Precalculus Practice Test: Graphs of Trig Functions
- Graphs of Trig Functions
- 10 questions
- 15 minutes
These 10 questions cover amplitude, period, phase shift, midline and range of sine, cosine and tangent graphs. For y = A sin(B(x − C)) + D, the period is 2π/|B|, but for tangent it is π/|B|.
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Question 1
What is the period of y = 3 sin(4x)?
Correct answer: D
The period of sin(Bx) is 2π/B = 2π/4 = π/2. The number 3 is the amplitude and does not change the period; 8π comes from multiplying 2π by 4.
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Question 2
What is the amplitude of y = −5 cos(x2) + 1?
Correct answer: B
The amplitude is |A| = |−5| = 5; the negative sign only reflects the graph. An amplitude is a distance, so it cannot be −5, and the +1 shifts the midline without changing the amplitude.
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Question 3
What is the phase shift of y = sin(2x − π3)?
Correct answer: A
Factor out the 2: sin(2(x − π/6)), so the graph moves π/6 to the right. Reading π/3 directly forgets that the 2 also acts on the shift.
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Question 4
What is the range of y = 2 cos x − 3?
Correct answer: C
The midline is y = −3 and the amplitude is 2, so the outputs run from −3 − 2 = −5 to −3 + 2 = −1. The interval [−2, 2] ignores the vertical shift.
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Question 5
What is the period of y = tan(3x)?
Correct answer: D
The tangent function repeats every π, so tan(3x) repeats every π/3. The value 2π/3 uses the sine and cosine period by mistake.
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Question 6
What is the maximum value of y = 4 sin x + 2?
Correct answer: B
The largest value of sin x is 1, so the maximum is 4(1) + 2 = 6. The value 4 is the amplitude, which ignores the upward shift of 2.
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Question 7
Which equation has amplitude 3, period π, and passes through the origin while increasing there?
Correct answer: C
Period π requires 2π/B = π, so B = 2, and a positive sine starts at 0 and increases. With −3 in front, the graph decreases as it leaves the origin.
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Question 8
Where are the vertical asymptotes of y = tan x? (k is any integer.)
Correct answer: A
Since tan x = sin x/cos x, asymptotes occur where cos x = 0, at π/2 + kπ. At x = kπ the tangent equals 0, so those are zeros, not asymptotes.
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Question 9
The height in feet of a rider on a Ferris wheel is h(t) = 20 − 18 cos(πt5), where t is in minutes. What is the greatest height the rider reaches?
Correct answer: D
The cosine term ranges from −1 to 1, so −18 cos is at most 18 and the greatest height is 20 + 18 = 38 feet. The value 2 feet is the lowest height, and 20 feet is the center.
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Question 10
What are the zeros of y = cos(2x) on the interval [0, π]?
Correct answer: B
Solve 2x = π/2 or 3π/2 (the zeros of cosine in [0, 2π]), so x = π/4 or 3π/4. The value π/2 is a zero of cos x, not of cos(2x), where it gives cos π = −1.
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