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1 Pre-‐Calculus Name: ______________________ Final Exam Review Packet Chapter 1: Functions and Graphs 1. Find the slope-‐intercept form of the equation of the line passing through the point (5, -‐2) and
perpendicular to the line 3x – 2y = 12. 2. Find the point-‐slope form of the equation of the line passing through the point (3, 10) and parallel
to the line x – 3y = 1. 3. Find the domain and range of the function (write your answers in interval notation):
a. b. c.
4. Find the intervals on which the given function is increasing, decreasing, or constant.
a. b. c.
5. Described the transformation(s) of the graphs of the equations below. Let f(x) equal the “parent”
function and g(x) equal the “transformed” function.
a. b. c.
6. Determine the inverse of the function
7. Determine f(g(x)) given:
y = x − 7 y = 3x2 − 4 y = 5x − 2
y = x2 y = −2x −1
f (x) = 2 x < 3x2 x ≥ 3
⎧⎨⎪
⎩⎪
f (x) = x2
g(x) = 5(x −1)2f (x) = xg(x) = x +1 − 8
f (x) = x3
g(x) = −3 f (x −1)3
f (x) = −(x +1)2 − 3
f (x) = 5x2 +1g(x) = 4x − 7
Chapter 2: Polynomial, Power, and Rational Functions 8. Find all real zeros of the function: 9. Write the polynomial in completely factored form given that (x + 2) is a factor. 10. Given that 3i is root of the function , find the remaining roots. 11. Find a 4th degree polynomial with zeros at 5, -‐2, and . 12. Determine the horizontal and vertical asymptotes of the functions below:
a. b. c.
f (x) = x3 − 7x + 6
f (x) = x3 − x + 6
f (x) = x4 − 6x3 +14x2 − 54x + 45
2 − i
f (x) = x + 5x2 − 4
f (x) = x2 − 9x2 + x −12
f (x) = x2 +1x2 − 6x + 9
d. e.
13. Find the slant asymptote of the function
Chapter 3: Exponential and Logarithmic Functions 14. Evaluate: a. log2 16 b. lne2 c. log5 7 15. Write the exponential form of the logarithmic equation: log3 81 = 4 16. Write the logarithmic form of the exponential equation: 53 = 125 17. Sketch the graph of the equations. a. y = log2 x b. y = ex 18. Write each expression in condensed form.
a. log 5 + log 2 – log 3 b. log2 x −12log2 y c.
153log(x +1) + 2 log(x −1) − log 7[ ]
f (x) = x3
(x − 2)(x + 5)f (x) = 5x2
(x − 2)(x +1)
f (x) = x2 + 2x + 22x −1
19. Write each expression in expanded form.
a. log3x2
yz b. ln 5x
x2 +13
20. Solve for x:
a. log317x⎛
⎝⎜⎞⎠⎟= 4 b. 3x+3 = 243 c. e2x+1 = 9
d. log(3x + 7) + log(x − 2) = 1 e. ln(7 − x) + ln(3x + 5) = ln(24x) Chapters 4 and 5: Trigonometric Functions 21. The terminal side of an angle in standard form contains the point (-‐5, 12). Give the sine and cosine
values of the angle. 22. Convert from radian to degree measure.
a. 3π2 b. −
43π c. π
6 d. 5π
12
23. Convert from degree to radian measure. a. 135° b. -‐240° c. 45° d. 720°
24. Evaluate:
a. sin cos−1 − 32
⎛
⎝⎜⎞
⎠⎟ b. tan(sin−1 0) c. sin(arctan 2x)
25. Determine period, amplitude, phase shift and vertical shift of the functions listed below. a. f (x) = sin(2x) + 5
b. f (x) = −4 cos x4− π⎛
⎝⎜⎞⎠⎟
26. Simplify the following expressions:
a. 1− cos4 x1+ cos2 x
b. sin2 x cot2 x + sin2 x
c. 1− csc xcsc x
d. 2sin2 x + cos2 x −1
27. A ladder is leaning up against the side of the house. The base of the ladder is 5 feet from the wall
and makes an angle of 39° with the ground. Find the length of the ladder. 28. From a point on a cliff 75 feet above water level an observer can see a ship. The angle of
depression to the ship is 4°. How far is the ship from the base of the cliff?
29. Solve the trig equations for the domain 0,2π[ ).
a. 6cos2 x − 5sin x − 2 = 0 b. 8sin x2
⎛⎝⎜
⎞⎠⎟− 8 = 0
c. 2sin2 x − 5sin x = −3 d. tan x4
⎛⎝⎜
⎞⎠⎟=
33
e. tan2 x csc x = tan2 x f. sin x = cos x g. cos2 x − cos2x = 0 30. Given sin x = 3/5 and x terminates in quadrant II, find cos 2x. 31. Given cos x = ½ and x terminates in quadrant IV, find sin 2x.
Chapter 6: Parametric and Polar Equations 32. Sketch the curve given by the parametric equations:
33 1 2 ≤≤−+== ttytx= 𝑡
t -‐3 -‐2 -‐1 0 1 2 3 x
y 33. Sketch the curve given by the parametric equations: πθθ 20 sin5 cos2 ≤≤== tyx
θ x
y 34. Determine two different parametric equations to represent the equation 35 −= xy
35. Convert each point or equation from polar to rectangular coordinates.
a. )6/5,5( π− b. 6πθ = c. θsin6=r
36. Convert each point or equation from rectangular to polar coordinates.
a. )4,3( b. 0622 =−+ xyx c. 0532 =++− yx
37. Sketch the graphs of the polar equations. a. )3sin(4 θ=r b. )cos(63 θ+=r Chapter 8: Topics in Analytic Geometry 38. Find the vertex of the parabola: 4y2 + 4y −16x = 0 39. Find an equation of the parabola, opening down, with vertex (-‐3,1) and solution point (4,-‐5). 40. Find the center of the ellipse: 9x2 + 4y2 − 36x − 24y − 36 = 0 41. Find the equation of the ellipse with minor axis of length 8 and vertices (±9, 3). 42. Find the center of the hyperbola: 25y2 −144x2 +150y − 576x − 3951 = 0
43. Graph the hyperbola: x2
9−y2
12= 1
44. What type of conic is represented by the equations? a. 2x2 − 5y2 + 4x − 6 = 0 b. 3x2 + 3y2 − 4x + 5y −16 = 0 c. y2 +10y − 20x + 37 = 0 Chapter 11 / 1: Limits and an Introduction to Calculus Determine the limits using a numerical, graphical or analytical method. 45. lim
x→23x − 4( )
46. limx→0
9x2
⎛⎝⎜
⎞⎠⎟
47. limx→0
9x
⎛⎝⎜
⎞⎠⎟
48. lim
x→7100
49. limx→3
x − 3x2 − 9
50. limx→∞
x − 3x2 − 9
51. limx→5
x2 − 3x −10x − 5
⎛⎝⎜
⎞⎠⎟
52. limx→3
x − 3x − 3
53. limx→−2+
x + 2x + 2
54. limx→0
x + 4 − 2x
55. limx→0
sin 5xx
56. limx→∞
3x3 − x +15x − 2x2 + 2x3
57. limx→∞
5x5 −17x + 5
58. limx→2−
f (x) for f (x) = 2x +1 x ≤ 25x x > 2
⎧⎨⎩
59. For the function, f(x), find the value(s) of discontinuity and label as removable or non-‐removable.
f (x) = x2 + 5x + 6x + 2
60. Given limx→c
f (x) = −5 and limx→c
g(x) = 2 , evaluate the following limits.
a. lim
x→cf (x) − 2g(x)[ ]
b. limx→ c
f (x)g(x)
⎡
⎣⎢
⎤
⎦⎥
c. lim
x→c4 f (x) + 3g(x)[ ]
Precalculus Final Exam Review Packet Answers
1. 3
4
3
2+−= xy
2. y −10 = 13(x − 3)
3. a. D: [7,∞ ), R: [0, ∞ ) b. D: (-‐∞ ,∞ ) R: [-‐4, ∞ ) c. D: (-‐∞ ,2) (2, ∞ ) R: (-‐∞ ,0) (0, ∞ ) 4. a. decreasing on (-‐∞ , 0) increasing on (0, ∞ ) b. increasing on (-‐∞ ,1) (1, ∞ ) c. constant on (-‐∞ , 3) increasing on (3, ∞ ) 5. a. shift right 1, narrower b. shift left 1, down 8 c. shift right 1, inverted, narrower 6. f −1(x) = ± −x − 3 −1 7. f (g(x)) = 80x2 − 280x + 246 or 5(4x − 7)2 +1 8. x = -‐3, 1, 2
9. x + 2( ) x −1− i 2( ) x −1+ i 2( ) or(x + 2)(x2 − 2x + 3)
10. x = ±3i, 5,1 11. f (x) = x4 − 7x3 + 7x2 + 25x − 50 12. a. x = ±2, y = 0 b. x = -‐4, y = 1 (Hole at x = 3 which is NOT a VA) c. x = 3, y = 1 d. x = 2, x = -‐5 e. x = 2, x = -‐1, y = 5
13. y = 12x + 5
4
14. a. x = 4 b. x = 2 c. x = 1.209 15. 34 = 81 16. log5 125 = 3 17. a. b.
18. a. log103
b. log2xy
⎛
⎝⎜⎞
⎠⎟
c. logx +1( )3 x −1( )2
7⎡
⎣⎢⎢
⎤
⎦⎥⎥
1/5
19. a. log3 x −12log3 y −
12log3 z
b. ln 5 + ln x − 13ln x2 +1( )
20. a. x = 567, b. x = 2, c. x = .599 or (-‐1+ln9)/2 d. x = 8/3, e. x = 7/3 21. sinθ = 12 /13 cosθ = −5 /13 22. a. 270°, b. -‐240°, c. 30°, d. 75° 23. a. 3π/4, b. -‐4π/3, c. π/4, d. 4π
24. a. ½, b. 0, c. 2x4x2 +1
25. a. period = π, amp. = 1, v.s. = 5 up, p.s. = 0 b. period = 8π, amp. = 4, v.s. = 0, p.s. = 4π 26. a. sin2 x b. 1 c. sin x – 1 d. sin2 x 27. 6.434 feet 28. 1072.550 feet
29. a. x = π6, 5π6 b. x = π c. x = π/2
d. x = 2π/3 e. ∅ f. x = π/4, 5π/4 g. 0, π 30. 7/25 31. − 3 / 2 32. 33. 34. Answers will vary
35. a. 5 32, −52
⎛
⎝⎜⎞
⎠⎟ b. y = 1
3x
c. x2 + y2 − 6y = 0 36. a.
5,53.130( ) or 5,0.927rad( )
b. r = 6cosθ
c. r = −5
−2cosθ + 3sinθor
r = 52cosθ − 3sinθ
37. a. b.
38. Vertex: −116,−12
⎛⎝
⎞⎠
39. y = − 649
x + 3( )2 +1 40. center: (2, 3)
41. x2
81− (y − 3)
2
16= 1
42. C(2, -‐3) 43.
44. a. hyperbola, b. circle, c. parabola 45. 2 46. ∞ 47. DNE 48. 100 49. 1/6 50. 0 51. 7 52. DNE 53. 1 54. ¼ 55. 5 56. 3/2 57. ∞ 58. 5 59. removable discontinuity at x = -‐2 60. a. -‐9, b. -‐5/2, c. -‐14
