ( ( ))f g x f g x . The domain of the composite function f g is the set of

all x such that x is in the domain of g and ( )g x is in the domain of f. For 1 & 2, use functions above: 1. (2)f g 2. (2)g f

For 3-9, use graph: 3. ( )( 3)f g

4. ( )( 5)g f

5. ( )(0)f g

6. ( )(3)g f

7. ( )(7)f g

8. ( )(2)f f

**9. ( )( 7)g f

x ( )g x

2

5

6

Page 115

Find ( )f g x and ( )g f x

9. Given ( ) 3 1f x x and 2( ) 2g x x

10. Given 2( )f x x and ( ) 2g x x

Find ( )f g x :

11. Given ( )3

xf x

x

and

2( )g x

x

Page 116

Composite Functions

Use the given functions f and g to find f g x and g f x to

1. 2( ) 2 7f x x , ( ) 3 4g x x

2. 2( ) 4 3f x x x , ( ) 7g x x

3. 2( ) 2f x x , ( ) 4g x x

4. ( ) 3f x x , 1

( )g xx

5. ( ) 2 3f x x , 3

( )2

xg x

Use the given functions f and g to find f g x

6. 2

( )3

f xx

, 1

( )g xx

7. ( )1

xf x

x

,

4( )g x

x

8. ( )f x x , ( ) 2g x x

9. 2( ) 4f x x , ( ) 1g x x Use the graphs to evaluate the expressions below. 10. ( )(3)f g

11. ( )(1)f g

12. ( )(1)g f

13. ( )(0)g f

14. ( )(5)f f

15. ( )(4)f f

16. ( )(0)g g

17. ( )(2)g g

f(x)

g(x)

Page 117

Composite Functions-Answers

1. 2

2

18 48 39

6 17

( )( )

( )( )

f g x x x

g f x x

2. 2

2

10 24

4 4

( )( )

( )( )

f g x x x

g f x x x

3. 2

2

2

( )( )

( )( )

f g x x

g f x x

4.

3 1

1

3

( )( )

( )( )

xf g x

x

g f xx

5.

( )( )( )( )f g x xg f x x

6. 2

1 3

( )( )

xf g x

x

7. 4

4

( )( )f g x

x

8. 2 ( )( )f g x x 9. 5 ( )( )f g x x

10. 2

11. 2

12. 2

13. 5

14. 1

15. 3

16. 2

17. 1

Page 118

y

x

y

x

y

x

y

x

y

x

y

x

Inverse Functions

Review:DefinitionofaFunctionA function is a set of ordered pairs where for every x-value there is a unique y-value. Graphically: Use the vertical line test to determine if the graph is a function. Determine if the following graphs are functions?

OnetoOneFunctions(1–1functions)A function is said to be one to one if each y-value corresponds to only one x-value. Graphically: Use the horizontal line test to determine if the following functions are One to One Functions. Determine if the following funcitons are One to One Functions

WhatisanInverseFunction? Only one-to-one functions have inverse functions. A function and its inverse can be described as the "DO" and the "UNDO" functions. A function takes a starting value, performs some operation on this value, and creates an output answer. The inverse of this function takes the output answer, performs some operation on it, and arrives back at the original function's starting value.

Page 119

( ) 2 3f x x The inverse of ( ) 2 3f x x ? Domain of f Range of f

DefinitionofInverseFunctions:

If f is a one to one function, then 1( )f x is the inverse function of f if: 1( ( ))f f x x , for every x in the domain

And 1( ( ))f f x x , for every x in the domain

VerifyingInverses:Find ( ( ))f g x and ( ( ))g f x to determine whether each pair of functions f and g are inverses of each other.

1. 9

( )4

xf x

and ( ) 4 9g x x

2. 3

( )2

xf x

and ( ) 3 2g x x

Note: 1f x is notation for the inverse of f .

Pronounced: “ f inverse of x ”.

1f x is NOT “ f to the negative 1 exponent”

Page 120

3. 2

( )5

f xx

and 2

( ) 5g xx

FindingtheInverseFunction Process: 1. Replace ( )f x with y . 2. Switch x and y . 3. Solve for y .

4. Re-write y as 1( )f x . For the following problems:

a) Find the inverse of the given function. b) VERIFY your equation is the inverse by showing 1( ( ))f f x x and 1( ( ))f f x x .

1. ( ) 3 1f x x 2. 3( ) 4f x x

Page 121

3. ( ) 5f x x 4. 4

( ) 9f xx

5. 2 3

( )5

xf x

x

Page 122

GraphingInverseFunctionsIn order to graph the inverse of a function, you need to switch the domain and the range. In other words, reverse the order of the ordered pairs.

Page 123

Page 124

Inverse Functions

Find an equation for 1( )f x . Verify that your equation is correct by

showing 1( ( ))f f x x and 1( ( ))f f x x 1. ( ) 3f x x 2. ( ) 2f x x 3. ( ) 2 3f x x

4. 3( ) 2f x x 5. 3( ) ( 2)f x x 6. 1

( )f xx

7. ( )f x x 8. 7

( ) 3f xx

9. 2 1

( )3

xf x

x

Which graphs represent functions that have inverse functions? 10. 11. 12.

Use the graph of the given function to graph its inverse function. 13. 14. 15.

Page 125

Inverse Functions-Answers

1. 1( ) 3f x x 2. 1 1( )

2f x x 3. 1 3

( )2

xf x

4. 1 3( ) 2f x x 5. 1 3( ) 2f x x 6. 1 1( )f x

x

7. 1 2( )f x x 8. 1 7( )

3f x

x

9. 1 3 1

( )2

xf x

x

10. No 11. Yes 12. No 13. 14. 15.

 

Page 126

ExponentialFunctions

An exponential function is a function where a positive number is raised to a power.

( ) xf x b where b > 0 and b ≠1

Exponential Functions NOT Exponential Functions

Reviewofexponents:

0x 1x 1x 1

a

b

2x

GraphthefollowingExponentialFunctions:

( ) 2xf x 1

( )2

x

f x

Domain:___________________ Domain:___________________

Range:____________________ Range:____________________

X-int:_____________________ X-int:_____________________

Y-Int:_____________________ Y-Int:_____________________

Asymptote:________________ Asymptote:________________

Page 127

Transformations

( ) xf x b

All exponential functions (in “basic” form) have 2 point on their graphs:

(1, ____) (0, ____)

Vertical Shift ( ) xf x b c

( ) xf x b c

Horizontal Shift ( )( ) x cf x b ( )( ) x cf x b

Reflections ( ) xf x b

( ) xf x b

Vertical Stretch and Compress ( ) xf x c b

Note that ( )x xc b cb

TransformationsExamples:

Sketch ( ) 3xf x Then, sketch the following

4( ) 3xf x

( ) 3 4xf x

( ) 3xf x

( ) 2 3xf x

Page 128

NaturalNumbere:Of all possible choices of bases, the most preferred or most natural base it the number e. The number e has important significance in science and mathematics. It is often called Euler’s number named after Leonhard Euler.

2.7182818284590452353602874713527...e

The number e is defined by:

If n is a positive integer, then 1

1n

en

as n (discussed in Calculus)

NOTE: The number e is a number, not a variable.

Sketch ( ) xf x e Then, sketch the following

4( ) xf x e

( ) 3xf x e

( ) 3 xf x e

1( ) 2xf x e

Page 129

Page 130

Exponential Functions In exercises 1-8, graph each function by making a table of coordinates.

1. 5( ) xf x 2. 4( ) xf x 3. 1

3

( )x

f x

4. 1

2

( )x

f x 5. 3

2

( )x

f x 6. 4

3

( )x

f x

7. 0 6( ) .x

f x 8. 0 9( ) .x

f x

By translating, reflecting, and stretching the graph of 2( ) xf x , obtain the graphs of the following functions. Give the domain, range, and equation of any asymptotes of the function.

9. 12 ( ) xf x 10. 22 ( ) xf x 11. 2 2 ( ) xf x

12. 2 1 ( ) xf x 13. 22 3 ( ) xf x 14. 12 3 ( ) xf x

15. 2( ) xf x 16. 2 1 ( ) xf x 17. 2 ( ) xf x

18. 12 ( ) xf x 19. 12 ( ) xf x 20. 12 3 ( ) xf x

21. 1

2 32

( ) xf x 22. 112

2 ( ) xf x 23. 12 2 1 ( ) xf x

By translating, reflecting, and stretching the graph of 3( ) xf x , obtain the graphs of the following functions. Give the domain, range, and equation of any asymptotes of the function.

24. 23 ( ) xf x 25. 13 ( ) xf x 26. 3 1 ( ) xf x

27. 3 2 ( ) xf x 28. 13 3 ( ) xf x 29. 23 3 ( ) xf x

30. 3( ) xf x 31. 3 1 ( ) xf x 32. 3 ( ) xf x

33. 13 ( ) xf x 34. 13 ( ) xf x 35. 13 2 ( ) xf x

36. 1

3 32

( ) xf x 37. 12 3 ( ) xf x 38. 12 3 1 ( ) xf x

By translating, reflecting, and stretching the graph of ( ) xf x e , obtain the graphs of the following functions. Give the domain, range, and equation of any asymptotes of the function.

39. 1( ) xf x e 40. 2( ) xf x e 41. 2 ( ) xf x e

42. 1 ( ) xf x e 43. 1 2 ( ) xf x e 44. 2 1 ( ) xf x e

45. ( ) xf x e 46. 2 ( ) xf x e 47. ( ) xf x e

48. 1 ( ) xf x e 49. 2 ( ) xf x e 50. 2 1 ( ) xf x e

51. 2 3 ( ) xf x e 52. 11

2 ( ) xf x e 53. 11

12

( ) xf x e

Page 131

Answers—Exponential Functions 1. 3. 5.

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

7. 9. D: ( , ) , R: (0, ) 11. D: ( , ) , R: (2, ) Asymptote: 0y Asymptote: 2y

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

13. D: ( , ) , R: ( 3, ) 15. D: ( , ) , R: (0, ) 17. D: ( , ) , R: ( ,0) Asymptote: 3y Asymptote: 0y Asymptote: 0y

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

Page 132

19. D: ( , ) , R: ( ,0) 21. D: ( , ) , R: ( 3, ) 23. D: ( , ) , R: (1, ) Asymptote: 0y Asymptote: 3y Asymptote: 1y

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

-4 -3 -2 -1 1 2 3 4

-6-5-4-3-2-1

12

-4 -3 -2 -1 1 2 3 4

-2-1

123456

25. D: ( , ) , R: (0, ) 27. D: ( , ) , R: (2, ) 29. D: ( , ) , R: ( 3, ) Asymptote: 0y Asymptote: 2y Asymptote: 3y

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

-4 -3 -2 -1 1 2 3 4

-2-1

123456

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

31. D: ( , ) , R: (1, ) 33. D: ( , ) , R: ( ,0) 35. D: ( , ) , R: ( , 2) Asymptote: 1y Asymptote: 0y Asymptote: 2y

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

Page 133

37. D: ( , ) , R: (0, ) 39. D: ( , ) , R: (0, ) 41. D: ( , ) , R: ( 2, ) Asymptote: 0y Asymptote: 0y Asymptote: 2y

-4 -3 -2 -1 1 2 3 4

-2-1

123456

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

43. D: ( , ) , R: (2, ) 45. D: ( , ) , R: (0, ) 47. D: ( , ) , R: ( ,0) Asymptote: 2y Asymptote: 0y Asymptote: 0y

-4 -3 -2 -1 1 2 3 4

-2-1

123456

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

49. D: ( , ) , R: ( ,0) 51. D: ( , ) , R: ( 3, ) 53. D: ( , ) , R: (1, ) Asymptote: 0y Asymptote: 3y Asymptote: 1y

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

-4 -3 -2 -1 1 2 3 4

-6-5-4-3-2-1

12

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

Page 134

GraphsofLogarithmicFunctions

Review: Find the inverse of 2

( )3

xf x

Find the inverse of ( ) 2xf x

The exponential function has an inverse called _______________________

DefinitionofLogarithmicFunction:

For 0x and 0, 1b b , yb x is equivalent to logby x

The function ( ) logbf x x is the logarithmic function with base b.

GraphthefollowingInverseFunctions:

( ) 2xf x 2( ) logf x x

Domain:___________________ Domain:___________________

Range:____________________ Range:____________________

X-int:_____________________ X-int:_____________________

Y-Int:_____________________ Y-Int:_____________________

Asymptote:________________ Asymptote:________________

Page 135

GraphthefollowingInverseFunctions:

( ) 4xf x 4( ) logf x x

Domain:___________________ Domain:___________________

Range:____________________ Range:____________________

x-int:_____________________ x-int:_____________________

y-int:_____________________ y-int:_____________________

Asymptote:________________ Asymptote:________________

Note: Some bases are used frequently, and have simplified notation. Common Log (Base 10): 10log x = Natural Log (Base e): loge x =

Inverse Functions Logarithm Form Exponential Form ( ) xf x b

( ) 10xf x

( ) xf x e

Transformations:

Parent Function: logbf x x All logarithmic functions (in “basic” form) have 2 points on their

graphs: (1,0) and ( ,1)b

Vertical Shift logbf x x c

logbf x x c

Horizontal Shift logbf x x c

logbf x x c

Reflections logbf x x

Vertical Stretch and Compress

logbf x c x

Page 136

Sketch ( ) logf x x Then, sketch the following

( ) log 2f x x

( ) log( 2)f x x

( ) log( 1) 4f x x

( ) 2logf x x

( ) log( 2) 3f x x

Page 137

Sketch ( ) lnf x x Then, sketch the following

( ) ln 3f x x

( ) ln 3f x x

( ) ln( 2) 4f x x

1( ) ln2f x x

( ) ln( 1) 2f x x

Page 138

Logarithmic Functions In exercises 1-8, sketch the graphs of each pair of functions on the same set of axes. Label all asymptotes.

1. 5( ) xf x and 5( ) logg x x

2. 4( ) xf x and 4( ) logg x x

3. 1

4

( )x

f x and 1

4

( ) logg x x

4. 1

2

( )x

f x and 1

2

( ) logg x x

5. ( ) xf x e and ( ) lng x x

6. 10( ) xf x and ( ) logg x x By translating, reflecting, and stretching the graph of ( ) logf x x , obtain the graphs of the following functions. Give the domain, range, and equation of any asymptotes of the function. 7. 1 ( ) log( )f x x 8. 2 ( ) log( )f x x

9. 1 ( ) logf x x 10. 2 ( ) logf x x

11. 2 3 ( ) log( )f x x 12. 1 4 ( ) log( )f x x

13. ( ) logf x x 14. 2( ) logf x x

15. 1 ( ) logf x x 16. 2 ( ) logf x x

17. 1 2 ( ) log( )f x x 18. 2 3 ( ) log( )f x x By translating, reflecting, and stretching the graph of ( ) lnf x x , obtain the graphs of the following functions. Give the domain, range, and equation of any asymptotes of the function. 19. ( ) ( 1)f x ln x 20. ( ) ( 3)f x ln x

21. 4f ( x ) ln x 22. ( ) 3f x lnx

23. ( ) ( 2) 1f x ln x 24. ( ) ( 2) 4f x ln x

25. ( ) lnf x x 26. 12( ) logf x x

27. 2 3 ( ) ln( )f x x 28. 1 2 ( ) ln( )f x x

Page 139

Answers—Logarithmic Functions 1. 3. 5.

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

-4 -3 -2 -1 1 2 3 4

-4-3-2-1

1234

7. D: (1, ) , R: ( , ) 9. D: (0, ) , R: ( , ) 11. D: ( 2, ) , R: ( , ) Asymptote: 1x Asymptote: 0x Asymptote: 2x

-2 -1 1 2 3 4 5 6

-4-3-2-1

1234

-2 -1 1 2 3 4 5 6

-4-3-2-1

1234

-2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

12

13. D: (0, ) , R: ( , ) 15. D: (0, ) , R: ( , ) 17. D: (1, ) , R: ( , ) Asymptote: 0x Asymptote: 0x Asymptote: 1x

-2 -1 1 2 3 4 5 6

-4-3-2-1

1234

-2 -1 1 2 3 4 5 6

-4-3-2-1

1234

-2 -1 1 2 3 4 5 6

-2-1

123456

Page 140

19. D: ( 1, ) , R: ( , ) 21. D: (0, ) , R: ( , ) 23. D: (2, ) , R: ( , ) Asymptote: 1x Asymptote: 0x Asymptote: 2x

-2 -1 1 2 3 4 5 6

-4-3-2-1

1234

-2 -1 1 2 3 4 5 6

-2-1

123456

-2 -1 1 2 3 4 5 6

-4-3-2-1

1234

25. D: (0, ) , R: ( , ) 27. D: ( 2, ) , R: ( , ) Asymptote: 0x Asymptote: 2x

-2 -1 1 2 3 4 5 6

-4-3-2-1

1234

-2 -1 1 2 3 4 5 6

-2-1

123456

Page 141

Page 142

Properties of Logarithms

Review of exponent properties:

32 05 7

3 11 1

6

Compare exponents to logarithms:

Exponential Form Log Form

32 8

2 13

9

81 9

3 27 3

2log 16 4

3

1 1log

23

Evaluate the following expressions:

7log 49 3log 27 6

1log

6 3

1log

9

6log 6 3

1log

3 81log 9 12log 1

Common logarithms: Base Ten

2log10

5log 10

Page 143

Natural Logarithms: Base e

2lne

1ln

e ln e

Since logs and exponents are inverse functions, they undo one another.

The following properties show this:

log xa a x and

loga xa x

25log 5 3ln e 7log 57

ln8e 4log 114 log1000

PropertiesofLogarithms For 0M and 0N :

1. Product Rule: log log logb b bM N MN

2. Quotient Rule: log log logb b b

MM N

N

3. Power Rule: log log pb bp M M

Use the properties of logs to expand each expression as much as possible.

3

4

2loga

p

qr

2

53

loga

u

v

Express as a single logarithm:

33log 2log log

2a a ax y z 2log log ( 3) log ( 3)a a ax x x

Page 144

Properties of Logarithms

Evaluate each expression without using a calculator.

1. 4log 16

2. 2log 64

3. 5

1log

5

4. 2

1log

8

5. 7log 7

6. 2

1log

2

7. 64log 8

8. 5log 5

9. 4log 1

10. 75log 5

11. 8log 198

12. log100

13. 7log10

14. log3310

15. ln1

16. 6ln e

17. 6

1ln

e

18. ln125e

19. 9ln xe

20. 2ln5xe

21. 3log10 x

Use properties of logarithms to express each of the following as sums or differences of simpler logarithms.

22. 3

loga

xy

z 23.

2 4

6log

3a

w z

x 24.

22 4log

1a

x

x

Use properties of logarithms to express each of the following as a single logarithm.

25. 12log log 1

2a ax x 26. 1

log 4log log3 a a ax y z

27. log log 1 log 1a a ax x x 28. 4ln 2lnx y

29. 2 2 2log log logx y z 30. 3 3 3

1log 2 log log

2x y z

Page 145

Properties of Logarithms-Answers

1. 4log 16 2

2. 2log 64 6

3. 5

1log 1

5

4. 2

1log 3

8

5. 7

1log 7

2

6. 2

1 1log

22

7. 64

1log 8

2

8. 5log 5 1

9. 4log 1 0

10. 75log 5 7

11. 8log 198 19

12. log100 2

13. 7log10 7

14. log3310 33

15. ln1 0

16. 6ln 6e

17. 6

1ln 6

e

18. ln125 125e

19. 9ln 9xe x

20. 2ln5 25xe x

21. 3log 310 x x

22. 1

3 32

log log loga a ax y z

23. 2 4 3 6 log log log loga a a aw z x

24. 2 12 4 1

2 log ( ) log ( )a ax x

25. 2 1log ( )a x x

26. 4 3

loga

y xz

27. 1 1

log( )( )a

xx x

28. 4 2ln x y

29. 2

logxyz

30. 3 2

logx

y z

Page 146

Exponential Equations

An exponential equation has the variable in the exponent. The easiest way to solve

is to use the same base. ( x ya a )

Process: 1. Isolate the base. 2. Re-write each side of the equation with the same base. 3. Equate the exponents. 4. Solve.

Solve the following:

2 16x 2 14 64x

1 316 8x x 1

497

x

3 23

1xee

2 19 27x

Page 147

An exponential equation has the variable in the exponent. When you cannot get the bases to be the same, you have to use logarithms to solve. We make use of the following logarithmic property:

log log pa ap M M or ln ln pp M M

Process:

1. Isolate the base. 2. Take the log (ln) of both sides of the equation. 3. Use the log property (above) to re-write the exponents as coefficients. 4. Solve. 5. Use a calculator to approximate the solution.

Solve the following:

7 60x 2 140 8 x

2 1 7 13xe 35 11 35x

1 54 3x x 1 2 53 7x x

Page 148

Exponential Equations

Solve each equation.

If =x ya a , then =x y

1. 4 8x =

2. 13 81x+ =

3. 1

28

x =

4. 19 27x+ =

5. 25 125x =

6. 1 125

5x− =

7. 3 1 110

100x+ =

8. 4 116 4x− =

9. 1

366

x⎛ ⎞= ⎜ ⎟⎝ ⎠

10. 3 54 8x x+ =

11. 2 121xee

− =

12. 2xe e −=

13. 1 49 27x =

14. 6 4125 0.2x+ =

15. 2 8 3 12 8x x+ −=

16. 223 81x =

Solve. Round answers to 3

decimal places.

17. 4 21x =

18. 7 35x =

19. 52 11x =

20. 17 20x+ =

21. 13 16x+ =

22. 3 25 7x+ =

23. 29 17 6x− − =

24. 4 1 11xe + =

25. 9 107xe =

26. 53 25xe =

27. 2 34 120xe − =

28. 3 41000 3000xe − =

29. 4 13 1 19xe + + =

30. 23(2 9 ) 11x+ =

31. 3 4 23xe − =

32. 1 2 53 7x x+ −=

33. 8 2 5 22 3x x+ −=

34. 3 1 5 45 7x x+ +=

Page 149

Exponential Equations-Answers

1. 3

2⎧ ⎫⎨ ⎬⎩ ⎭

2. { }3

3. { }−3

4. ⎧ ⎫−⎨ ⎬⎩ ⎭

1

4

5. ⎧ ⎫⎨ ⎬⎩ ⎭

3

2

6. ⎧ ⎫⎨ ⎬⎩ ⎭

3

2

7. { }−1

8. ⎧ ⎫⎨ ⎬⎩ ⎭

3

8

9. { }−2

10. ⎧ ⎫−⎨ ⎬⎩ ⎭

10

3

11. ⎧ ⎫−⎨ ⎬⎩ ⎭

1

2

12. ⎧ ⎫⎨ ⎬⎩ ⎭

5

2

13. ⎧ ⎫⎨ ⎬⎩ ⎭

1

6

14. ⎧ ⎫−⎨ ⎬⎩ ⎭

13

18

15. ⎧ ⎫⎨ ⎬⎩ ⎭

11

7

16. { }± 2

17. ln21ln4

x = ≈2.196

18. ln35ln7

x = ≈ 1.827

19. ln115ln2

x = ≈ 0.692

20. ln20

1ln7

x = − ≈ 0.540

21. ln16

1ln3

x = − ≈1.524

22. ln7 2ln5

3ln5x

−= 0.264≈ −

23. ln23

2ln9

x = + ≈3.427

24. 1 ln11

4x

− += ≈0.349

25. 107

ln9

x ⎛ ⎞= ⎜ ⎟⎝ ⎠

≈2.476

26. 1 25

ln5 3

x ⎛ ⎞= ⎜ ⎟⎝ ⎠

≈0.424

27. 3 ln30

2x

+= ≈3.201

28. 4 ln3

3x

+= ≈1.700

29. 1 ln6

4x

− += ≈0.198

30. ( )5ln 3

2ln9x = ≈0.116

31. 3 ln23

4x

−= ≈ − 0.034

32. ln3 5ln7

ln3 2ln7x

− −=

−≈3.877

33. 2ln3 2ln2

8ln2 5ln3x

− −=

−≈ − 68.760

34. 4 ln7 ln5

3ln5 5ln7x

−=

−≈ − 1.260

Page 150

Logarithmic Equations—Form 1

log log constantb bM N 1. Combine logs using log properties. 2. If the coefficients do not equal to one, then use the log property:

log log pb bp M M

2. Now your equation should have the form of: log constantb D

3. Re-write the equation in exponential form to get rid of the log: constantb D 4. Solve for variable. 5. Check your answer back into the original equation. (You can only take the log of numbers that are greater than zero.)

Solve the following:

5log (3 2) 2x 4 2 3ln( x )

log( 2) log( 1) 1x x 2

2 2log ( 3 ) log ( 2) 2x x x

Page 151

Logarithmic Equations—Form 2

log log logb b bM N C 1. Combine logs using log properties. 2. If the coefficients do not equal to one, then use the log property:

log log pb bp M M

2. Now your equation should have the form of: log logb bD C

3. Using property of equality, you can now say that D C 4. Solve for variable. 5. Check your answer back into the original equation. (You can only take the log of numbers that are greater than zero.)

2 2 2log log ( 5) log 6x x ( 4) ( 1)ln x ln x ln x 2log log7 log100x 3 3 3log (4 ) log ( 8) log (2 13)x x x

( 4) ( 1) (3 12)ln x ln x ln x

Page 152

Logarithm Equations

Solve the following: (Check your solutions!)

1. 3log (4 7) 2x

2. 2log (4 7) 3x

3. ln(5 2 ) 2x

4. 3 3log log ( 2) 1x x

5. 4 4log log ( 12) 3x x

6. 3 3log log ( 24) 4x x

7. 4 4log ( 3) log ( 3) 2x x

8. 5 5log (4 15) log 2x x

9. 4 4log ( 2) log ( 1) 1x x

10. 2 2log (4 10) log ( 1) 3x x

11. 4 4log (3 1) log ( 1) 1x x

12. log(3 2) log( 1) 1x x

13. log(2 1) log( 3) 1x x

14. log ( ) log ( )7 71 5 1x x

15. 22 2log ( 6 ) log (1 ) 3 * *x x x

**new problem as of 7/30/15

16. 6 6 6log 3 log 4 log 24x

17. 2 2 2log ( 5) log log 4x x

18. 8 8 8log ( 1) log log 4x x

19. 1

log( 4) log(3 10) logx xx

20. 3 3 3log log ( 6) log 27x x

21. 3 3 3log ( 9) log ( 6) log 126x x

22. 7 7 7log log (3 11) log 4x x

23. 6 6 61 1

log log 9 log 272 3

x

24. ln( 2) ln( 4) ln3x x

25. 2log log5 log1000x

26. log log( 7) 3log2x x

Page 153

Logarithm Equations-Answers

1. 4x

2. 154

x

3. 25

1.1952e

x

4. 1x

5. 16x

6. 27x

7. 5x

8. 54

x

9. 2x

10. 12

x

11. No Solution

12. 127

x

13. 298

x

14. 6x

15. 5x

16. 2x

17. 53

x

18. 13

x

19. 5x

20. 3x

21. 12x

22. 4x

23. 9x

24. 7x

25. 50 2x

26. 1x

Page 154

Matrices

Review:

Solve the system by elimination method:   2 9

4 3 14

x y

x y

 

 

 

 

We will be using matrices to solve Systems of Linear Equations.   A matrix is a rectangular array of numbers arranged in rows and columns placed inside brackets. The numbers inside the brackets of a matrix are called elements. 

        Example:

13 15 21

7 5 6

12 1 41

 

 An Augmented Matrix is a matrix that is used to represent a system of Linear Equations. It has a vertical bar separating the columns of the matrix into 2 groups (one for the coefficients of the variables in the linear system and the other for the answers).    o Note: If a variable is missing, we assign it the coefficient of ZERO.   Rewrite the following system of equations in an augmented matrix System  

System of Equations  Augmented Matrix 

3 9

3 3

2 0

x y z

x y z

x y z

 

      

3 5 12

4 5

2 3 4

x z

y z

x y

 

      

One method to solve a system of linear equations using a matrix is to get the matrix in Echelon Form, which means to have only 1s in your main diagonal (going from upper‐left to lower‐right) and 0s below the ones.  

Once you are in Echelon Form, you will use back substitution to solve your system.  

    Example:  

1 2 1 0

0 1 2 3

0 0 1 2

 

Page 155

 

RowOperations

There are 3 row operations that produce matrices that represent systems with the same solution set. 

Row Operation   Notation   Example  

Interchange 2 Rows  1 3R R   1 1 3 3

3 18 12 21

1 2 1 0

 Multiply a Row by a non‐zero number  

2 22R R   1 2 1 0

3 18 12 21

1 1 3 3

 Multiply a Row by a non‐zero number and then add the product to any other row  

2 3 316 R R R   1 2 1 0

6 36 24 42

1 1 3 3

 

 To solve a system of linear equations using matrices we would  

1. Write system as an augmented matrix  2. Use row operations to get row equivalent matrix in Row Echelon Form  3. Use Substitution to solve for the variables. Answer solution in an ordered triple.  

Solve the following System of Equations: 

1. 3 9

3 3

2 0

x y z

x y z

x y z

Page 156

2. 2 2

2 2 6

3 2 15

x y z

x y z

x y z

 

Page 157

Matrices

Solve the following using matrices.

1.

x y zx y zx y z

2

2 3 2 4

4 3 1

2.

x y zx zx y z

2 9

2 2

3 5 2 22

3.

2

2 5

2 2 1

x y z

x y z

x y z

4.

3 0

1

3 11

x y

x y z

x y z

5.

3 4 3

2 2 8

2 3 9

x y z

x y z

x y z

                 Answers in RANDOM order:   (3, 1, 1) , ( 1,5,0) , ( 1,2, 2) , (3,4, 5) , (1, 1,2)

Page 158

Midpoint and Distance

Distance Formula: Use to find the distance between two points .

Example 1: Find the distance between A(4,8) and B(1,12)

Example 2: Find the distance between A ( 1,4) and B (3, 2)

2 22 1 2 1distance ( ) ( )x x y y

Page 159

Midpoint Formula: Use to find the center of a line segment

Example 3: Find the midpoint of A ( 2, 1) and B ( 8,6)

2 1 2 1midpoint ,2 2

x x y y

Page 160

Circles

Circles

Definition of Circle: A circle is a set of points in a plane that are located a fixed

distance, called the radius from a given point in the plane called the center.

Standard Form of the Equation of a Circle:

The standard form of the equation of a circle with

the center ( , )h k and the radius r is

2 2 2( ) ( )x h y k r

Example 1: Write the equation of the circle in standard form given:

Center (2, 1) and 4r

Example 2: Write the equation of the circle in standard form given:

Center ( 5,3) and 3 2r

Page 161

Example 3: Give the center and radius of the circle:

2 2( 1) ( 4) 25x y

Example 4: Give the center and radius of the circle:

2 2( 3) 20x y

Example 5: Give the center and radius of the circle:

2 2 2 8 6 0x y x y

Example 6: Give the center and radius of the circle:

2 2 8 2 19 0x y x y

Page 162

Example 7: Find an equation, in standard form, of the circle that satisfies the

given conditions.

a) Center (4, 2) and tangent to the

x-axis.

b) Center ( 3,5) and tangent to the

y-axis.

c) Center at the origin; passes through (4,1)

d) Center is (6,7) ; passes through (1, 3)

Page 163

e) Endpoints of the diameter are ( 8,1) and (2,7)

f) Endpoints of the diameter are ( 9, 8) and ( 3,0)

Page 164

Circles

Determine the length and the midpoints of the line segments with the given endpoints. 1. 7,2 and 1, 4

2. 9, 3 and 6,5

3. 5,8 and 1, 2

4. 6, 3 and 1,4

Find an equation, in standard form, of the circle that satisfies the given conditions.

5. Center 0,0 ; 7r

6. Center 3,2 ; 5r

7. Center 1,4 ; 2r

8. Center 3, 1 ; 3r

9. Center 4,2 ; 3 2r

Find the center and radius of the circle described by the equation.

10. 2 2 16x y

11. 2 2( 3) ( 1) 36x y

12. 2 2( 3) ( 2) 4x y

13. 2 2( 2) ( 2) 49x y

14. 2 2( 1) 12x y

15. 2 2( 1) 20x y Complete the square and write equation in standard form. State the center and the radius.

16. 2 2 6 2 6 0x y x y

17. 2 2 8 4 16 0x y x y

18. 2 2 10 6 30 0x y x y

19. 2 2 4 12 9 0x y x y

20. 2 2 8 2 8 0x y x y

21. 2 2 12 6 4 0x y x y

22. 2 2 2 15 0x y x

23. 2 2 6 7 0x y y Find an equation, in standard form, of the circle that satisfies the given conditions.

24. Center 8, 3 ; tangent to the x-axis.

25. Center 4,5 ; tangent to the x-axis.

26. Center 8, 3 ; tangent to the y-axis.

27. Center 4,5 ; tangent to the y-axis.

28. Center at the origin; passes through 5, 3

29. Center at the origin; passes through 2,7

30. Center at the 1,3 ; passes through 5,5

31. Center at the 2, 5 ; passes through 1, 3

32. Endpoints of the diameter are P 1,1 and Q 5,5

33. Endpoints of the diameter are P 1,3 and Q 7, 5

34. Endpoints of the diameter are P 7,1 and Q 3,9

35. Endpoints of the diameter are P 6, 2 and Q 0,6

Page 165

Circles

Answers: 1. 10, (3, 1)d M

2. 3

17, ( ,1)2

d M

3. 2 29, (3,3)d M

4. 5 1

7 2, ,2 2

d M

5. 2 2 49x y

6. 2 2( 3) ( 2) 25x y

7. 2 2( 1) ( 4) 4x y

8. 2 2( 3) ( 1) 3x y

9. 2 2( 4) ( 2) 18x y

10. Center 0,0 ; 4r

11. Center 3,1 ; 6r

12. Center 3,2 ; 2r

13. Center 2, 2 ; 7r

14. Center 0,1 ; 2 3r

15. Center 1,0 ; 2 5r

16. 2 2( 3) ( 1) 4x y ;

Center 3, 1 ; 2r

17. 2 2( 4) ( 2) 4x y ;

Center 4, 2 ; 2r

18. 2 2( 5) ( 3) 64x y ;

Center 5,3 ; 8r

19. 2 2( 2) ( 6) 49x y ;

Center 2,6 ; 7r

20. 2 2( 4) ( 1) 25x y ;

Center 4,1 ; 5r

21. 2 2( 6) ( 3) 49x y ;

Center 6,3 ; 7r

22. 2 2( 1) 16x y ;

Center 1,0 ; 4r

23. 2 2( 3) 16x y ;

Center 0,3 ; 4r

24. 2 28 3 9x y

25. 2 24 5 25x y

26. 2 28 3 64x y

27. 2 24 5 16x y

28. 2 2 34x y

29. 2 2 53x y

30. 2 21 3 20x y

31. 2 22 5 5x y

32. 2 22 3 13x y

33. 2 23 1 32x y

34. 2 22 5 41x y

35. 2 23 2 25x y

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