Algebra 1 – Final Review Packet
Name: _________________________ Per.: _____ Date: _______________
Algebra 1 FINAL EXAM REVIEW Spring Semester Material (by chapter)
Your Algebra 1 Final will be on _________________________ at ________. You will need to bring your textbook and number 2 pencils with you to the final exam. The final exam will cover the entire year. Re-review the material from the fall semester as well.
Do not lose this packet. Replacement packets will cost $$$.
1
Algebra 1 – Final Review Packet
What You Must Memorize For Final
1) Quadratic Formula:
x=
–b ±
b2 – 4ac 2a
2) Standard Form: ax2 + bx + c = 0 3) Perfect Squares from 0 to 169: √0 , √1 , √4 , √16 … , 4) Discriminant:
b2 - 4ac
5) Complete the Square: x2
5x + ___ 5
2
25
2
4
6) Direct Variation: y=kx 7) Inverse Variation:
8) Vertex: 9) Pythagorean Theorem: a2 + b2 = c2 10) X and Y Intercepts: To find y-intercept, set x’s equal to zero. To find, x-intercepts, set y equal to zero and solve for x. 2
Algebra 1 – Final Review Packet *** show your work wherever applicable for full credit ***
Chapter 5 Match each of the following polynomials with its special term name. 1.
4x2 + 27x – 8
_____
a)
trinomial
2.
5x3y10
_____
b)
binomial
3.
3x + 5x3
_____
c)
monomial
Add or subtract the following polynomials. 4. (8t2 – 10t + 2) + (8t + 13)
_________________________
5.
(x2 + 5x – 1) – (7x2 + 2)
_________________________
6.
(x4 + 7x3 + 7) – (2x4 – 4x3 + 1)
_________________________
7.
(3n3 + n2 – n – 4) + (5n3 – 4n2 + 11)
_________________________
Write in decimal form (standard notation). 8. 8 × 104
10.
9 × 10-3
11.
7.24 × 10-6
Write in scientific notation. 12. 400,000
14.
0.00056
13.
15.
0.0000814
9.
9.82 × 105
5412
3
Algebra 1 – Final Review Packet Simplify the following. Leave all answers with positive exponents. 22. (2w2x4)3 16. x2 • x7
17.
(a4)12
18.
(3b)3
23.
28x 2 y −9
24.
x14 x6
25.
⎛x⎞ ⎜⎜ ⎟⎟ ⎝y⎠
8
19.
9770
20.
1 5−3
26.
7d4 d6 • df 5 f 5
21.
–6x8y–8
27.
3t4 v3 21t2 v6
Multiply or Divide. Express your answers in scientific notation. 28. (2.3 × 102)(4.5 × 10–7) 4.8 × 102 29. 1.2 × 105
4
Algebra 1 – Final Review Packet Multiply. 30. 3x(4x – 9)
_________________________
31.
4x2(x + 6)
_________________________
32.
2x2(15x3 – 10)
_________________________
Multiply the following. Use your choice of methods, but show your work!! 33. (x + 9)(x – 6) (x – 6)(x – 8) 36.
34.
(x + 3)(4x + 5)
37.
(7x + 3)(7x – 2)
35.
(3x – 1)(8x + 1)
38.
(3x3 – 2x2 + 6)(x + 5)
5
Algebra 1 – Final Review Packet
Chapter 6 Factor out the largest possible monomial. 39. 5x2 – 15 40. 8a + 10b – 16
41.
3c4 – 6c2 – 15c
Factor completely (remember – they are not always “ready” to go…). 42. x2 + 9x + 14 43. y2 – 15y + 54 44. t2 + 8t + 15
45.
m2 + 23m – 24
46.
x2 – x – 12
Factor the following differences of squares completely. 48. x2 – 121 49. 100a2 – 144
6
47.
x2 + xy – 42y2
50.
5m2 – 20
Algebra 1 – Final Review Packet Factor the following perfect square trinomials completely. 51. x2 + 18x + 81 52. 25m2 + 30m + 9
53.
4y3 – 16y2 + 16y
Factor completely (remember – they are not always “ready” to go…). 54. 3y2 – 20y + 12 55. 2x2 – 13x – 45 56. 18n3 + 33n2 – 6n
Solve for the given variable. 57. (a – 5)(a + 2) = 0
58.
x(x – 3) = 0
7
59.
y2 + 23y – 24 = 0
Algebra 1 – Final Review Packet
Chapter 10 Multiply or Divide. 2 3 60. • 3 16
62.
5x x • 15 2
Add or Subtract. 3 7 64. + 4 4
66.
61.
3 7 ÷ 4 8
63.
5x x ÷ 15 2
65.
(5x – 7) – (8x – 12)
67.
Simplify Completely. 12x4y6 68. 8x7y2
69.
8
1 5 – 7 3
(4x + 3) (12x + 4) – 5 5
3x + 9 3x
Algebra 1 – Final Review Packet 70.
3a + 9b 12a2
72.
14a2 − 14b2 21a − 21b
Multiply. 5 4x 74. • 2x 17
76.
4x 4x + 4 • 2x + 2 8x
9
71.
6y2 + 3y 3y2 + 6y
73.
b2 − 10b + 21 b2 − 11b + 28
75.
2x2 5 • 2x x
77.
m2 − 4 4m2 • 5m m+2
Algebra 1 – Final Review Packet Divide. 2x 12x 78. ÷ 7 21
80.
7 26 ÷ x+4 x+4
Add or Subtract. 4x 6x 82. + 7 7
84.
2w2 + w 9 – 3 w w3
79.
10x3 2x3y ÷ 6x2 5x
81.
4x − 6 6x − 9 ÷ 25 5
83.
5x + 3 3x + 7 + x+3 x+3
85.
(4x + 2) (7x − 6) − (3x + 1) (3x + 1)
10
Algebra 1 – Final Review Packet 3x2 + 2x − 5 2x2 − x + 6 86. + 5x + 1 5x + 1
87.
1 5 – 3x x
88.
5m 3 − m−1 m(m − 1)
89.
2u v − 2 2 uv3 u v
90.
3a −1 + a+2 a
91.
1 x+4 − 2 x−4 x − x – 12
93.
15 7 − b −9 2b − 6
92.
3 3 + 2 x–2 x + 4x – 12
11
2
Algebra 1 – Final Review Packet
Chapter 11 94.
Estimate which two integers each square root is between:
a) 95.
96.
73 is between ____ and ____
13 is between ____ and ____
Simplify (show work):
a)
180
b)
252
c)
x2y3
d)
196b5
b)
700 7
Simplify to find (show work):
81 •
a)
c)
3
25
7 +8
7
d)
5 3
e)
97.
b)
5
24 − 4
5 (5 +
f)
6
5
)
Find the missing side length for each triangle (show your steps): a) b)
13
u
2
12
v
5
12
Algebra 1 – Final Review Packet 98.
99.
Solve (show work): 4x + 7 = 15 a)
b)
x + 5 + 8 = 19
To hang a math poster in his office, Mr. Zito leaned a 10 foot ladder against the wall, placing the bottom of the ladder on the floor 3 feet away from the wall. How high up on the wall was the ladder? (Show work. A labeled picture is also required)
Chapter 13 100.
Write the quadratic formula.
Solve the following using the QUADRATIC FORMULA. Complete all blanks.
101.
3x2 + 4 = 8x
_____________________ (standard form) a = _____ b = _____
substitution
c = _____
solution(s) for x:
13
Algebra 1 – Final Review Packet 102.
6x + 5 = -2x2
_____________________ (standard form) a = _____ b = _____
substitution
c = _____
solution(s) for x:
103.
Write the formula for the discriminant
For questions 5 – 7, the blanks provided are for the following information:
a) b) c) 104.
105.
106.
Substitute values for a, b, & c into the discriminant formula. Find the discriminant. Tell how many solutions the quadratic has.
3x2 + 4 = 3x
4x + 2 = 3x2
x2 = 8x – 16
a)
d = __________________ (substitution)
b)
d = __________________ (simplified)
c)
_____________________ (# of solutions)
a)
d = __________________ (substitution)
b)
d = __________________ (simplified)
c)
_____________________ (# of solutions)
a)
d = __________________ (substitution)
b)
d = __________________ (simplified)
c)
_____________________ (# of solutions)
14
Algebra 1 – Final Review Packet 107.
108.
Solve each of the following using the zero-product property. SHOW WORK.
a)
(x + 10)(x + 24) = 0
b)
(2x – 6)(5x + 2) = 0
c)
x(x + 158) = 0
d)
5x(4x – 16)(x + 9) = 0
Solve each of the following equations by factoring: SHOW WORK.
a)
109.
b)
x2 + 4x – 21 = 0
b)
u2 – 26u __________
Complete the square for the following:
a) 110.
x2 – 8x + 16 = 0
x2 + 20x __________
Solve using any method you want… show work!! b) x2 + 10x = –4 a) x2 – 49 = 0
15
Algebra 1 – Final Review Packet
Chapter 12 Identify the domain and range for each. Tell if the relation is a function.
111.
h: {(3, 1), (2, 4), (3, 5), (4, 8)}
112.
j: {(2, 7), (3, 6), (4, 5), (5, 4), (6, 3)}
a)
domain: ____________________
a)
domain: ____________________
b)
range:
b)
range:
c)
function / not a function
c)
function / not a function
113.
____________________
k: {(1, 2), (2, 3), (3, 2), (4, 1)}
114.
____________________
m: {(4, 5), (4, 2), (4, 1), (4, 3), (1, 6)}
a)
domain: ____________________
a)
domain: ____________________
b)
range:
b)
range:
c)
function / not a function
c)
function / not a function
____________________
____________________
Find the indicated outputs for the following functions.
115.
f(x) = –4x2 – 2
116.
f(1) =
g(–2) =
f(–3) =
g(–1) =
f(0) =
g(4) =
16
g(x) = –| x – 3 | + 6
Algebra 1 – Final Review Packet Determine which of the following graphs represent functions. 117. 118. 119. y
y
x
function / not a function
y
x
function / not a function
function / not a function
State the domain and range for the following graphs. 120. 121. y y
x
x
122.
y
x
x
Domain: _________
Domain: _________
Domain: _________
Range: __________
Range: __________
Range: __________
Graph the following functions.
123.
f(x) = | x |
124.
y
g(x) = | x – 3 | – 4
125.
y
h(x) = –
2 x+2 3
y
x
x
17
x
Algebra 1 – Final Review Packet Graph each of the following quadratic functions, finding all indicated information. 126. f(x) = x2 b Vertex – 2a
y-intercept
127. –
f(x) = x2 – 5
b 2a
Vertex
y-intercept
128. –
x-intercept(s)
x-intercept(s)
f(x) = x2 – 4x – 12
b 2a
y-intercept
Vertex
x-intercept(s)
18
Algebra 1 – Final Review Packet 129.
Find an equation of variation where y varies directly as x for each pair of values given. a) y = 3 when x = 24 b) y = 50 when x = 25
130.
Find an equation of variation where y varies inversely as x for each pair of values given. 2 a) y = when x = 27 b) y = 4 when x = 8 3
19
Algebra 1 – Final Review Packet
132. Write the equation in Standard Form and then identify a, b, and c.
131. Find the Vertex: f(x) = 4x2 + 8x + 1
3x2 = -2x + 1 133. Solve using the quadratic formula: 134. Find the x and y intercepts: 3x2 -7x +4 = 0 f(x) = x2 – 3x – 10
135.
136. What is the Domain and the Range of a relation defined by: {(5,2), (5,3), (6,2), (3,1)}
y
x
Function? Yes or No
Domain: _________________
Domain: ________________
Range: __________________
Range: _________________ 137. Find the indicated outputs for the function: f(x) = 3x2 – 1 for f(-2)
138. Find the equation of variation where y varies directly as x, and y = 14 and x = 2.
20
Algebra 1 – Final Review Packet
140. 139. Find the equation of variation where y varies inversely with x, and y = 14 and x = 2.
141. ( x + 3) 2 = 16
x −1 = 4
142. What number should be added to complete the square? X2 + 6x + __
143.
4 25
144.
25 3
145.
12 − 2 =
146. (2 5 ) 2
147. Find the length of side a. 9 a
3
21
Algebra 1 – Final Review Packet
ANSWER KEY
17
a48
18
27b3
19
1
20
53 or 125
21
6
22
8w6x12
23
28x2y9 or 28y9x2
24
X8
25
26 27
7
7
28
1.035x10‐4
29
4.4x10‐3
30
12x2‐27x
31
4x3+24x2
32
30x5‐20x2
33
x2+3x‐54
Page 3‐21
34
4x2+17+15
1
a
35
24x2+5x‐1
2
c
36
X2‐14x=48
3
b
37
45x2+7x‐6
4
8t2‐2t+15
38
3x4+13x3‐10x2+6x+30
5
‐6x2+5x‐3
39
5(x2‐3)
6
‐x4+11x3+6
40
2(4a+5b‐8)
7
8n ‐3n ‐n+7
41
3c(c3‐2c‐5)
8
80,000
42
(x+2)(x+7)
9
982,000
43
(y‐6)(8‐9)
10
.009
44
(t+3)(t+5)
11
.00000724
45
(m‐1)(m+24)
46
(x+3)(x‐4)
12
3
2
5
4 x 10 3
13
5.412 x 10
47
(x‐6y)(x+7y)
14
‐4
48
(x‐11)(x+11)
49
(10a‐12)(10a+12)
50
5(m‐2)(m+2)
15 16
5.6 x 10
‐5
8.14 x 10 9
X
22
Algebra 1 – Final Review Packet
ANSWER KEY 81
10 3
2
82
10 7
4y(y‐2)
2
83
54
(3y‐2)(y‐6)
84
55
(2x+5)(x‐9)
85
56
3n(6n‐1)(n+2)
86
57
a=5 ; a=‐2
87
58
X=0 ; x=+3
88
59
Y=1 ; y=‐24
89
60
1 8
90
61
6 7
91
2
51
(x+9)
52
(5m+3)
53
8
10 3
2
9 3 3
8 1
5
1
5
1
14 3 5
3 1
2
3
2 2
1 4 3
3
2
21
92
63
75 2
93
64
5 2
94
65
8 21
95
a) 6√5 ; b) 6√7 ; c) xy
66
‐3x+5
96
a) 45 ; b) 10 ; c) 11√7 ; d) 6√6 ; e)
62
6
8
67
1 5 3 2
68
3
69
2
78 79 80
8 and 9 3 and 4 ; d) 14
97
a) 5 b) √29 or 5.4
98
a) 16 b) 116
99
√91 or 9.5 4
√ 2
√
102
NO SOLUTION; can’t have a negative inside radical
4
103
2 5 1 2 6
x = 2 & x =
10 17 5 1
75 76
3
101
3 4
74
a) b)
7
1 2 3
73
7 3
100
2
72
2
4
71
77
3
70
6
104 105 106
b) 40 ; c) 2 solutions b) 0 ; c) 1 solution
107
7 26
a) ‐10 & ‐24 ; b) 3 & d) 0 & 4 & ‐9
; c) 0 & ‐158
108
a) 4 ; b) ‐7 & 3
109
a) +100 ; b) +169
110
23
4 b) ‐39 ; c) zero solutions
a) 7 & ‐7 ; b) 5
√21
√
Algebra 1 – Final Review Packet
ANSWER KEY
111
a) {2,3,4} ; b) {1,4,5,8} ; c) not a function
112
a) {2,3,4,5,6} ; b) {3,4,5,6,7} ; c) function
113
a) {1,2,3,4} ; b) {1,2,3} ; c) function
114
a) {1,4} ; b) {1,2,3,5,6} ; c) not a function
115
‐6 ; ‐38 ; ‐2
116
1 ; 2 ; 5
117
Not a function
118
Not a function
119
function
120
Domain: {x|x≤2} ; Range: {y|all real numbers}
121
Domain: {x|1≤x≤5} ; Range: {y|1≤y≤4}
122
Domain: all real numbers ; Range: all real numbers
123
Graph should form a v:
124
Graph should form a v:
125
Graph should be a negative sloped line:
126
0 ; vertex (0,0) ; y‐intercept (0,0) ; x‐intercepts (0,0)
127
0 ; vertex (0,‐5) ; y‐intercept (0,‐5) ; x‐intercepts = √5 √5 , 0 & √5 , 0
128
2 ; vertex (2,‐16) ; y‐intercept (0,‐12) ; x‐intercepts = 6 & ‐2 or (6,0) & (‐2,0)
129
a)
130
a)
; b) y=2x ; b)
131
(‐1, ‐3)
132
3x2 + 2x 1 = 0 a=3 ; b=2 ; c= 1
133
& 1
134
‐2 & 5 ; or (‐2, 0) & (5,0)
135
Yes ; all real numbers; {y|y
136
{3,5,6} ; {1,2,3}
137
11
138
y=7x 28
139
4
140
17
141
1 & 7
24
Algebra 1 – Final Review Packet 142
9
143
2 5
144
5√3 3
145
2√3
146
20
147
√78
√2
25
