Download Algebra 2 Semester 2 Final Review Name: : Hour: ______ 7.1

Algebra 2
Semester 2 Final Review
Name: _______________________________________________________________Date: _______________________________ Hour: ______
7.1 Finding Rational Solutions of Polynomial Equations
Find all rational zeros for each equation.
1. 4𝑥 3 + 𝑥 2 − 4𝑥 − 1 = 0
2. 𝑥 5 − 2𝑥 4 − 24𝑥 3 = 0
3. 3𝑥 5 − 18𝑥 4 − 21𝑥 3 = 0
4. −𝑥 4 + 2𝑥 3 + 8𝑥 2 = 0
Identify the rational zeros of each function, then write the function in factored form.
5. 𝑓(𝑥) = 𝑥 3 + 3𝑥 2 + 3𝑥 + 1
6. 𝑓(𝑥) = 𝑥 3 + 5𝑥 2 − 8𝑥 − 48
7.2 Finding Complex Solutions of Polynomial Equations
7. 𝑥 4 − 2𝑥 3 − 14𝑥 2 − 2𝑥 − 15 = 0
8. 𝑥 4 − 16 = 0
9. 𝑥 4 + 4𝑥 3 + 4𝑥 2 + 64𝑥 − 192 = 0
10. 𝑥 3 − 64 = 0
11. 5𝑥 3 + 5𝑥 2 + 𝑥 + 1 = 0
12. 4𝑥 4 − 8𝑥 3 − 5𝑥 2 + 10𝑥 = 0
13. 5𝑥 4 − 11𝑥 2 + 6 = 0
14. 3𝑥 3 − 10𝑥 2 + 7𝑥 + 2 = 0
8.1 Graphing Simple Rational Functions
𝑔(𝑥) = 𝑎 (1
𝑏
1
)+𝑘
(𝑥−ℎ)
𝟏
Tell the transformations applied to the graph of 𝒇(𝒙) = 𝒙 to produce the graph of g(x). Then, give the
equations of the asymptotes and domain and range in set notation.
1
16. 𝑔(𝑥) = (𝑥+4) − 6
1
1
18. 𝑔(𝑥) = −3(𝑥−1) + 7
15. 𝑔(𝑥) = 2 (𝑥−3) + 5
1
17. 𝑔(𝑥) = − (𝑥+9) − 1
Graph 𝒈(𝒙) by finding its asymptotes, domain and range, and two points on each curve.
1
)+
𝑥−1
2
20. 𝑔(𝑥) =
1
−
(𝑥+3)
1
)−
𝑥+1
3
22. 𝑔(𝑥) =
1
+
−0.5(𝑥−2)
19. 𝑔(𝑥) = 3 (
21. 𝑔(𝑥) = − (
𝟏
Rewrite the function in 𝒈(𝒙) = 𝒂 ((𝒙−𝒉)) + 𝒌 or 𝒈(𝒙) = (𝟏
domain and range.
23. 𝑔(𝑥) =
3𝑥−4
𝑥−1
𝒃
𝟏
(𝒙−𝒉)
1
1
) + 𝒌 form and graph it. Then, state the
4𝑥−7
24. 𝑔(𝑥) = −2𝑥+4
𝟏
25. Write a function for the graph in the form 𝒈(𝒙) = 𝒂 ((𝒙−𝒉)) + 𝒌.
9.1 Adding and Subtracting Rational Expressions
Write an equivalent expression with the new denominator given.
1
26. 𝑥 ; den = 𝑥(𝑥 − 6)
27.
𝑥−1
; den = (𝑥
𝑥+5
𝑥2
28. 𝑥−3; den =𝑥(𝑥 − 3)
+ 5)(𝑥 − 1)
Simplify each expression. Identify any excluded values.
29.
2𝑥 2 −12𝑥+16
7𝑥 2 −28𝑥
30.
𝑥 2 −5𝑥
31.
𝑥 2 +2𝑥−35
𝑥 2 −1
𝑥−1
Find the Least Common Denominator (LCD) for each pair of expressions. Leave factored.
−14
32. 𝑥 2 −11𝑥+24 and
9
𝑥 2 −6𝑥+9
12𝑥
5
33. 15𝑥+60 = 𝑥 2 +9𝑥+20
Add or Subtact. Identify any excluded values.
34.
3𝑥+1
−7𝑥−4
+ 𝑥−2
𝑥−2
3𝑥 2 −1
𝑥+2
36. 𝑥 2 −3𝑥−18 − 𝑥−8
2𝑥 2
35. 𝑥 2 −5𝑥 +
𝑥+6
𝑥 2 +3𝑥−4
𝑥2
2𝑥
37. 𝑥 2 −7𝑥−18 − 𝑥−9
9.2 Multiplying and Dividing Rational Expressions
Multiply. Identify any excluded values.
38.
3𝑥 2
2𝑥 2 −6𝑥−20
∙
𝑥 2 −2𝑥−8 𝑥 2 −3𝑥−10
𝑥 2 −9
𝑥−8
40 𝑥 2 −5𝑥−24 ∙ 2𝑥2 −18𝑥
39.
𝑥 2 −8𝑥
7𝑥+35
∙
14(𝑥 2 +8𝑥+15) 𝑥+8
41.
𝑥
3𝑥−27
∙ 𝑥+1
𝑥−9
Divide. Identify any excluded values.
42.
44.
(𝑥+7)2
𝑥2
÷
𝑥 2 +9𝑥+14
𝑥 2 +𝑥−2
𝑥+11
2𝑥+6
÷ 𝑥 2 +2𝑥−3
4𝑥
43.
6𝑥
9𝑥 2 −27𝑥−36
÷
3𝑥−30
𝑥 2 −10𝑥
20
5𝑥 2 −40𝑥
45. 𝑥 2 −7𝑥 ÷ 𝑥 2 −15𝑥+56
9.3 Solving Rational Equations
Solve algebraically.
1
46. 𝑥 −
4
𝑥−2
3𝑥
4
= 3𝑥
1
48. 𝑥 2 −4 = 𝑥−2
5𝑥−5
5
1
47. 𝑥 2 −4𝑥 − 𝑥 2 −4𝑥 = 𝑥
49.
3𝑥+7
𝑥−5
5𝑥+17
= 2𝑥−10
10.1 Inverses of Simple Quadratic and Cubic Functions
Graph the function f(x) for the domain {𝒙|𝒙 ≥ 𝟎}. Then write and graph its inverse function, 𝒇−𝟏 (𝒙).
50. 𝑓(𝑥) = 0.25𝑥 2
51. 𝑓(𝑥) = 𝑥 2 + 3
Graph the function f(x). Then write and graph its inverse function, 𝒇−𝟏 (𝒙).
52. 𝑓(𝑥) = 0.5𝑥 3
53. 𝑓(𝑥) = 𝑥 3 − 2
Write the rule for the inverse of each function.
54. 𝑓(𝑥) = 𝑥 2 − 9, 𝑥 ≥ 0
55. 𝑓(𝑥) = 5𝑥 2 , 𝑥 ≥ 0
56. 𝑓(𝑥) = 0.75𝑥 3
57. 𝑓(𝑥) = 𝑥 3 + 7
10.2 Graphing Square Root Functions
Find the endpoint and two additional points to graph each function. Identify the domain and range.
58. 𝑓(𝑥) = √𝑥 − 4
59. 𝑓(𝑥) = 2√𝑥 + 1
Describe the transformations applied to the graph of 𝒇(𝒙) = √𝒙 to produce the graph of 𝒈(𝒙).
60. 𝑔(𝑥) = 4√𝑥 + 8
61. 𝑓(𝑥) = −√3𝑥 + 2
62. Write the function that matches the graph using the given transformation format.
𝑔(𝑥) = 𝑎√𝑥 − ℎ + 𝑘
10.3 Graphing Cube Root Functions
Tell the transformations that have been applied to the parent graph of 𝒇(𝒙) = 𝟑√𝒙 to produce the graph
of 𝒈(𝒙). Then graph each cube root function by finding the point of symmetry and two points on each
side.
13
3
63. 𝑔(𝑥) = √𝑥 − 3 + 2
64. 𝑔(𝑥) = 2 √𝑥 + 2 − 3
3
65. Write a function of the form 𝑔(𝑥) = 𝑎 √𝑥 − ℎ + 𝑘 that matches the graph.
11.1 Radical Expressions and Radical Exponents
Write each expression in radical form. Simplify numerical expressions when possible.
5
66. 646
2
69. 273
3
4
67. (6𝑥)2
68. (−8)3
1
70. (100𝑎)2
1
71. (7𝑥)−3
Write each expression by using rational exponents. Simplify numerical expressions when possible.
7
4
73. √514
2
5
3
5
72. (√2)
75. (√𝑛2 )
76.
74. (√169)
1
77.
3
(√3𝑚)
1
7
4
( √5𝑝)
11.2 Simplifying Radical Expressions
Simplify each expression. Assume all variables are positive.
3
78. −3√12𝑟
−
81. (4𝑥)
1
2
5
2
79. 42 ∙ 42
80. (27 ∙ 64)3
3
1
2
82.
∙ (9𝑥)
272
3
83. −5 √−500𝑥 5 𝑦 3
2
273
11.3 Solving Radical Equations
84.
x 6 7
85.
87.
14 x  2  x  3
88.
2x  5  3 x  1
86.
1
 x  42  6
89.
3
x  6  3 3 x  24
x 4 39
12.1 Arithmetic Sequences
Write an explicit and a recursive rule for each sequence. Give the constraints on n.
90.
n
0
1
2
3
4
f(n)
8
12
16
20
24
91.
n
1
2
3
4
…
f(n)
11
7
3
1

92. Given the explicit rule for the sequence, write its recursive rule. Then give the first four terms of the
sequence. 𝑓(𝑛) = 14 + 1.5𝑛, for 𝑛 ≥ 0.
93. Give the recursive rule for a sequence, write its explicit rule. Then give the first four terms of the
sequence. 𝑓(1) = 7, 𝑓(𝑛) = 𝑓(𝑛 − 1) + 47, 𝑓𝑜𝑟 𝑛 ≥ 1
12.2 Geometric Sequences
Write an explicit rule and a recursive rule for each sequence.
94.
n
0
1
2
3
4
f(n)
8
4
2
1
0.5
95.
n
1
2
3
4
…
f(n)
.2
1
5


Given each pair of terms of a geometric sequence, find the common ratio, r, and the first term, noted as
either f(0) or f(1), to write an explicit rule and a recursive rule for each sequence.
96. First term 𝑓(1); 𝑓(1) = 14, 𝑓(2) = 42
97. First term 𝑓(0); 𝑓(2) = 108, 𝑓(4) = 243
98. Given the explicit rule for the sequence, write its recursive rule. Then give the first four terms of the
sequence. f(n) = 7(3)n−1 , for 𝑛 ≥ 1
99. Give the recursive rule for a sequence, write its explicit rule. Then give the first four terms of the
sequence. f(1) = 8; f(n) = 2.5 ∙ f(n − 1) for n  2
12.3 Finite Geometric Series
Find the number of terms in the sequence, then find its sum.
100. -3 + 18 – 108 + … – 5038848
101. 4 + 20 + 100 … + 312500
13.1 and 13.2 Exponential Growth and Decay Functions
1
2
102. Given the function 𝑔(𝑥) = (2)𝑥−1 + 3 complete the following:
a) Tell the transformations that have been applied to the graph of 𝑓(𝑥) = 2𝑥 to produce the graph of 𝑔(𝑥).
b) Identify the asymptote, reference point, and at least two other points to
graph the function.
c) Give the domain and range in set notation.
d) Describe the end behavior.
1 𝑥+1
3
103. Given the function 𝑔(𝑥) = −2 ( )
− 2 complete the following:
1𝑥
a) Tell the transformations that have been applied to the graph of 𝑓(𝑥) = 3 to produce the graph of 𝑔(𝑥).
b) Identify the asymptote, reference point, and at least two other points to
graph the function.
c) Give the domain and range in set notation.
d) Describe the end behavior.
Write the exponential function that is represented by the graph.
104.
105.
13.3 The Base e
106. Given the function 𝑔(𝑥) = −(𝑒)𝑥+2 + 1 complete the following:
a) Tell the transformations that have been applied to the graph of 𝑓(𝑥) = 𝑒 𝑥 to produce the graph of 𝑔(𝑥).
b) Identify the asymptote, reference point, and at least two other points to
graph the function.
c) Give the domain and range in set notation.
d) Describe the end behavior.
13.4 Compound Interest
For each investment described, write the exponential growth function that models the value as a
function of time t, then solve.
107. The principal amount, $6325, earns 4.15% interest compounded annually. How long will it take for the
account’s value to surpass $9500?
108. The principal amount, $5200, earns 3.75% interest compounded quarterly. How long will it take for the
account’s value to surpass $15,000?
109. The principal amount, $13,000, earns 4.7% interest compounded continuously. How long will it take for
the account’s value to reach $52,000?
110. Hannah plans to make a deposit in one of two accounts. Account A has a 3.24% nominal rate with
interest compounded continuously, and Account B has a 3.25% nominal rate with interest compounded
semiannually.
a) Find the effective annual interest rate for Account A.
b) Find the effective annual interest rate for Account B.
c) Which account should Hannah choose?
15.1 Defining and Evaluating a Logarithmic Function
Write each exponential function in logarithmic form.
111.
34 = 81
114.
(6) = 𝑑
1 𝑐
1 −3
112.
(4)
115.
𝑥𝑦 = 𝑧
= 64
113.
5𝑎 = 𝑏
116. 2𝑥+5 = 48
Write each logarithmic function in exponential form.
8
27
3
117.
𝑙𝑜𝑔6 7776 = 5
118.
𝑙𝑜𝑔𝑥 125 = 3
119.
𝑙𝑜𝑔2
120.
𝑙𝑜𝑔𝑑 16 = ℎ
121.
ln 7 ≈ 1.946
122.
𝑙𝑜𝑔6 216 = 3
=3
Evaluate without a calculator.
123. 𝑙𝑜𝑔2 8
124. 𝑙𝑜𝑔8 64
126. 𝑙𝑜𝑔6 1
127. 𝑙𝑜𝑔8 8
125. 𝑙𝑜𝑔7 7
1
1
128. 𝑙𝑜𝑔7 49
Evaluate each by writing the logarithmic equation as an exponential equation with common bases on
each side.
129. Given 𝑓(𝑥) = 𝑙𝑜𝑔4 𝑥, find the following:
1
a) 𝑓 (16)
4
b) 𝑓(√64)
3
c) 𝑓(4√256)
130. Given 𝑓(𝑥) = 𝑙𝑜𝑔3 𝑥, find the following:
1
a) 𝑓(9√27)
b) 𝑓 (81)
c) 𝑓(243)
15.2 Graphing Logarithmic Functions
Tell the transformations applied to the parent graph of each function. Then, identify the asymptote,
reference point, and 1-2 other points to graph each function on the same graph as the parent function.
Give the domain and range in set notation.
1
131. 𝑔(𝑥) = − 2 𝑙𝑜𝑔2 (𝑥 + 1) − 3
132. 𝑔(𝑥) = 2 log(𝑥 − 2) + 1
16.1 Properties of Logarithms
Use the properties of logarithms to expand each expression.
133. 𝑙𝑜𝑔6 3𝑥
136. 𝑙𝑜𝑔4
𝑥𝑦
3
𝑥
134. 𝑙𝑜𝑔2 5
135. log 𝑥𝑦 2
137. 𝑙𝑜𝑔3 𝑥 2 𝑦𝑧
138. 𝑙𝑜𝑔5 2𝑥
Use the properties of logarithms to condense each expression.
139. 𝑙𝑜𝑔3 7 − 𝑙𝑜𝑔3 𝑥
140. 2𝑙𝑜𝑔5 𝑥 − 𝑙𝑜𝑔5 3
141. 3 log 𝑥 − log 4
142. ln 5 + ln 𝑥 + ln 𝑦
143. 𝑙𝑜𝑔4 5 + 2𝑙𝑜𝑔4 𝑥 − 𝑙𝑜𝑔4 5
144. −3 log 𝑥 + log 𝑦
Use the properties of logarithms to simplify and evaluate when possible.
𝟏𝟒𝟓. 𝑙𝑜𝑔3 27 − 𝑙𝑜𝑔3 81
1
146. 𝑙𝑜𝑔5 (25) + 𝑙𝑜𝑔5 125
147. 𝑙𝑜𝑔4 643
148. 𝑙𝑜𝑔8 128 − 𝑙𝑜𝑔8 2
150. 𝑙𝑜𝑔6 6𝑥−1 + 𝑙𝑜𝑔4 643
1
151. 𝑙𝑜𝑔𝑥 𝑥 3 − 𝑙𝑜𝑔3 (243)
16.2 Solving Exponential Equations
Solve each exponential equation algebraically. (Use logs!!!)
152. 3𝑥 = 68
153. 6𝑥 − 10 = 41
154. 5(18)6𝑥 = 26
155. 𝑒 𝑥−1 − 5 = 5
156. 9𝑥+10 + 3 = 81
157. −2𝑒 9𝑥−1 + 6 = −58
158. 5(6)3𝑥 = 20
159. 16𝑥−7 + 5 = 24
160. 8−5𝑥 − 5 = 53
19.1 Probability and Set Theory
Set A is the set of factors of 12, set B is the set of even natural numbers less than 13, set C is the set of
odd natural numbers less than 13, and set D is the set of even natural numbers less than 7. The
universal set for these questions is the set of natural numbers less than 13. So, 𝐴 = {1, 2, 3, 4, 6, 12}, 𝐵 =
{2, 4, 6, 8, 10, 12}, 𝐶 = {1, 3, 5, 7, 9, 11}, 𝐷 = {2, 4, 6}, 𝑎𝑛𝑑 𝑈 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. Answer each
question.
161. Is 𝐷 ⊂ 𝐴? Explain why or why not.
162. Is 𝐵 ⊂ 𝐶? Explain why or why not.
163. What is 𝐴 ∩ 𝐵?
164. What is 𝐴 ∩ 𝐶?
165. What is 𝐴 ∪ 𝐵?
166. What is 𝐴 ∪ 𝐶?
167. What is 𝐴𝐶 ?
168. What is 𝐵𝐶 ?
You have 10 cards numbers 1 to 10. You choose a card at random. Event A is choosing a number less
than 7. Event B is choosing an odd number. Calculate each probability.
169. 𝑃(𝐴)
170. 𝑃(𝐵)
171. 𝑃(𝐴 ∩ 𝐵)
172. 𝑃(𝐴 ∪ 𝐵)
173. 𝑃(𝐴𝐶 )
174. 𝑃(𝐵𝐶 )
19.2 Permutations
Find the number of permutations for each situation.
175. You are setting the combination on your gym lock. You want to use your favorite 4 numbers: 1, 0, 9, and
6, but you do not care what order they are in. How many different lock combinations are possible?
176. How many different batting orders are possible with a 9-person baseball team?
177. A group of 45 people are running a 5k race. The top three runners earn gold, silver, and bronze medals.
How many ways can the medals be won?
178. There are 35 applicants for three jobs: computer programmer, software tester, and manager. In how
many different ways can the jobs be given?
19.3 Combinations
Find the number of combinations for each situation.
179. There are twelve students on the debate team. The coach can choose 4 to send to the state competition.
How many different groups does she have to choose from to send to state?
180. How many different combinations of eight cards can be selected from a standard deck of 52?
181. A team plans to choose 3 of the colors represented of the Olympic Flag for its team uniforms. The colors
are blue, red, yellow, black, and green. How many different three-color combinations can be chosen?
182. Tony’s Tasty Pizza offers pepperoni, ham, sausage, onions, olives, and mushrooms as pizza toppings.
Adam wants to order a two-topping pizza. How many different two-topping combinations can he choose
from?
19.4 Mutually Exclusive and Overlapping Events
Given a standard deck of playing cards, find each probability.
183. Drawing an ace or a face card.
184. Drawing a red card or a black 10.
185. Drawing a King or and even number.
186. Drawing a face card or a seven
Given a regular 6-sided die, find each probability.
187. Rolling an even number or rolling a number less than 4.
188. Rolling a number greater than 3 or rolling an even.
189. Rolling an odd number or rolling a number greater than 1.
Given a standard deck of playing cards, find each probability.
190. P(drawing a spade or drawing a seven)
191. P(drawing a face card or drawing a black card)
20.2 and 20.3 Independent and Dependent Events
There are 5 tiles with the letters A, B, C, D, and E in a bag. You choose a tile without looking, put it
aside, and then choose another tile without looking. Use the Multiplication Rule to find the specified
probability, writing it as a fraction.
192. Find the probability that you choose a vowel followed by a consonant.
193. Find the probability that you choose a vowel followed by another vowel.
A bag holds 4 white marbles and 2 blue marbles. You choose a marble without looking, put it aside and
then choose another marble without looking. Use the Multiplication Rule to find the specified
probability, writing it as a fraction.
194. Find the probability that you choose a white marble followed by a blue marble.
195. Find the probability that you choose a white marble followed by another white marble.
Your mom brings home a bag of fruit from the Farmer’s Market. It contains 6 apples, 3 oranges, 3
pears, and 4 peaches. You choose a piece of fruit, decide you do not want that type, return it and draw
a new piece. Find each probability.
196. You choose an apple then an orange.
198. You choose an orange both times.
197. You choose a pear then a peach.