1. No category

Bremen District 228 – Mathematics Department
Topics in Algebra 2 – Second Semester Formula Sheet
Quadratic Formula
 b  b 2  4ac
x
2a
Discriminant
b 2  4ac
( x2  x1 ) 2   y 2  y1 
2
Distance Formula
y
Inverse Variation
k
x
y  a(b) x
Exponential Growth/Decay
Conic Sections
 x  h   y  k 
2
 x  h
a2
2
2
y k

b2
y  a  x  h  k
2
 r2
2
1
or
or
 x  h
b2
2
y k

a2
x  a y  k  h
2
2
1
Topics in Algebra 2
Semester 2 Final Exam Study Guide
Name _________________________________
Date _____________________Period ___
Chapter 5
1. Which of the following is a quadratic function?
A. 𝑓(𝑥) = −5 − 5𝑥
B. 𝑓(𝑥) = −8 − 7𝑥
C. 𝑓(𝑥) = 7 + 5𝑥 3 − 4𝑥 2 + 6𝑥
D. 𝑓(𝑥) = 6 − 3𝑥 2
2. Show that 𝑦 − 9 = 3𝑥 − 3𝑥 + 6 − 4𝑥 2 + 7 is a quadratic function by writing it in the form
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 and identifying a, b, and c.
3. Factor the quadratic expression 𝑥 2 − 15𝑥 + 54
A. (𝑥 + 6)(𝑥 + 9)
B. (𝑥 − 6)(𝑥 − 9)
C. (𝑥 + 1)(𝑥 + 54)
D. (𝑥 − 3)(𝑥 − 18)
𝑓(𝑥) =
4. Factor the quadratic expression 3𝑥 2 − 31𝑥 + 56
A. (3𝑥 + 7)(𝑥 + 8)
B. (3𝑥 − 7)(𝑥 + 8)
C. (3𝑥 − 7)(𝑥 − 8)
D. (𝑥 − 8)(3𝑥 + 7)
5. Factor the quadratic expression 𝑥 2 − 11𝑥 + 18
6. Identify the vertex and direction of opening for the graph of the given function: 𝑓(𝑥) = −3𝑥 2 − 30𝑥 + 50.
7. Use factoring and the Zero-Product Property to find the zeros of the quadratic function.
𝑓(𝑥) = 𝑥 2 − 7𝑥 − 8
A. 1,-8
B. 2,-4
C. 8,-1
D. 4,-2
8. Use factoring and the Zero-Product Property to solve the quadratic equation 4𝑥 2 + 16𝑥 + 16 = 0
9. Use the quadratic formula to solve the equation 5𝑥 2 + 36𝑥 + 36 = 0.
9
81
A. 10 , − 10
9
81
B. − 10 , 10
C.
D.
6
,6
5
6
− , −6
5
10. Use the quadratic formula to solve the equation 2𝑥 2 − 5𝑥 − 1 = 0.
A.
B.
C.
D.
−5±√33
4
5±√33
4
−5±√17
4
5±√17
4
11. Use the quadratic formula to solve the equation 4𝑥 2 + 7𝑥 − 15 = 0.
PAGE 1
12. Write 𝑓(𝑥) = (3𝑥 + 1)(𝑥 − 6) in standard form.
13. Find the length of the missing side to the nearest tenth.
5
13
14. Find the value of x to the nearest hundredth, given 5x2 + 3 = 18.
15. Simplify (3 – 4i) + (-6 + 7i).
16. Simplify (2 + 3i)(5 – 4i).
17. Use the discriminant to determine the number of real solutions in 3𝑥 2 + 6𝑥 − 24.
Chapter 7
18. Evaluate 𝑥 4 − 10𝑥 2 + 25 when x = 3.
A. 16
B. -23
C. 4
D. 9
19. Simplify 3𝑥 2 + 3𝑥 − 8 + 4𝑥 2 − 7𝑥 + 2.
A. 7𝑥 2 + 10𝑥 − 6
B. 7𝑥 2 − 4𝑥 − 6
C. – 𝑥 2 + 10𝑥 − 10
D. −𝑥 2 − 4𝑥 − 10
20. Evaluate 2𝑥 2 + 3𝑥 − 3 when x = 3.
21. Simplify −4𝑥 2 − 4𝑥 + 2𝑥 2 − 3𝑥 + 3 + 4.
22. Simplify −3𝑥(5𝑥 3 − 3𝑥 2 + 2𝑥 − 6).
23. Solve the polynomial equation 𝑥 3 + 13𝑥 2 − 𝑥 − 13 = 0.
24. Divide using synthetic division: (𝑥 2 + 7𝑥 − 5) ÷ (𝑥 − 2).
PAGE 2
Chapter 6
25. A population of 380 animals decreases at an annual rate of 18%. Find the multiplier for the rate of
exponential decay.
A.
B.
C.
D.
0.82
1.82
0.18
1.18
26. A town with a current population of 553,100 has a growth rate of 2.4%. Find the multiplier for the rate of
exponential growth.
27. The function 𝐸(𝑡) = 36 ∙ 3𝑡 approximates the number of nematodes in a certain sample of fresh compost
7
after t days. Find the initial number of nematodes when t=0. How many nematodes after 2 days?
A.
B.
C.
D.
36 ; 17,046
0; 34,092
0; 17,046
36 ; 34,092
28. Evaluate 5 ∙ 4𝑥 for 𝑥 = 2.9 to the nearest hundredth.
29. Which function represents exponential decay?
A.
B.
C.
D.
𝑦(𝑥) = 4𝑥 2 + 2𝑥 + 4
𝑦(𝑥) = 4 ∙ 0.23𝑥
𝑦(𝑥) = 4 ∙ 18𝑥
𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒𝑠𝑒
1 𝑥
𝑏
30. Describe the value of b for the function 𝑦(𝑥) = 5 ∙ ( ) to represent exponential decay.
A.
B.
C.
D.
b has any value greater than 1 or less than -1
b has any value greater than 5 or less than -5
b has any value between -1 and 1
b has any value between -5 and 5
31. Determine whether the function 𝑦(𝑥) = 7 ∙ 4.6𝑥 represents exponential growth or exponential decay.
32. Tell whether the function 𝑦(𝑥) = 8 ∙ 6𝑥 represents exponential growth or exponential decay.
PAGE 3
𝑟 𝑛𝑡
Use the formula 𝐴 = 𝑃 (1 + 𝑛)
33. Find the final amount of the investment for $1600 at a 4% interest rate compounded monthly for 4 years.
1
34. Write the equation log 4 256 = −4 in exponential form.
A. 44 = −256
B. 4−4 = 256
1
C. 44 = − 256
1
D. 4−4 = 256
1
35. Write the equation 3−2 = 9 in logarithmic form.
1
A. log −3 9 = −2
1
9
1
log 3 9
B. log 3 = −2
C.
D.
=2
1
log −2 9
=3
36. Write the equation 5−4 =
37. Write the equation log 6
1
36
1
in logarithmic
625
form.
= −2 in exponential form.
38. Solve the equation 𝑥 = log 3 81.
A. 5
1
B. 4
C. 4
1
D. − 4
39. Solve the equation
1
3
= log 27 𝑥.
40. Use the change of base formula to evaluate to four decimal places log15 427.
41. Write as a single logarithm (condense): 7 log 𝑥 + 3 log 𝑦 − 4 log 𝑧.
PAGE 4
Chapter 9
42. Find the distance between points E (2,8) and B (-8,1) and the coordinates of the midpoint of ̅̅̅̅
𝐸𝐵.
A. Distance = 3√13 ≈ 10.82
7
Midpoint = (5, 2)
B. Distance = 3√13 ≈ 10.82
9
Midpoint = (−3, 2)
C
D. Distance = √149 ≈ 12.21
7
Midpoint = (5, 2)
Distance = √149 ≈ 12.21
9
Midpoint = (−3, 2)
1
43. State the vertex of the graph 𝑦 = 8 (𝑥 + 6)2 − 3
44. Write the standard equation for the parabola with the given set of characteristics. Then graph the
parabola.
Focus: (3,2)
Vertex: (3,-2)
45. Find the standard equation of a circle with radius 5 and center: (0,0).
A. 𝑥 2 + 𝑦 2 = 25
B. 𝑥 2 + 𝑦 2 = 5
C. 𝑥 2 + 𝑦 2 = 10
D.
𝑥2
10
+
𝑦2
10
=1
46. Write the standard equation of the circle in
the graph.
PAGE 5
47. Graph the equation (𝑥 + 3)2 + (𝑦 − 1)2 = 16.
Label the center and radius.
49. Find the standard equation for the ellipse
using either the given characteristics, or
characteristics taken from the graph.
48. Find the standard equation for the ellipse
using either the given characteristics, or
characteristics taken from the graph.
50. Find the standard equation for the ellipse
using either the given characteristics, or
characteristics taken from the graph.
A.
B.
C.
D.
𝑥2
7
𝑥2
2
𝑥2
49
𝑥2
4
+
+
+
+
𝑦2
2
𝑦2
7
𝑦2
4
𝑦2
49
=1
=1
=1
=1
PAGE 6
51. Which is the graph of the ellipse represented by the equation
A.
(𝑥+5)2
16
+
(𝑦−4)2
25
= 1.
C.
Center: (5,-4)
Foci: (5,-7), (5,-1)
Vertices: (5,-9), (5,1)
Co-Vertices: (1,-4), (9,-4)
B.
Center: (5,-4)
Foci: (2,-4), (8,-4)
Vertices: (0,-4), (10,-4)
Co-Vertices: (5,-8), (5,0)
D.
Center: (-5,4)
Foci: (-5,1), (-5,7)
Vertices: (-5,-1), (-5,9)
Co-Vertices: (-9,4), (-1,4)
52. Graph the ellipse
(𝑥+5)2
16
+
(𝑦−5)2
9
Center: (-5,4)
Foci: (-8,4), (-2,4)
Vertices: (-10,4), (0,4)
Co-Vertices: (-5,0), (-5,8)
= 1. Label the center, foci, vertices and co-vertices.
PAGE 7
Chapter 10
53. Teesha is in the bowling club. There are 23 students in the club. Five of them will be picked at random to
attend an award banquet. What is the probability that Teesha will not be randomly chosen to attend the
banquet.
A.
5
23
B.
C.
18
23
D.
23
18
23
5
54. A spinner is evenly dividing into 8 equation areas and numbered 1 through 8. What is the probability of
spinning a number less than 4 in a single spin?
3
A.
B.
C.
D.
8
1
4
5
8
1
2
55. A box contains 7 green, 2 yellow, and 6 purple balls. Find the probability of obtaining a yellow ball in a
single random draw.
56. A lunch menu consists of 6 different kinds of sandwiches, 4 different kinds of soup, and 4 different drinks.
How many choices are there for ordering a sandwich, a bowl of soup, and a drink?
57. How many different arrangements can be made using all the letters of the word MATH?
58. How many different arrangements can be made using all the letters of the word ALGEBRA?
59. Find 7P4.
60. Twelve skiers are competing in the final round of the Olympics freestyle skating aerial competition. In how
many ways can 3 of the skiers finish first, second, and third to win the gold, silver, and bronze medals?
61. Eight members of a school marching band are audition for 3 drum major positions. In how many ways can
students be chosen to be drum majors?
PAGE 8
Chapter 8
Use the information to write the appropriate variation equation, and find y for the given values.
4
62. y varies directly as x and inversely as z. y = when x = 4 and z = 6. Find y when x =5 and z = 4.
A. 𝑦 =
B. 𝑦 =
C. 𝑦 =
D. 𝑦 =
9
2𝑧 8
;
3𝑥 15
3𝑧 6
;
2𝑥 5
3𝑥 15
;
𝑧 4
2𝑥
5
;8
3𝑧
6
63. y varies directly as x and inversely as z. y = -6 when x = -3 and z = 4. Find y when x = 3 and z = -9.
8𝑥
8
A. 𝑦 = 𝑧 ; − 3
8
8
B. 𝑦 = 𝑥𝑧 ; − 27
C. 𝑦 =
D. 𝑦 =
𝑥𝑧
27
;− 8
8
𝑧
; −1
3𝑥
3
64. y varies inversely as x. y = 2 when x = 2. Find y when x = 7.
65. y varies jointly as w and x and inversely as z. y = -12 when w = 6, x = 5, and z = -5.
Find y when w = 6, x = 9, and z = 2.
66. Designer Dolls, Inc. found that the number of dolls sold, N, varies directly as their advertising budget A, and
inversely as the price of each doll, P. Designer Dolls, Inc. sold 14,000 dolls when $60,000 was spent on
advertising and the price of a doll was set at $60. Determine the number of dolls sold when the amount
spent on advertising is increased to $120,000.
A. 27,533
B. 2,000
C. 3,266
D. 28,000
67. The wattage rating of an appliance is given by W, in watts, and varies jointly as the resistance, R, in ohms,
and as the square of the current, I, in amperes. If the wattage is 4 watts when the resistance is 100 ohms
and the current is 0.2 amperes, find the wattage when the resistance is 300 ohms and the current is 0.4
amperes.
A. 36,000 watts
B. 120 watts
C. 240 watts
D. 48 watts
68. A drama club is planning a bus trip to New York City to see a Broadway play. The cost per person for the
bus rental varies inversely as the number of people going on the trip. It will cost $36 per person if 67
people go on the trip. How much will it cost per person if 98 people go on the trip?
PAGE 9
4𝑥𝑦 3
69. What is the simplest form of
𝑥 2 −𝑥−6
70. What is the simplest form of
71. Add and simplify.
3𝑥+2
𝑥 2 −9𝑥+8
72. Solve the equation
3𝑥
𝑥+1
+
𝑥2𝑦
4𝑥 3
+
6
2𝑥
∙
∙
𝑦
∙
3𝑥
8𝑥 𝑦 2
𝑥+1
?
?
𝑥 2 +5𝑥+6
−2−6𝑥
𝑥 2 −9𝑥+8
=
7
𝑥
73. Simplify √72.
74. Rationalize the denominator
3
√7
75. Multiply. (2 + √3)(4 − 2√3)
76. Simplify. √8 + 6√2
PAGE 10
Topics in Algebra 2
Second Semester Review Guide ANSWER KEY
1. D
2. 𝑓(𝑥) = −4𝑥 2 + 22
𝑎 = −4, 𝑏 = 0, 𝑐 = 22
3. B
4. C
5. (x-9)(x-2)
6. (-5,125) down
7. C
8. x = -2
9. D
10. B
5
4
11. 𝑥 = , 𝑥 = −3
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
𝑓(𝑥) = 3𝑥 2 − 17𝑥 − 6
12
±1.73
-3+3i
22+7i
2 real solutions
A
B
24
−2𝑥 2 − 7𝑥 + 7
−15𝑥 4 + 9𝑥 3 − 6𝑥 2 + 18𝑥
𝑥 = 1, 𝑥 = −1, x = -13
13
24. 𝑥 + 9 + 𝑥−2
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
A
1.024
D
278.58
B
A
Exponential growth
Exponential growth
$1877.12
D
B
36.
1
log 5
=
625
1
6−2 = 36
37.
38. C
39. x = 3
40. 2.2366
41. log
𝑥 7𝑦3
𝑧4
−4
42. C
43. (-6,-3)
1
44. 𝑦 = 16 (𝑥 − 3)2 − 2
45.
46.
47.
48.
49.
50.
A
(𝑥 − 3)2 + (𝑦 + 2)2 = 9
Center: (-3,1), Radius: 4
C
(𝑥−1)2
25
(𝑥+5)2
25
+
+
(𝑦−3)2
4
(𝑦−1)2
9
=1
=1
51. 𝐵
52. Center: (-5,5)
Foci: (−5 ± √7, 5)
Vertices: (-1,5), (-5,5)
Co-Vertices: (-5,8), (-5,2)
53. C
54. A
55.
2
15
56.
57.
58.
59.
60.
61.
62.
63.
96
24
2520
840
1320
56
D
A
3
64. 𝑦 = 7
65.
66.
67.
68.
𝑦 = 54
D
D
$24.61
69.
3𝑦
2𝑥
(𝑥−3)(𝑥+1)
𝑥 2 −2𝑥−3
=
3
(𝑥+3)
4𝑥
4𝑥 4 +12𝑥 3
−3𝑥
𝑥 2 −9𝑥+8
2
𝑥 = −3,𝑥 = 2
70.
71.
72.
73. 6√2
74.
3√7
7
75. 2
76. 8√2
PAGE 11