Review of chpts P and 1

Name___________________________________

SHORT ANSWER. Write the word or phrase that best

completes each statement or answers the question.

Simplify the exponential expression.

1) (-10x3y8)2

2) (x8y)2

3)xy5

x4y

-2

4)12x-4y-2z3

3xy-2z-3

-3

Use the quotient rule to simplify the expression.

5)96x4

4x

6)48x4

2x

Perform the indicated operations.

7) (6x4 - 9x3 + 9x2 - 6) - (8x4 + 7x3 - 6x2 + 3)

8) (5x6 + 15x3 - 10) - (9x6 + 10x3 + 9)

Find the product.

9) (x - 11)(x2 + 6x - 3)

10) (x + 7)(9x2 + 4x + 6)

Factor the trinomial

11) x2 + 2x - 63

12) x2 - x - 54

13) 2x2 + 3x - 5

14) 5x2 - 9x - 2

15) 4x2 - 9y2

Find all numbers that must be excluded from the

domain of the rational expression.

16)x + 5

x2 - 11x + 30

Simplify the rational expression. Find all numbers that

must be excluded from the domain of the simplified

rational expression.

17)x2 + 10x + 24

x2 + 14x + 48

1

Multiply or divide as indicated.

18)2x + 8

15 ÷

5x + 20

20

19)15x - 15

9 ÷

5x - 5

36

Add or subtract as indicated.

20)5x + 5

5x + 9 +

5x + 13

5x + 9

21)x2 - 7x

x - 2 +

10

x - 2

22)2

x + 3 -

4

x - 3

23)4

x2 - 3x + 2 +

5

x2 - 1

Solve and check the linear equation.

24) (7x + 9) - 4 = 8(x - 9)

Add, subtract multiply or divide as indicated

25) (6 + 9i) - (-5 + i)

26) (2 - 7i) + (6 + 5i) + (2 + 5i)

27) (9 + 4i)(3 - 2i)

28) (7 - 6i)(-3 + 2i)

29)3i

1 + 7i

30)4 - 6i

5 - 8i

Perform the indicated operations and write the result in

standard form.

31) -36 + -64

32) -7 - -100

Solve the linear inequality.

33) 12x + 21 > 3(3x + 8)

34) -4(5x - 3) < -24x + 28

2

Answer KeyTestname: CHPT P AND 1 REVIEW

1) 100x6y16

2) x16y2

3)x6

y8

4)x15

64z18

5) 2 x 6x

6) 2 x 6x

7) -2x4 - 16x3 + 15x2 - 9

8) -4x6 + 5x3 - 19

9) x3 - 5x2 - 69x + 33

10) 9x3 + 67x2 + 34x + 42

11) (x - 7)(x + 9)

12) prime

13) (2x + 5)(x - 1)

14) (5x + 1)(x - 2)

15) (2x + 3y)(2x - 3y)

16) x ≠ 6, x ≠ 5

17)x + 4

x + 8, x ≠ -8, -6

18)8

15

19) 12

20) 2

21) x - 5

22)-2x - 18

(x + 3)(x - 3)

23)9x - 6

(x - 1)(x + 1)(x - 2)

24) {77}

25) 11 + 8i

26) 10 + 3i

27) 35 - 6i

28) -9 + 32i

29)21

50 +

3

50i

30)68

89 +

2

89i

31) 14i

32) i( 7 - 10)

33) (1, ∞)

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

34) (-∞, 4)

-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11

3

CHAPTER 1 FINAL REVIEW

1.

2. 4-(5x+7) = 4x-3(5x-9)

3.

4. (-7+8i) - (3+8i)

5. (6+i)(7-i)

6. √ √

7.

8.

1.5 Review: Solving Quadratics

Name _________________________________________ Hour _______

Solve using factoring

1. x2

+ 5x + 6 = 0 2. 10x2 – 35 = 65x

Solve using the square root property

3. 3x2 = 27 4. (x – 4)

2 = -25

Solve by completing the square

5. x2 + 6x + 5 = 0 6. 2x

2 + 8x + 40 = 0

Solve by using the quadratic formula

7. 4x2 = -4x + 2 8. x

2 + 2x – 24 = 0

Solve using any method 9. 3x

2 – 6x = -1 10. 6x

2 – 15x – 36 = 0

11. (3x + 2)2 = 36 12. 8x

2 + 6x = -5

13. 2 22 12 3 6x x x x 14. 2 4 4x x

15. 2 14 38 0x x 16. 2 4 1 5x x

17. 28 16 42 0x x 18. 2 18 40x x

Compute the discriminant. Then determine the number and type of solutions:

1 real solution, 2 real solutions, or 2 complex solutions

19. 2 5 2 0x x 20. 29 3 2 0x x

21. 2 9 6x x 22. 24 4 6x x

23. 22 5 4 0x x 24. 29 6 6 5x x

1.5 Review: Solving Quadratics Solve using factoring

1. x2

+ 5x + 6 = 0 2. 10x2 – 35 = 65x

{-2, -3} 1

7,2

Solve using the square root property

3. 3x2 = 27 4. (x – 4)

2 = -25

{ 3} {4 5 }i

Solve by completing the square

5. x2 + 6x + 5 = 0 6. 2x

2 + 8x + 40 = 0

1, 5 2 4i

Solve by using the quadratic formula

7. 4x2 = -4x + 2 8. x

2 + 2x – 24 = 0

1 3

2

6,4

Solve using any method 9. 3x

2 – 6x = -1 10. 6x

2 – 15x – 36 = 0

3 6

3

3

4,2

11. (3x + 2)2 = 36 12. 8x

2 + 6x = -5

8 4,

3 3

3 31

8

i

13. 2 22 12 3 6x x x x 14. 2 4 4x x

3,4 2

15. 2 14 38 0x x 16. 2 4 1 5x x

7 87 2 2i

17. 28 16 42 0x x 18. 2 18 40x x

3 7,

2 2

2,20

19. 2 5 2 0x x 20. 29 3 2 0x x

17; 2 real solutions -63; 2 complex solutions

21. 2 9 6x x 22. 24 4 6x x

0; 1 real solution -80; 2 complex solutions

23. 22 5 4 0x x 24. 29 6 6 5x x

57; 2 real solutions 0; 1 real solution

Chapter 2 Review for Final

Name___________________________________

Give the domain and range of the relation.

1) {(-4, -7), (-1, -4), (6, 8), (6, -8)}

2) {(4, -8), (-4, 9), (-1, 4), (-6, 1)

3) {(-7, -3), (-4, 8), (4, 5), (9, 9)}

Determine whether the relation is a function.

4) {(-6, -7), (-1, 4), (3, 4), (3, -7)}

5) {(-6, -3), (-6, -9), (2, -6), (4, 4), (8, -3)}

6) {(-3, -4), (3, 3), (5, -6), (8, 1), (11, -5)}

Does the equation define y as a function of x.

7) x + y = 49

8) x2 + y = 25

9) x + y2 = 25

Evaluate the function and simplify

10) f(x) = -5x + 8; f(-3)

11) f(x) = x + 11; f(-2)

12) f(x) = 3x2 - 4x - 3; f(x - 1)

Solve the problem.

13) The function P(x) = 0.45x - 97 models the

relationship between the number of pretzels

x that a certain vendor sells and the profit

the vendor makes. Find P(1000), the profit

the vendor makes from selling 1000 pretzels.

Use the vertical line test to tell if they are functions.

14)

x

y

x

y

15)

x

y

x

y

1

16)

x

y

x

y

17)

x

y

x

y

Use the given conditions to write an equation for the

line in point-slope form.

18) Slope = -4, passing through (4, 5)

19) Passing through (6, 4) and (8, 5)

20) Passing through (2, 8) and (3, 4)

State how the graph is moved, graph it

21) h(x) = (x - 7)2 + 7

22) g(x) = - x + 2 + 2

23) g(x) = x - 3 - 4

24) g(x) = x - 3 - 2

25) Draw a graph of a

a) a quadratic

b) a line

c) absolute value

d) square root

e) cubic

2

Answer KeyTestname: CHAPTER 2 REVIEW

1) domain = {6, -4, -1}; range = {8, -7, -4, -8}

2) domain = {-6, -4, 4, -1}; range = {1, 9, -8, 4}

3) domain = {-4, 4, 9, -7}; range = {8, 5, 9, -3}

4) Not a function

5) Not a function

6) Function

7) y is a function of x

8) y is a function of x

9) y is not a function of x

10) 23

11) 3

12) 3x2 - 10x + 4

13) $353

14) not a function

15) not a function

16) function

17) not a function

18) y - 5 = -4(x - 4)

19) y - 4 = 1

2(x - 6) or y - 5 =

1

2(x - 8)

20) y - 8 = - 4(x - 2) or y - 4 = - 4(x - 3)

21)

x-10 -8 -6 -4 -2 2 4 6 8 10

y10

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8 10

y10

8

6

4

2

-2

-4

-6

-8

-10

22)

x-10 -8 -6 -4 -2 2 4 6 8 10

y10

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8 10

y10

8

6

4

2

-2

-4

-6

-8

-10

23)

x-10 -8 -6 -4 -2 2 4 6 8 10

y10

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8 10

y10

8

6

4

2

-2

-4

-6

-8

-10

24)

x-10 -8 -6 -4 -2 2 4 6 8 10

y10

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8 10

y10

8

6

4

2

-2

-4

-6

-8

-10

25)

3

CHAPTER 3.1 QUADRATIC FUNCTIONS

List the 3 forms of the quadratic equation:

How do you determine if the function has a maximum or minimum?

Shifting quadratic equations: Graph the parent function x2. Then, show the shift on each graph.

y = x2 y = (x+5)2 + 1 (y-3) = (x-4)2

Find the x-intercepts of the function f(x) = 3x2 – 17x -6

Determine the end behavior of f(x) = 6x2 + 4x + 3

[A] rises to left and rises to right

[B] falls to left and rises to right

[C] rises to left and falls to right

[D] falls to left and falls to right

Does f(x) = -7x2 + 14x + 1 have a maximum or minimum? What is it?

a. Maximum (-2, 0) c. Maximum (1, 8)

b. Minimum (-1, 0) d. Minimum (-2, -55)

Find the x - intercepts of the function. ( )

a. (3, 0) (1, 0) c. (-1, 0) (-3, 0)

b. (-1/3, 0) (-1, 0) d. (1, 0) (1/3, 0)

Write one possible function of the given zeros: x = 4 and x = -5

Convert each to standard form y = ax2 + bx + c: y =(x+4)2 + 6 y = 3 (x+2)(x-4)

GRAPHING QUADRATIC FUNCTIONS:

1. ( ) – – 2. ( ) ( ) 3. ( )

( )( )

AOS _______________ AOS _______________ AOS _______________

zeros ______________ zeros ______________ zeros ______________

vertex _____________ vertex _____________ vertex _____________

y intercept _________ y intercept _________ y intercept _________

D: D: D:

R: R: R:

Ends Ends Ends

SKETCH GRAPH HERE: SKETCH GRAPH HERE: SKETCH GRAPH HERE:

1. A field bordering a straight stream is to be enclosed. The side bordering the stream won’t be fenced. 200 yards of fencing

material is to be used.

a. What is an equation to describe the possible dimensions of the field?

b. What is the maximum area? What would the dimensions be?

2. In a football game, a punter punts the ball downfield. The height of the ball in feet as a function of the number of seconds since

the ball was kicked can be measured by the function 2( ) 6 10 4f t t t . f(t)=the height of the ball t = time in

second

A. How many seconds until the ball reaches it’s maximum height?

b. How many seconds until the ball hits the ground again?

c. Where is the ball after 1.5 seconds?

3. A diver goes off a diving board . The function is f(x) = -16t2 + 64t + 160, where f(x) = his height and t = time in seconds.

a. How many seconds until the diver reaches his maximum height? What is his maximum height?

b. How many seconds until the diver hits the water?

c. After 6 seconds where is he?

CHAPTER 3.1 QUADRATIC FUNCTIONS

List the 3 forms of the quadratic equation: General/standard: y = ax2 + bx + c Vertex: y = a(x-h)2 +k, where (h, k) is the vertex form Factored: y = a(x-p)(x-q), where p and q are the x-intercepts

How do you determine if the function has a maximum or minimum?

If “a” is positive, the parabola opens upward The function has a minimum ex: y = 3x2 y = (x+2)2

If “a” is positive, the parabola opens downward The function has a maximum ex: y = -5x2 y = - (x+2)2 -3

Shifting quadratic equations: Graph the parent function x2. Then, show the shift on each graph.

y = x2 y = (x+5)2 + 1 (y-3) = (x-4)2

y = (x-4)2 +3

Find the x-intercepts of the function f(x) = 3x2 – 17x -6

x-intercepts occur when f(x) = 0. 0 = (3x + 1) (x – 6) 0 = 3x+1 0 = x-6 X = -1/3 x = 6

Determine the end behavior of f(x) = 6x2 + 4x + 3 A: Parabola and +6 = opens up.

[A] rises to left and rises to right [B] falls to left and rises to right [C] rises to left and falls to right [D] falls to left and falls to right

Does f(x) = -7x2 + 14x + 1 have a maximum or minimum? What is it? Opens down Maximum.

Maximums/minimums occur at the vertex. So, now, find the vertex at ( -b/2a, f(-b/2a)) x = -b / 2a x = -14/(2*-7) x = 1 y = -7(1)2 + 14(1) + 1 = -7 +14 +1 = 8

a. Maximum (-2, 0) c. Maximum ( 1, 8)

b. Minimum (-1, 0) d. Minimum (-2, -55)

Find the x - intercepts of the function. ( ) 0 = (3x+1)(x+1) x= -1/3 or x=-1 B

a. (3, 0) (1, 0) c. (-1, 0) (-3, 0)

b. (-1/3, 0) (-1, 0) d. (1, 0) (1/3, 0)

Write one possible function of the given zeros: x = 4 and x = -5 f(x) = (x-4)(x+5)

Convert each to standard form y = ax2 + bx + c: y =(x+4)2 + 6 y = 3 (x+2)(x-4)

Y = (x+4)(x+4) + 6 y = 3 (x2 -2x -8) Y = x2 + 8x +16 + 6 y = 3 (x2 -2x -8) Y = x2 + 8x +22 y = 3x2 -6x -24 GRAPHING QUADRATIC FUNCTIONS:

( ) ( ) ( ) ( )

( )( )

AOS X= - b/2a = 4 / 2(2) = 4/4 X = 1

X –value of the vertex X = -4

2 ways to find: 1) Average of the zeros

2) Convert back to y=ax2 + bx

+ c and use x = -b/2a form. Y = -1/2 (x2 + 6x – 16) Y = -1/2 x2 - 3x + 8

( )

X = -3

VERTEX g(1) = 2(1)2 – 4(1) – 6 = 2 – 4 – 6 = -8 (1, -8)

At (h,k) (-4, -9)

g(-3) = -1/2 (-3+8) (-3 -2) = -1/2 (5) (-5) = 12.5 (-3, 12.5)

ZEROS 0 = 2(x2 – 2x – 3) 0 = 2 (x-3)(x+1) divide by 2 0 = (x-3)(x+1) 0 = (x-3) 0 =(x+1) zero product X = 3 or x = -1 property (3, 0) (-1,0)

0 = - (x+4)2 - 9 9 = - (x+4)2 -9 = (x+4)2

3i = x+4 X = -4 + 3i or x = -4 – 3i

( )( )

0 = (x+8) (x-2) 0 = x +8 0 = x-2 -8 = x 2 = x

(-8, 0) (2, 0)

Y-INT g(0) = 2(0)2 – 4(0) – 6 = -6 (0, -6)

h(0) = - (0+4)2 -9 = -16-9 = -27 (0, -27)

g(0) = -1/2 (0+8) (0-2) = -1/2 (8) (-2) = 8 (0, 8)

D: ( ) ( ) ( )

R: ( ) ( ) ( )

Ends Up Down Down

Graph: Check using your calculator

Check using your calculator

Check using your calculator

1. A field bordering a straight stream is to be enclosed. The side bordering the stream won’t be fenced. 200 yards of fencing

material is to be used.

Label the sides with the dimensions in terms of x. Then, answer the following questions:

x

200 – 2x

a. What is an equation to describe the possible dimensions of the field? A(x) = x (200-2x)

b. What is the maximum area? What would the dimensions be?

Plug formula into calculator and find the maximum at (50, 5000), where x = 50 means the length of the field was 50. And

y=5000 represents a(x) or area. Then, find the width with the formula 200 – 2(50) = 100.

DIMENSIONS 50 X 100 Area: 5,000 yds2

2. In a football game, a punter punts the ball downfield. The height of the ball in feet as a function of the number of seconds since

the ball was kicked can be measured by the function 2( ) 6 10 4f t t t . f(t)=the height of the ball t = time in

second

A. How many seconds until the ball reaches it’s maximum height?

Plug formula into calculator and find max. x = time. Y = height. So, the max point at (0.833, 8.166) means that The ball reaches it’s maximum height at 8.166 feet high.

b. How many seconds until the ball hits the ground again?

Find the x-intercepts using the zero function on the calculator. The ball hits the ground after 2 seconds

c. Where is the ball after 1.5 seconds? f(1.5) = -6 (1.5)2 + 10(1.5) + 4 = 5.5

The ball is 5.5 feet in the air.

3. A diver goes off a diving board . The function is f(x) = -16t2 + 64t + 160, where f(x) = his height and t = time in seconds.

a. How many seconds until the diver reaches his maximum height? What is his maximum height?

Plug formula into calculator and find max. x = time. Y = height. So, the max point at (2, 224) means that The diver reaches it’s maximum height of 224 after 2 seconds.

c. How many seconds until the diver hits the water?

Find the x-intercepts using the zero function on the calculator. The diver hits the water at 5.7 seconds after diving.

c. After 6 seconds where is he? In the pool. F(6) = -32. So, she’s dove beneath the water, In reality, she is probably

swimming out of the pool or something like that. So, after hitting the water, the equation no longer correctly describes the

divers motion.

length

width

3.2 - 3.4 Review for Final

Name___________________________________

Are these functions? If not circle what makes it not a

function.

1) f(x) = 3x + 4x4

2) f(x) = 9 - 1

x6

3) f(x) = 4

x5 - x4 - 3

4) f(x) = 2x3 + 3x2 - 2x-3 + 27

Find the degree of the polynomial function.

5) f(x) = 9 - x5

4

6) 16x4 + 6x3 + 4x + 3y5 + 5

Find the zeros - factor

7) f(x) = x3 + x2 - 12x

8) f(x) = x3 - 6x2 + 9x

9) f(x) = x3 + 7x2 - x - 7

10) f(x) = 5(x + 1)(x - 3)3

Divide using synthetic division.

11) (x2 + 6x + 5) ÷ (x + 4)

12) (x2 + 10x + 15) ÷ (x + 4)

13)-3x3 - 11x2 + 24x + 20

x + 5

14) (x4 + 81) ÷ (x - 3)

Solve the polynomial equation. In order to obtain the

first root, use synthetic division to test the possible

rational roots.

15) x3 + 2x2 - 5x - 6 = 0

16) x3 + 6x2 - x - 6 = 0

Find an nth degree polynomial function with real

coefficients satisfying the given conditions.

17) n = 3; 3 and i are zeros; f(2) = 30

18) n = 3; - 5 and i are zeros; f(-3) = 60

1

Answer KeyTestname: 3.2 REVIEW FOR FINAL

1) Yes

2) No

3) No

4) No

5) 5

6) 5

7) x = 0, x = - 4, x = 3

8) x = 0, x = 3

9) x = -1, x = 1, x = - 7

10) x = -1, x = 3,

11) x + 2 - 3

x + 4

12) x + 6 - 9

x + 4

13) -3x2 + 4x + 4

14) x3+ 3x2 + 9x + 27 + 162

x - 3

15) {-3, -1, 2}

16) {1, -1, -6}

17) f(x) = -6x3 + 18x2 - 6x + 18

18) f(x) = 3x3 + 15x2 + 3x + 15

2