Review of chpts P and 1
Name___________________________________
SHORT ANSWER. Write the word or phrase that best
completes each statement or answers the question.
Find the product.
9) (x - 11)(x2 + 6x - 3)
Simplify the exponential expression.
1) (-10x3y8)2
10) (x + 7)(9x2 + 4x + 6)
2) (x8y)2
Factor the trinomial
11) x2 + 2x - 63
3)
xy5 -2
x4 y
12) x2 - x - 54
13) 2x2 + 3x - 5
12x-4y-2z3 -3
4)
3xy-2z-3
14) 5x2 - 9x - 2
Use the quotient rule to simplify the expression.
96x4
5)
4x
6)
15) 4x2 - 9y2
Find all numbers that must be excluded from the
domain of the rational expression.
x+5
16)
2
x - 11x + 30
48x4
2x
Simplify the rational expression. Find all numbers that
must be excluded from the domain of the simplified
rational expression.
x2 + 10x + 24
17)
x2 + 14x + 48
Perform the indicated operations.
7) (6x4 - 9x3 + 9x2 - 6) - (8x4 + 7x3 - 6x2 + 3)
8) (5x6 + 15x3 - 10) - (9x6 + 10x3 + 9)
1
Multiply or divide as indicated.
2x + 8 5x + 20
18)
÷
15
20
26) (2 - 7i) + (6 + 5i) + (2 + 5i)
27) (9 + 4i)(3 - 2i)
19)
15x - 15 5x - 5
÷
9
36
28) (7 - 6i)(-3 + 2i)
29)
3i
1 + 7i
30)
4 - 6i
5 - 8i
Add or subtract as indicated.
5x + 5 5x + 13
+
20)
5x + 9
5x + 9
x2 - 7x
10
+
21)
x 2
x-2
22)
2
4
x+3 x-3
23)
4
5
+
2
2
x - 3x + 2 x - 1
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)
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)
34) -4(5x - 3) < -24x + 28
2
Answer Key
Testname: CHPT P AND 1 REVIEW
1) 100x6y16
2) x16y2
3)
x6
y8
4)
x15
64z18
34) (-∞, 4)
-3
5) 2 x 6x
6) 2 x 6x
7) -2x4 - 16x3 + 15x2 - 9
8) -4x6 + 5x3 - 19
9) x3 - 5x2 - 69x + 33
9x3 + 67x2 + 34x + 42
(x - 7)(x + 9)
prime
(2x + 5)(x - 1)
(5x + 1)(x - 2)
(2x + 3y)(2x - 3y)
x ≠ 6, x ≠ 5
x+4
17)
, x ≠ -8, -6
x+8
10)
11)
12)
13)
14)
15)
16)
18)
8
15
19) 12
20) 2
21) x - 5
-2x - 18
22)
(x + 3)(x - 3)
23)
9x - 6
(x - 1)(x + 1)(x - 2)
24)
25)
26)
27)
28)
{77}
11 + 8i
10 + 3i
35 - 6i
-9 + 32i
21 3
+
29)
i
50 50
30)
68 2
+
i
89 89
31) 14i
32) i( 7 - 10)
33) (1, ∞)
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
3
-2
-1
0
1
2
3
4
5
6
7
8
9
10 11
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. 2x2 + 8x + 40 = 0
Solve by using the quadratic formula
7. 4x2 = -4x + 2
8. x2 + 2x – 24 = 0
Solve using any method
9. 3x2 – 6x = -1
10. 6x2 – 15x – 36 = 0
11. (3x + 2)2 = 36
12. 8x2 + 6x = -5
13. 2 x2  x  12  3x2  6 x
14. x2  4 x  4
15. x2  14 x  38  0
16. x2  4 x  1  5
17. 8x2  16 x  42  0
18. x2  18x  40
Compute the discriminant. Then determine the number and type of solutions:
1 real solution, 2 real solutions, or 2 complex solutions
19. x2  5x  2  0
20. 9 x2  3x  2  0
21. x2  9  6 x
22. 4 x2  4 x  6
23. 2 x2  5x  4  0
24. 9 x2  6 x  6  5
1.5 Review: Solving Quadratics
Solve using factoring
1. x2 + 5x + 6 = 0
2. 10x2 – 35 = 65x
 1 
7, 
 2
{-2, -3}
Solve using the square root property
3. 3x2 = 27
{3}
4. (x – 4)2 = -25
{4  5i}
Solve by completing the square
5. x2 + 6x + 5 = 0
1, 5
6. 2x2 + 8x + 40 = 0
2  4i
Solve by using the quadratic formula
7. 4x2 = -4x + 2
 1  3 


 2 
Solve using any method
9. 3x2 – 6x = -1
 3  6 


 3 
11. (3x + 2)2 = 36
8. x2 + 2x – 24 = 0
6, 4
10. 6x2 – 15x – 36 = 0
 3 
4, 
 2
13. 2 x2  x  12  3x2  6 x
3, 4
12. 8x2 + 6x = -5
 3  i 31 


8


2
14. x  4 x  4
2
15. x2  14 x  38  0
16. x2  4 x  1  5
 8 4 
 , 
 3 3
7 
87

2  i 2
17. 8x2  16 x  42  0
 3 7 
 , 
2 2 
18. x2  18x  40
19. x2  5x  2  0
17; 2 real solutions
20. 9 x2  3x  2  0
-63; 2 complex solutions
21. x2  9  6 x
0; 1 real solution
22. 4 x2  4 x  6
-80; 2 complex solutions
23. 2 x2  5x  4  0
57; 2 real solutions
24. 9 x2  6 x  6  5
0; 1 real solution
2, 20
Chapter 2 Review for Final
Name___________________________________
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.
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)
Use the vertical line test to tell if they are functions.
14)
3) {(-7, -3), (-4, 8), (4, 5), (9, 9)}
y
Determine whether the relation is a function.
4) {(-6, -7), (-1, 4), (3, 4), (3, -7)}
x
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
15)
y
8) x2 + y = 25
x
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)
1
State how the graph is moved, graph it
21) h(x) = (x - 7)2 + 7
16)
y
x
22) g(x) = - x + 2 + 2
17)
y
23) g(x) = x - 3 - 4
x
24) g(x) = x - 3 - 2
Use the given conditions to write an equation for the
line in point-slope form.
18) Slope = -4, passing through (4, 5)
25) Draw a graph of a
a) a quadratic
b) a line
c) absolute value
d) square root
e) cubic
19) Passing through (6, 4) and (8, 5)
20) Passing through (2, 8) and (3, 4)
2
Answer Key
Testname: CHAPTER 2 REVIEW
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
domain = {6, -4, -1}; range = {8, -7, -4, -8}
domain = {-6, -4, 4, -1}; range = {1, 9, -8, 4}
domain = {-4, 4, 9, -7}; range = {8, 5, 9, -3}
Not a function
Not a function
Function
y is a function of x
y is a function of x
y is not a function of x
23
3
3x2 - 10x + 4
$353
not a function
not a function
function
not a function
y - 5 = -4(x - 4)
1
1
19) y - 4 = (x - 6) or y - 5 = (x - 8)
2
2
23)
10
8
6
4
2
-10 -8 -6 -4 -2-2
-4
10
8
6
-10 -8 -6 -4 -2-2
-4
-6
-8
-10
2
4
6 8 10 x
-6
-8
-10
22)
y
6
4
2
-10 -8 -6 -4 -2-2
8 10 x
2
-4
10
8
4 6
y
4
2
6 8 10 x
2
24)
4
4
8 10 x
-8
25)
2
4 6
-10
y
-10 -8 -6 -4 -2-2
2
-6
20) y - 8 = - 4(x - 2) or y - 4 = - 4(x - 3)
21)
10
8
6
y
-4
-6
-8
-10
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
2
Find the x-intercepts of the function f(x) = 3x – 17x -6
2
Determine the end behavior of f(x) = 6x + 4x + 3
[A]
[C]
rises to left and rises to right
rises to left and falls to right
[B]
[D]
falls to left and rises to right
falls to left and falls to right
2
Does f(x) = -7x + 14x + 1 have a maximum or minimum? What is it?
a. Maximum (-2, 0)
b. Minimum (-1, 0)
Find the x - intercepts of the function.
a. (3, 0) (1, 0)
b. (-1/3, 0) (-1, 0)
c. Maximum (1, 8)
d. Minimum (-2, -55)
( )
c. (-1, 0) (-3, 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
f (t )  6t 2  10t  4 . 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?
2
3. A diver goes off a diving board . The function is f(x) = -16t + 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
2
Find the x-intercepts of the function f(x) = 3x – 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
2
Determine the end behavior of f(x) = 6x + 4x + 3
[A] rises to left and rises to right
[C] rises to left and falls to right
Does f(x)
A: Parabola and +6 = opens up.
[B] falls to left and rises to right
[D] falls to left and falls to right
= -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.
a. (3, 0) (1, 0)
b. (-1/3, 0) (-1, 0)
( )
0 = (3x+1)(x+1)  x= -1/3 or x=-1  B
c. (-1, 0) (-3, 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 = x2 + 8x +16 + 6
Y = x2 + 8x +22
y = 3 (x2 -2x -8)
y = 3 (x2 -2x -8)
y = 3x2 -6x -24
GRAPHING QUADRATIC FUNCTIONS:
( )
( )
AOS
X= - b/2a
= 4 / 2(2) = 4/4
X=1
(
)
X –value of the vertex
( )
(
)(
)
2 ways to find:
1) Average of the zeros
X = -4
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
ZEROS
Y-INT
D:
R:
Ends
Graph:
g(1) = 2(1)2 – 4(1) – 6
= 2–4–6
= -8
(1, -8)
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
At (h,k)
g(-3) = -1/2 (-3+8) (-3 -2)
= -1/2 (5) (-5)
= 12.5
(-3, 12.5)
(-4, -9)
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)
(3, 0) (-1,0)
g(0) = 2(0)2 – 4(0) – 6 = -6
(0, -6)
h(0) = - (0+4)2 -9 = -16-9 = -27
(0, -27)
(
(
(
(
Up
Down
g(0) = -1/2 (0+8) (0-2)
= -1/2 (8) (-2)
= 8
(0, 8)
(
)
(
)
Down
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
length
200 – 2x
width
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 yds
2
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
f (t )  6t 2  10t  4 . 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?
2
f(1.5) = -6 (1.5) + 10(1.5) + 4 = 5.5
The ball is 5.5 feet in the air.
2
3. A diver goes off a diving board . The function is f(x) = -16t + 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.
3.2 - 3.4 Review for Final
Name___________________________________
13)
Are these functions? If not circle what makes it not a
function.
1) f(x) = 3x + 4x4
2) f(x) = 9 -
-3x3 - 11x2 + 24x + 20
x+5
1
x6
14) (x4 + 81) ÷ (x - 3)
3) f(x) =
4
x5 - x4 - 3
4) f(x) = 2x3 + 3x2 - 2x-3 + 27
Find the degree of the polynomial function.
9 - x5
5) f(x) =
4
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
6) 16x4 + 6x3 + 4x + 3y5 + 5
Find the zeros - factor
7) f(x) = x3 + x2 - 12x
16) x3 + 6x2 - x - 6 = 0
8) f(x) = x3 - 6x2 + 9x
9) f(x) = x3 + 7x2 - x - 7
Find an nth degree polynomial function with real
coefficients satisfying the given conditions.
17) n = 3; 3 and i are zeros; f(2) = 30
10) f(x) = 5(x + 1)(x - 3)3
Divide using synthetic division.
11) (x2 + 6x + 5) ÷ (x + 4)
18) n = 3; - 5 and i are zeros; f(-3) = 60
12) (x2 + 10x + 15) ÷ (x + 4)
1
Answer Key
Testname: 3.2 REVIEW FOR FINAL
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
Yes
No
No
No
5
5
x = 0, x = - 4, x = 3
x = 0, x = 3
x = -1, x = 1, x = - 7
x = -1, x = 3,
3
11) x + 2 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