C Challenge

49. Geometry Use the drawing below and similar triangles. Justify the statement that tan θ = sin θ cos θ . tangent theta equals . fraction sine theta , over cosine theta end fraction . .

50. 

    1. Graph y = tan x , y = a tan x y equals tangent x comma y equals eh tangent x  (with a > 0 eh greater than 0 ), and y = a tan x y equals eh tangent x  (with a < 0 eh less than 0 ) on the same coordinate plane.
    2. Reasoning Recall the pattern of five elements for graphing a tangent function: asymptote- ( − 1 ) open negative 1 close -zero-(1)-asymptote. How does the value of a affect this pattern?
51. Writing How many solutions does the equation x = tan x x equals tangent x  have for 0 ≤ x < 2 π ? 0 less than or equal to x less than 2 pi question mark  Explain.

Standardized Test Prep

SAT/ACT

52. Which value is NOT defined?
    1. tan 0
    2. tan π
    3. tan 3 π 2 tangent . fraction 3 pi , over 2 end fraction
    4. 1 tan π 4 fraction 1 , over tangent , pi over 4 end fraction
53. What is the exact value of tan 7 π 6 ? fraction 7 pi , over 6 end fraction , question mark
    6. − 3 negative square root of 3
    7. − 3 3 negative , fraction square root of 3 , over 3 end fraction
    8. 3 3 fraction square root of 3 , over 3 end fraction
    9. 3 square root of 3
54. Which equation does NOT represent a vertical asymptote of the graph of y = tan θ ? y equals tangent theta question mark
    1. θ = − π 2 theta equals negative , pi over 2
    2. θ = 0
    3. θ = π 2 theta equals , pi over 2
    4. θ = 3 π 2 theta equals , fraction 3 pi , over 2 end fraction
55. Which function has a period of 4π?
    6. y = tan 4 θ y equals tangent 4 theta
    7. y = tan 2 θ y equals tangent 2 theta
    8. y = tan 1 2 θ y equals tangent , 1 half , theta
    9. y = tan 1 4 θ y equals tangent , 1 fourth , theta

Short Response

56. Does a tangent function have amplitude? Explain.

Mixed Review

Solve each equation in the interval from 0 to 2π. Round your answer to the nearest hundredth. See Lesson 13-5.

57. cos t = 1 4 cosine t equals , 1 fourth
58. 10 cos t = − 2 10 cosine t equals negative 2
59. 3 cos t 5 = 1 3 cosine , t over 5 , equals 1
60. 5 cos π t = 0 . 9 5 cosine pi t equals 0 . 9

61. Find the mean, median, and mode for the set of values. See Lesson 11-5.

    9 6 8 1 3 4 5 2 6 8 4 9 12 3 4 10 7 6

Find the 27th term of each sequence. See Lesson 9-2.

62. 5, 8, 11, …
63. 59, 48, 37, …
64. − 11 , − 5 , 1 , … negative 11 comma negative 5 comma 1 comma dot dot dot
65. 6 , − 7 , − 20 , … 6 comma negative 7 comma negative 20 comma dot dot dot

Get Ready! To prepare for Lesson 13-7, do Exercises 66–68.

Identify each horizontal and vertical translation of the parent function y = | x | . y equals vertical line x vertical line .  See Lesson 2-6.

66. y = | x − 2 | + 5 y equals vertical line x minus 2 vertical line plus 5
67. y = | x + 5 | − 4 y equals vertical line x plus 5 vertical line negative 4
68. y = | x + 2 | + 1 y equals vertical line x plus 2 vertical line plus 1

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