1. (Ch 7) Show different ways to simplify the complex fraction:
Method I: Method II:
2. (Ch 7) Simplify:
3. (Ch 7) Solve:
4. (Ch 7 & 12) Which graph(s) will have a vertical asymptote and what will be the equation of the asymptote(s)?
A. ( )
B. f(x) = x
2 + y
2 = 9 C. ( )
D. ( )
5. (Ch 7) A river has a current of 4 km/hr. find the speed of Lynn’s boat in still water if it goes 40 km
downstream in the same time that it takes to go 24 km upstream.
Algebra 2 w/ Trig
Review for Final Exam
Chapters 7, 10, 11, 12, 13, 17, 18 & 19, 20.4
NAME _____________________________
Review Problem Due Dates: Mon 5/5: Problems 1-9 Due on Tues 5/6: Problems 10-26 Due on Wed 5/7: Problems 27-37 Due on Thurs 5/8: Problems 38-42 Due on Fri 5/9: Problems 43-55 Due on Mon 5/12: Problems 56-60 Due on Tues 5/13: Problems 61-89 Due on Wed 5/14: Work day – Class meets in the Eannelli Lab Due on Thurs 5/15: Problems 90-99 Due on Fri 5/16: Problem100 Class meets in the Eannelli Lab Turn in your list to Ms. H (10 pts)
Exam Format: True/False
Multiple Choice
Show your work problems
Graphs
Extra Credit
Small calculator only - Ms. H will provide them.
6. (Ch 10) Given f(x) = 2x2 – 8x + 1, use any method of your choice to fill in the blanks below and then graph the
function.
A. Opens (circle one): Up Down Left Right B. Shape near vertex (circle one): Basic Wider Narrow
C. Vertex: (________ , _______)
D. Equation of Axis: _____________
E. Domain: ____________________
F. Range: ______________________
7. Given x = -0.5(y + 3)2 + 2, use any method of your choice to fill in the blanks below and then graph the function.
A. Opens (circle one): Up Down Left Right B. Shape near vertex (circle one): Basic Wider Narrow
C. Vertex: (________ , _______)
D. Equation of Axis: _____________
E. Domain: ____________________
F. Range: ______________________
8. What is the most efficient way you can explain to a friend that there are no x-intercepts for f(x) = x2 + x + 10? Are there other ways to recognize this as well?
9. A projectile is fired straight upward from ground level so that its distance (in feet) above the ground t seconds after
firing is given by s(t) = – 16t2 + 400 t.
a. What is the height after 5 seconds? b. What is the maximum height achieved? c. When does the projectile hit the ground? (Ch 11) Multiple Choice.
10. Which expression is equivalent to loga b = c?
A. ab = c B. b
c = a C. a
c = b D. b
a = c E. none of these
11. Simplify: log2 8 = ?
A. 0 B. 1 C. 2 D. 3 E. none of these
12. What is the base in log 100?
A. 0 B. 1 C. 10 D. 100 E. e
13. What is the base in ln 100?
A. 0 B. 1 C. 10 D. 100 E. e
14. Which of these values is closest to e?
A. 2 B. 3 C. 2.8 D. 2.7 E. 3.14
15. Simplify: ln e?
A. – 1 B. 0 C. 1 D. 2 E. 3
16. What is the common log of 0.001?
A. – 3 B. –2 C. 1 D. 2 E. 3
17.
=
A. – 3 B. –2 C. 1 D. 2 E. 3
18. log 27
=
A.
B.
C.
D.
E.
19. √ =
A. 6 B. 4 C. 8 D. 3 E.
20. Solve:
A. 4 B.
C.
D.
E.
21. For any value of x that is greater than zero,
A. – 1 B.
C. 0 D. 1 E. x
22. Solve: 1 – 2x
= 16 x – 3
A. 2 B. 0 C.
D.
E.
23. log
=
A. 10 log x B. log 10 + log x C. (log 10)(log x)
D. log x E. 1 – log x
24. log 3 2
x =
A. log 3 2 – log 3 x B. x log 3 2 C. 2x
D. log 2 2 + log 3 x E. x log 3 4
25. ln √ A. ½ B. ½ ln (2 – e) C. ½ (ln 2 – 1)
D. ½ ln 2 – ln e E. ½ – ln e-1/2
26. Graph each:
A. y = ex B. y = e
x – 2 + 1 C. y = 10
x D. y = 10
– x
E. y = ln x F. y = –ln (x + 5) G. y = log2x H. y = – log2(x – 1)
27. (Ch 12) Use synthetic division to find f(2) given f(x) = 3x3 – 12x
2 + x – 10.
28. (Ch 12) f(x) = 6x3 + 17x
2 – 63x + 0
A. List the possible Rational Zeros of f(x): ________________________________________________________
B. Describe the End Behavior of f(x): as x , y ________ and as x , y ________
C. What is the maximum number of zeros f(x) could have?______________
D. What is the maximum number of turning points the graph of f(x) could have?___________
E. Given that -5 is a zero of f(x), use synthetic division to find all the zeros:__________________
F. Write f(x) in factored form:______________________________________________________
29. (Ch 12) Show your work to find all the zeros of f(x) = x3 + x
2 + 6x – 8
30. (Ch 12) A. If 2 and 3 + 2i are zeros of some f(x), what else must be a zero?__________________
B. Multiply the factors to find a possible polynomial expression for f(x) above. _____________________________
31. (Ch 13) Circle the type of conic section for each and fill in the blanks where applicable.
A. (x – 2)2
+ (y – 3)2
= 49 Circle Ellipse Hyperbola Parabola Center:_____________ Radius: ___________
B. ( )
( )
Circle Ellipse Hyperbola Parabola Center:_____________
Distance from center to major vertices:______ Distance from center to minor vertices:______ Distance from center to foci:______
C. ( )
( )
Circle Ellipse Hyperbola Parabola Center:_____________
Distance from center to vertices:______ Distance from center to foci:______ Is transverse axis vertical or horizontal?___________
D. y = -8(x – 1)2
+ 3Circle Ellipse Hyperbola Parabola Vertex:______ Opens: Up Down Left Right Distance from vertex to focus:______ Equation of axis:___________ Equation of Directrix:___________
E. 2x 2 + 3y 2 – 4x + y = 49 Circle Ellipse Hyperbola Parabola
F. 2x 2 – 3y 2 – 4x + y = 49 Circle Ellipse Hyperbola Parabola
G. 2x 2– 4x + y = 49 Circle Ellipse Hyperbola Parabola
H. 3x 2
+ 3y 2
– 4x + y = 49 Circle Ellipse Hyperbola Parabola
32. (Ch 13) Give the equation of a parabola which has its focus at (2, 0) and directrix x = – 2. Graph and fill in the
other information below. Equation of parabola: Vertex: Equation of Axis: Focus: (2, 0) Equation of Directrix:
33. (Ch 13) A. Write the equation of a circle with center at the origin and radius of 1:_________________________ B. In trig, what name do we give to this circle? ______________________
34. (Ch 13) Graph (x – 3)2 + (y + 2)
2 = 25.
Center: Radius:
35. (Ch 13) ( )
( )
Graph the conic section labeling key pts and fill in the other information below.
Center: Intercepts: Foci: Length of Major Axis: Length of Minor Axis:
36. (Ch 13) ( )
Graph the conic section labeling key pts and fill in the other information below.
Center: Intercepts: Foci: Equation of Transverse Axis:
37. (Ch 13) Convert 4x2+ 4y
2 + 8x – 16y – 20 = 0 to standard/graph form. Then graph the conic section labeling
key pts. 38. (Ch 17) Graph each basic trig function labeling key pts and equations of asymptotes: A. y = sin x and y = csc x B. y = cos x and y = sec x C. y = tan x D. y = cot x
*For these inverse trig functions, be sure to restrict the domain and range to the accepted 1-1 sections!
E. y = sin -1
x F. y = arc cos x G. y = tan -1
x
39. (Ch 17) Fill in the blanks below and graph (labeling key pts): y = 1 –
cos 2x
Amplitude: Period: Phase Start: Phase End:
40. (Ch 17) Fill in the blanks below and graph (labeling key pts): y = 2 sin (
)
Amplitude: Period: Phase Start: Phase End:
41. (Ch 17) Fill in the blanks below and graph (labeling key pts): y = csc (
)
Period: Phase Start: Phase End: 42. (Ch 18) Write the definitions of the six trig rations in the unit circle (when hypotenuse = 1) and using SOHCAHTOA
(when hypotenuse ≠ 1).
sin cos tan cot csc sec
In Unit Circle
When hyp ≠ 1
43. (Ch 18) Write 3 different Pythagorean Identities so that all six trig ratios appear somewhere in your answers. A. B. C. 44. (Ch 18) Use a Pythagorean Identity to complete each equation below.
A. 1 – sin2x = B. csc2x = C. tan2x + 1 = 45. (Ch 18) Complete the Cofunction Identities:
A. sin (
) = B. sec (
) = C. csc ( ) =
D. Given sin 17o = a, what conclusion can you reach using a Cofunction Identity?______________________ 46. (Ch 18) Complete the Negative Angle Identities: A. sin ( ) = B. cos ( ) = C. tan ( ) = D. csc ( ) = E. sec ( ) = F. cot ( ) =
G. Given sin 17o = a, what conclusion can you reach using a negative angle identity?______________________
H. Given cos 147o = b, what conclusion can you reach using a negative angle identity?_____________________
47. (Ch 18) Use trig identities (on each side without crossing over) to verify the identity:
A. ( )
B.
Sum and Difference Identities (Section 18.3 - 18.4)
( ) ( )
( )
( ) ( )
( ) 48. (Ch 18) Which expression is equivalent to sin(50o) cos(12o) – cos(12o) sin(50o) 48. _________
A. sin(50o) B. sin(62o) C. sin(38o) D. cos(38o) E. tan(62o)
49. (Ch 18) Use a Sum Identity to find the value of cos (75o) Hint: 75o = 45o + 30o 49. __________________
50. (Ch 18) Given cos s =
and sin t =
and angles s & t are in quadrant IV. 50. sin s = ____________________
Place your answer in the blank for each value.
cos t = ______________________
sin (s – t) = ___________________
51. (Ch 18) Use your Trig Identity Handout or p834 of your textbook to name the type of identity used below: sin 38o = 2sin19ocos19o
52. (Ch 19) Simplify each. Be sure you are using the proper restrictions for each inverse trig function.
A. ( √
) B. cos
-1 ( ) = C. tan-1
( ) =
D. arc cos ( ) = E. arc tan ( √ ) = F. ( √
)
D. arc sin (
) = E. arc tan (
√ ) = F. (
)
53. An angle , in standard position, terminates at the point ( 7, 24). Find cot . 54. Given sin = ¼ and tan < 0. cos = _________ tan = _______ ends in quadrant ______ 55. (Ch 18 & 19) Solve each trig equation.
A. Find all x in [0, 2 ]: B. Find ALL solutions (using n form):
C. Find all x in [0, 2 ]: D. Find ALL solutions (using n form):
(Common Functions & Transformations – all chapters mixed)
56. Given the graph of f(x) shown:
A. Graph f(–x) + 2
B. Graph – f(x + 2)
57. Sketch a graph for each. Label key pts or axis tick marks and equations of asymptotes.
A. f(x) = 3 B. g(x) = x C. f(x)= |x| D. g(x) = –|x–1|
E. f(x) = √ F. g(x) =2 – √ G. f(x)= √
H. g(x) = x3
58. (Ch 12.4) Graph each rational function. Label all key pts and give equations of asymptotes.
A. f(x) =
B. f(x) =
C. f(x) =
1
1
-1 -1 3 -3
3
-3
f(x)
59. Graph each piece-wise defined function.
A. ( ) {
B. ( ) {
√
60. Give the Domain of 59A:_____________________ Range of 59A:______________________________ (Mixed Practice) True or False.
61. The logarithm of 32 in base 2 is 5. 62. Division is commutative. 63. Addition is commutative. 64. (0, 1) is a point on the graph of y = cos x. 65. The amplitude of y = 3 cos 2x is 2. 66. The period of y = 3 cos 2x is π.
67. The graph of y = cot x has asymptotes at x =
and x =
.
68. The range of y = cos x is * +.
66. y = 2– x
and y = (½)x have the same graph.
67. The graphs of y = e– x
and y = ex are reflections across the y-axis.
68. The graphs of y = – e x and y = e
x are reflections across the x-axis.
69. If a > 1 and ax = a
y, then x = y.
70. (0, 1) is a point on the graph of y = ex.
71. y = 0 is an asymptote of the graph of y = ex.
72. The graph of 4x2 – 4y
2 + x – 6y + 10 = 0 is a circle.
73. The graph of 4x2+ x – 6y + 10 = 0 is a parabola.
74. The length of the major axis of ( )
( )
is 10.
75. The equation of the axis of x = (y + 1)2 – 3 is y = – 1
76. log 5x2 is equivalent to (log 5 + 2log x).
77. For f(x) = 7x5 – 6x
3 + 2x – 1, as x , y .
78. For f(x) = 7x5 – 6x
3 + 2x – 1, as x , y .
79. For f(x) = –
, as x , y .
80. For f(x) = –
, the domain is * +.
81. f(x) = –
has a horizontal asymptote at y = 2.
82. f(x) = –
has a vertical asymptote at x = 5.
83. f(x) = –
has a y-intercept of
.
84. f(x) = –
has an x-intercept of
.
85. sin2x – 1 = cos
2x.
86. Which is the equation of the graph shown?
A.
B. √
C. √
D. √
E. none of these
87. Which is the graph of f(x) = (
)
?
88. In which of these graphs is y quadratic while x is linear? A. B. C. D. E. none of these 89. Find the balance if $12,000 is invested for 18 years at 4% annual interest. . . (You need not memorize these formulas)
A. compounded monthly:
B. compounded continuously:
90. Expand: (x + 2)5
91. Expand: (x + 3)3
(0, 3)
(– 5, 0) (5, 0)
𝐴 𝑃 ( 𝑟
𝑛)𝑛𝑡
𝐴 𝑃𝑒𝑟𝑡
92. Factor: f(x) = 8x3 – 27
93. Find all the complex zeros of f(x) = 8x3 – 27
94. (Ch 20.4) Use the figure to find: a + b, a – b, – a. Use (x, y) notation. 95. Given vectors a = – i + 2j, b = i – j. Find: 2a, 2a + 3b, b – 3a
96. Sketch the resultant for vectors u and w with angle between them. |u| = 8, |w| = 12, = 30o 97. Find the magnitude and direction angle for vector u: <–7, 24> 98. Write the vector in <–7, 24> form.
99. Given forces of 283 and 125 newtons, forming an angle 132o, find the magnitude of the resultant force.
Due Date: Friday, 5/16 Name__________________________________ ____/10 You will tear off this page and turn it in to be scored. 100. Find and watch 8 review videos that help you review topics that are difficult for you. Be sure the URL addresses
are complete, so Ms. H can view your videos.
URL Topic(s) Covered What did you learn or re-learn?
1.
2.
3.
4.
5.
6.
7.
8.