PRECALCULUS REVIEW SPRING FINAL



PRECALCULUS REVIEW SPRING FINAL NAME:_____________________________

Graph each function, state the y-intercept and the roots.

1.) f(x) = –1/2(x – 2)(x + 3) 2.) g(x) = x(x – 4)(x + 3)

3) y-int. ______ roots _______ 4) y-int. ______ roots _______

[pic] [pic]

Evaluate f(2), f(–2), and f(0) for each of the following functions:

5.) f(x) = x3 + 2x2 – 4x +7 6.) f(x) = 2x2 + 7x – 4 7.) f(x) = x6 –- 4x4 – x3 + 3x2 + 4x – 3

8.) If 3 + 8i is a root of a polynomial, name another root.

9.) List all the possible rational roots of: 9x4 – 2x + 12 = 0

10.) Solve x3 – 5x2 + 4x – 20 = 0, if one root is 5. 11.) Is 2 a factor of: 3x3 – 12x + 1 = 0?

For each function find the following, if possible: domain, range, vertical asymptotes, hole, horizontal asymptotes, y-intercept, zeros, and slant asymptote. Then GRAPH.

12.) f(x) = x2 + 6x – 7 13.) f(x) = x2 + 4x – 3

x2 – 1 x + 2

SOLVE:

13.) [pic] = x + 4 14.) [pic] = 2 15.) 13 + 3[pic] = 7

16.) (x2 – 9)(x2 + 6x – 7) = 0 17.) (x + 3)(x2 – 2x –3) = 0

Write in Interval Notation:

18.) x > –3 19 7 < x < 5 20 x < –3 or x > 7

21.) Simplify: a.) (3x-2)-1 b.) (8x3)2(3x) c.) (9x2)-2 d.) x-7(x9 + x7)

(3x-1)3 (4x2)3 3x4

22.) Solve for x: a.) 8x = 4 b.) 16x = 64 c.) 3-x · 9 = 271-x d.) 5(x2 + 1) = 50

23.) Solve for x: a.) 7x = 3.4 b.) 8 x+1 = 5

24.) Graph: a.) f(x) = 2x b.) f(x) = 2 x + 2 c.) f(x) = –2x d.) f(x) = 2-x + 1

25.) Find an exponential equation to model the data: hrs | 1 2 3 4 5

bacteria | 11 125 1400 14500 16000

26.) Solve: log 8 x = –2/3 27.) Solve: log 9 x = ½ 28.) Solve: 4ln x = 20

29.) Solve: log 5 25 = x 30.) Solve: log x 9 = 2

31.) Rewrite as a single logarithm: a.) 3 log 5 x – 2 log 5 y b.) log 2 3 + 3log 2 x – 5 log 2 y

32.) Graph: a.) y = log 2 x b.) y = log 2 (x – 3) c.) y = -log 2 x + 1 d.) y = log 2 –x

33.) Find the center and radius of the circle with equations:

a.) (x – 7)2 + (y – 3)2 = 50 b.) x2 + y2 + 4x – 8y – 5 = 0

34.) Graph: Label foci and vertices. a.) 9x2 + 16y2 = 144 b.) 4x2 + 9y2 = 36

Matching: 35.) 8x2 + 2y2 + 6y – 40 = 0 a.) circle

36.) 3x2 + 3y2 + 9x – 6y = 0 b.) ellipse

37.) 3x – 4y –12 = 0 c.) hyperbola

38.) 5x2 – 6y2 + 100x – 6y – 80 = 0 d.) line

39.) x2 + 8x – 16y + 64 = 0 e.) parabola

40.) Find the equation of an ellipse with center (1,2); vertices (6,2) & (- 4,2); focus (4,2)

41.) Find the equation of a parabola with focus (1, -1) and directrix y = 3.

42.) Find the equation of a ellipse with center (0,0); focus (0, [pic] ) and vertex (0,7).

Express each point in rectangular coordinates:

43.) A(3, 45˚) 44.) B(-2, π/2) 45.) C(4, 150°)

46.) Name the Quadrant where each point is located: a.) (3,150°) b.) (-2, π/3) c.) (-4, -30°)

d.) (-3, 5π/6) e.) (5, 60°) f.) (4, -120°)

47.) Find the rectangular form of: a.) r = 2cosθ b.) r = -3secθ

48.) If v = (4, 60°) Write 3v in polar and component form.

49.) Find a vector equation through (-3,4) parallel to (x,y) = (7,8) + t(-2,5).

For #s 50 to 53: m = (-3,5) n = (6,8) p = (-5,-3) v = (9,12) w = (-1,-2)

50.) Find the indicated dot product: a.) m • n b.) p • v c.) m • w

51.) Find: a.) || m || b.) || n + 2p || c.) || w + v/3||

52.) State whether the pair of vectors are parallel, perpendicular or neither.

a.) m, n b.) m, p c.) n, v d.) p, w

53.) Draw: a.) m b.) 3w c.) w + v/3

54.) If P = (3,-5) and PM = (-1, -1) find point M.

55.) A line passes through (-2,7) and (1,1). Find : a.) the vector equation of the line

b.) the parametric equations of the line c.) the rectangular equation of the line

56.) Determine k so that (-3,k) and (4,6) are a.) perpendicular and b.) parallel

57.) Find the sum of the first six terms of the geometric sequence: 128, 64, 32,…

20

58.) Evaluate: Σ (2k – 1)

k = 1

59.) Find the 8th term of the arithmetic sequence with a1 = 83 and a2 = 74.

60.) Find the 7th term of the geometric sequence with a1 = 5 and a3 = 10.

MULTIPLE CHOICE

61.) All of the following are possible rational roots of 4x3 + 25x2 – 3x – 6 = 0 EXCEPT:

a.) -1/4 b.) 3/2 c.) 2/3 d.) ½

62.) F(x) = 3x4 + x3 – 4x2 + 3x – 5 has at most how many real roots?

a.) 0 b.) 1 c.) 2 d.) 3

63.) Solve: x3 – 3x2 – 23x + 85 = 0 , given one root is –5.

a.) -5, 5, 3 b.) -5, -5, -3 c.) -5, -4 ± i d.) -5, 4 ± i

64.) If 3 + 4i is a root of P(x) = 0, then which of the following is also a root?

a.) 3 + 4i b.) 3 – 4i c.) -4 + 3i d.) -3 – 4i

65.) Evaluate f(-3), given f(x) = -4x3 + 3x2 + 7x – 4

a.) -106 b.) 110 c.) 83 d.) -118

66.) Solve, then check algebraically: [pic] = x + 11.

a.) –5 b.) –5 & –16 c.) –16 d.) 5 & 16

67.) Solve: [pic]

a ) 6 b.) 8 c.) 9 d.) 12

68.) log2 1/16 =

a.) 1/8 b.) 4 c.) -4 d.) ¼

69.) Solve: 3lnx = 15

a.) x = 5 b.) x = e5 c.) x = 105 d.) x = e-5

70.) Tomball had a population of 7000 in 1980. The town’s growth can be represented by the function:

N = N0ekt where k = .034. Find the population of Tomball in 2000.

a.) 144,841 b.) 13817 c.) 7238 d.) 4760

71.) Rewrite as a single logarithm: log7 3 + 4log7 x – 2log7 y

a.) log7 ( 3 + 4x – 2y) b.) log7 3x4 / y2 c.) log7 12x/2y d.) log7 (3x4)/2y

72.) Find the center and the radius of the circle with equation: x2 + y2 – 6x + 8y – 11 = 0

a.) (3, -4); 6 b.) (3, -4); 11 c.) (-3, 4); 6 d.) (-3, 4); 11

73.) Find the foci of the ellipse: 16x2 + 25y2 = 400

a.) (±5, 0) b.) (0, ±4) c.) (±3, 0) d.) (0, ±3)

74.) Find the equation of the circle with (-2, 5) and (4, -1) as endpoints of the diameter.

a.) (x+ 1)2 + (y + 2)2 = 18 b.) (x – 1)2 + (y – 2)2 = 3√ 2

c.) (x + 1)2 + (y + 2)2 = 3√ 2 d.) (x – 1)2 + (y – 2)2 = 18

75.) Find the equation of the parabola with focus (4, -3/2) and directrix y = 3/2.

a.) y + 3/2 = 1/12(x – 4)2 b.) y – 3/2 = 1/6(x – 4)2

c.) y – 3/2 = 1/12(x + 4)2 d.) y = 1/6(x –4)2

76.) Find the equation of the parabola with vertex (-2, 5) and focus ( 2, 5)

a.) x + 2 = 1/16 (y – 5)2 b.) x – 2 = 1/16(y – 5)2

c.) x – 2 = ¼(y – 5)2 d.) x + 2 = ¼(y – 5)2

77.) Identify the conic: 3x2 – 3y2 + 12x – 6y + 15 = 0

a.) circle b.) parabola c.) ellipse d.) hyperbola

78.) In what quadrant does (-3, 150˚) lie?

a.) I b.) II c.) III d.) IV

79.) Express (4, -4√ 3) in polar coordinates.

a.) (-8, π/6) b.) (8, - π/6) c.) (-8, π/3) d.) (8, - π/3)

80.) The graph of r = 6sinθ is a(n):

a.) circle with radius 6 b.) circle with center (0, 6)

c.) circle with diameter 6 c.) ellipse

81.) Express (5, -π/3) in rectangular coordinates.

a.) (5/2, -5√ 3/2) b.) (5√ 3/2, -5/2) c.) (5/2, 5√ 3/2) d.) (5√ 3/2, 5/2)

82.) Find the eleventh term of the arithmetic sequence with a1 = 62 and a2 = 54.5

a.) 144.5 b.) 20.5 c.) 13 d.) -13

83.) Find the sum of the first nine terms of the arithmetic series 3 + 8 +13 + …

a.) 43 b.) 207 c.) no sum d.) 1,464,843

84.) Find the tenth term of the geometric sequence with a1 = 4, a3 = 8 and r > 0.

a.) 24[pic] b.) 48[pic] c.) 64[pic] d.) 128

85.) Which of the following is equivalent to an = 4an-1 – 1 when a1 = 3.

a.) 1, 3, 11, … b.) 3, 11, 43, … c.) 3, 7, 11, … d.) 11, 43, 161, …

86.) Find the sum of the first nine terms of the geometric series 300 – 150 + 75 – …

a.) 12775/64 b.) 114975/256 c.) 12825/64 d.) 115425/256

87.) Evaluate: [pic]

a.) 121 b.) 2500 c.) 9.6 x 1023 d.) -60.5

88.) Which of the following is equivalent to 3 + 6 + 9 + …?

a.) [pic] b.) [pic] c.) [pic] d.) [pic]

89.) If M = (-3, 4) and N = (2,7) then || NM || is

a.) 22 b.) [pic] c.) [pic] d.) 5

90.) If v = (-8, 200˚) then ¼ v =

a.) (-8, 50˚) b.) (-2, 50˚) c.) (-2, 200˚) d.) (-4, 200˚)

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