Algebra 1 – Final Review Packet
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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 $$$.
Algebra 1 – Final Review Packet
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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
2
25
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.
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*** 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 _____
2. 5x3y10 _____
3. 3x + 5x3 _____
a) trinomial
b) binomial
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
9. 9.82 × 105
10. 9 × 10-3
11. 7.24 × 10-6 Write in scientific notation. 12. 400,000
13. 5412
14. 0.00056
15. 0.0000814
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Simplify the following. Leave all answers with positive exponents. 16. x2 • x7 17. (a4)12 18. (3b)3 19. 9770
20. 3−51
21. –6x8y–8
22. (2w2x4)3
23. 9
2
y28x
−
24. 6
14
xx
25. 8
yx⎟⎟⎠
⎞⎜⎜⎝
⎛
26. 5
6
5
4
fd
df7d
•
27. 3t4 v3
21t2 v6
Multiply or Divide. Express your answers in scientific notation. 28. (2.3 × 102)(4.5 × 10–7)
29. 4.8 × 102
1.2 × 105
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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) 34. (x + 3)(4x + 5) 35. (3x – 1)(8x + 1)
36. (x – 6)(x – 8) 37. (7x + 3)(7x – 2) 38. (3x3 – 2x2 + 6)(x + 5)
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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 47. x2 + xy – 42y2
Factor the following differences of squares completely. 48. x2 – 121 49. 100a2 – 144 50. 5m2 – 20
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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 59. y2 + 23y – 24 = 0
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Chapter 10 Multiply or Divide.
60. 23 •
316 61.
34 ÷
78
62. 5x2 •
x15 63.
5x2 ÷
x15
Add or Subtract.
64. 34 +
74 65.
13 –
57
66. (5x – 7) – (8x – 12) 67.
(12x + 4)5 –
(4x + 3)5
Simplify Completely.
68. 12x4y6
8x7y2 69. 3x + 9
3x
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70. 3a + 9b
12a2 71. 6y2 + 3y3y2 + 6y
72. 14a2 − 14b2
21a − 21b 73. b2 − 10b + 21b2 − 11b + 28
Multiply.
74. 52x •
4x17 75.
2x2
x • 52x
76. 4x
2x + 2 • 4x + 4
8x 77. m2 − 4
5m • 4m2
m + 2
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Divide.
78. 2x7 ÷
12x21 79.
10x3
5x ÷ 2x3y6x2
80. 7
x + 4 ÷ 26
x + 4 81. 4x − 6
5 ÷ 6x − 9
25
Add or Subtract.
82. 4x7 +
6x7 83.
5x + 3x + 3 +
3x + 7x + 3
84. 2w2 + w
w3 – 9w3 85.
(4x + 2)(3x + 1) −
(7x − 6)(3x + 1)
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86. 3x2 + 2x − 5
5x + 1 + 2x2 − x + 6
5x + 1 87. 13x –
5x
88. 5m
m − 1 − 3
m(m − 1) 89. 2uuv3 −
vu2 v2
90. 3a
a + 2 + −1
a 91. 1
x − 4 − x + 4
x2 − x – 12
92. 3
x – 2 + 3
x2 + 4x – 12 93. 15
b2 − 9 − 7
2b − 6
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Chapter 11 94. Estimate which two integers each square root is between:
a) 73 is between ____ and ____ b) 13 is between ____ and ____
95. Simplify (show work):
a) 180 b) 252 c) x2y3 d) 196b5
96. Simplify to find (show work):
a) 81 • 25 b) 700
7
c) 3 7 + 8 7 d) 5 24 − 4 6
e) 5
3 f) 5 ( )5 + 5
97. Find the missing side length for each triangle (show your steps):
a) b)
u 13 12
2 v 5
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98. Solve (show work):
a) 4x + 7 = 15 b) x + 5 + 8 = 19 99. 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 = _____ substitution
b = _____
c = _____
solution(s) for x:
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102. 6x + 5 = -2x2
_____________________ (standard form)
a = _____ substitution
b = _____
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) Substitute values for a, b, & c into the discriminant formula. b) Find the discriminant. c) Tell how many solutions the quadratic has.
104. 3x2 + 4 = 3x a) d = __________________ (substitution)
b) d = __________________ (simplified)
c) _____________________ (# of solutions)
105. 4x + 2 = 3x2 a) d = __________________ (substitution)
b) d = __________________ (simplified)
c) _____________________ (# of solutions)
106. x2 = 8x – 16 a) d = __________________ (substitution)
b) d = __________________ (simplified)
c) _____________________ (# of solutions)
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107. 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
108. Solve each of the following equations by factoring: SHOW WORK.
a) x2 – 8x + 16 = 0 b) x2 + 4x – 21 = 0
109. Complete the square for the following: a) x2 + 20x __________ b) u2 – 26u __________
110. Solve using any method you want… show work!!
a) x2 – 49 = 0 b) x2 + 10x = –4
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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. g(x) = –| x – 3 | + 6
f(1) = g(–2) =
f(–3) = g(–1) =
f(0) = g(4) =
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Determine which of the following graphs represent functions. 117. 118. 119. y y y
x x x
function / not a function function / not a function function / not a function State the domain and range for the following graphs. 120. 121. 122. y Domain: _________ Domain: _________ Domain: _________
Range: __________ Range: __________ Range: __________ Graph the following functions.
123. f(x) = | x | 124. g(x) = | x – 3 | – 4 125. h(x) = – 23 x + 2
x
y
x
y
x
y
x x
y y
x
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Graph each of the following quadratic functions, finding all indicated information. 126. f(x) = x2
– b 2a
Vertex
y-intercept x-intercept(s)
127. f(x) = x2 – 5
– b 2a
Vertex
y-intercept x-intercept(s)
128. f(x) = x2 – 4x – 12
– b 2a
Vertex
y-intercept x-intercept(s)
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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.
a) y = 23 when x = 27 b) y = 4 when x = 8
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131. Find the Vertex: f(x) = 4x2 + 8x + 1
132. Write the equation in Standard Form and then identify a, b, and c. 3x2 = -2x + 1
133. Solve using the quadratic formula: 3x2 -7x +4 = 0
134. Find the x and y intercepts: f(x) = x2 – 3x – 10
135. Function? Yes or No Domain: ________________ Range: _________________
136. What is the Domain and the Range of a relation defined by: {(5,2), (5,3), (6,2), (3,1)} Domain: _________________ 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.
y
x
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139. Find the equation of variation where y varies inversely with x, and y = 14 and x = 2.
140. 41 =−x
141. 16)3( 2 =+x 142. What number should be added to complete the square? X2 + 6x + __
143. 254
144. 325
145. =− 212
146. 2)52(
147. Find the length of side a.
9
a
3