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AP PreCalculus Practice Quiz: Sinusoidal Function Transformations

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 14 questions to check your progress.

Question 1 of 14

In the function f(θ) = 5 sin(θ) + 2, what is the amplitude?

All Questions (14)

In the function f(θ) = 5 sin(θ) + 2, what is the amplitude?

A) 5

B) 2

C) 1

D) 2π

Correct Answer: A

According to the content, for a function in the form f(θ) = a sin(b(θ+c)) + d, the amplitude is |a|. In this case, a = 5, so the amplitude is |5| = 5.

Which parameter in the general form g(θ) = a cos(b(θ+c)) + d determines the vertical shift of the midline?

A) a

B) b

C) c

D) d

Correct Answer: D

The content states that the graph of y = a sin(b(θ+c)) + d has a midline vertical shift of d units from y=0. This also applies to the cosine function.

What is the phase shift of the function f(θ) = sin(θ + π/2)?

A) π/2 units to the right

B) π/2 units to the left

C) 2 units to the left

D) 2 units to the right

Correct Answer: B

The content specifies that the phase shift is -c. In the function f(θ) = sin(θ + π/2), c = π/2. Therefore, the phase shift is -(π/2), which represents a shift of π/2 units to the left.

What is the period of the function g(θ) = cos(4θ)?

A) 4π

B) 2π

C) π/2

D) 8π

Correct Answer: C

The period of a sinusoidal function is given by the formula |1/b| * 2π. For g(θ) = cos(4θ), the value of b is 4. The period is (1/4) * 2π = 2π/4 = π/2.

The graph of g(θ) = sin(θ) - 3 is a transformation of the graph of f(θ) = sin(θ). How is the graph of f(θ) transformed?

A) Translated 3 units to the right

B) Translated 3 units to the left

C) Translated 3 units up

D) Translated 3 units down

Correct Answer: D

The content states that the graph of g(θ) = sin(θ) + d is a vertical translation of the graph of f by d units. In this case, d = -3, so the graph is translated 3 units down.

The transformation g(θ) = sin(bθ) is described as a horizontal dilation of f(θ) = sin(θ). This transformation affects which property of the graph?

A) Amplitude

B) Midline

C) Period

D) Phase Shift

Correct Answer: C

According to the provided content, the graph of the multiplicative transformation g(θ) = sin(bθ) is a horizontal dilation and differs in period by a factor of |1/b|.

For the function y = -4 cos(θ) + 1, what are the amplitude and the midline vertical shift?

A) Amplitude: -4, Midline shift: 1

B) Amplitude: 4, Midline shift: 1

C) Amplitude: 1, Midline shift: 4

D) Amplitude: 4, Midline shift: -1

Correct Answer: B

The amplitude is given by |a|. Here, a = -4, so the amplitude is |-4| = 4. The midline vertical shift is given by d, which is 1 in this function.

Identify the period and phase shift of the function f(θ) = 3 sin(2(θ - π/4)).

A) Period: π, Phase shift: π/4 to the left

B) Period: 4π, Phase shift: π/4 to the right

C) Period: π, Phase shift: π/4 to the right

D) Period: 2π, Phase shift: π/2 to the left

Correct Answer: C

The period is |1/b| * 2π. Here b=2, so the period is (1/2) * 2π = π. The phase shift is -c. In the form b(θ+c), we have 2(θ - π/4), so c = -π/4. The phase shift is -(-π/4) = π/4, which is a shift to the right.

How does the graph of g(θ) = 0.5 sin(θ) differ from the graph of f(θ) = sin(θ)?

A) It is horizontally compressed by a factor of 0.5.

B) It is vertically compressed by a factor of 0.5.

C) It is shifted down by 0.5 units.

D) It is shifted right by 0.5 units.

Correct Answer: B

The function g(θ) = a sin(θ) represents a vertical dilation of the graph of f(θ) = sin(θ). Since |a| = 0.5, which is less than 1, the graph is vertically compressed. The amplitude is changed by a factor of |a|.

A sinusoidal function has a midline at y = -5, an amplitude of 3, a period of π, and a phase shift of 2 units to the left. Which of the following functions matches this description?

A) f(θ) = 3 sin(2(θ - 2)) - 5

B) f(θ) = -5 sin(0.5(θ + 2)) + 3

C) f(θ) = 3 sin(0.5(θ - 2)) - 5

D) f(θ) = 3 sin(2(θ + 2)) - 5

Correct Answer: D

Amplitude |a|=3. Midline d=-5. Period |1/b|*2π = π, so |b|=2. Phase shift -c = -2, so c=2. Combining these gives the form f(θ) = 3 sin(2(θ + 2)) - 5.

The value |a| in the function f(θ) = a sin(b(θ+c)) + d corresponds to which feature of the graph?

A) Period

B) Phase Shift

C) Midline

D) Amplitude

Correct Answer: D

The provided content explicitly states that the graph of y = f(θ) = a sin(b(θ+c)) + d has an amplitude of |a| units.

The period of g(θ) = sin(bθ) differs from the period of f(θ) = sin(θ) by a factor of |1/b|. If the period of g(θ) is 4π, what is the value of b?

A) 2

B) 1/2

C) 4

D) 1/4

Correct Answer: B

The period of a standard sine function is 2π. The new period is given by |1/b| * 2π. We are given that the new period is 4π. So, |1/b| * 2π = 4π. Dividing both sides by 2π gives |1/b| = 2. This implies |b| = 1/2. Assuming b is positive, b = 1/2.

Which transformation is represented by the parameter 'c' in the function g(θ) = sin(θ + c)?

A) Vertical dilation

B) Vertical translation

C) Horizontal dilation

D) Horizontal translation

Correct Answer: D

The content states that the graph of g(θ) = sin(θ + c) is a horizontal translation, or phase shift, of the graph of f(θ) = sin(θ) by -c units.

Consider the function y = a sin(b(θ+c)) + d. If a < 0, how does this affect the graph compared to a case where a > 0, assuming all other parameters are identical?

A) The amplitude becomes negative.

B) The graph is reflected across the midline.

C) The period is inverted.

D) The phase shift is reversed.

Correct Answer: B

The parameter 'a' controls the vertical dilation (amplitude). The amplitude is |a|, which is always positive. A negative sign on 'a' reflects the graph vertically. In a sinusoidal function, this reflection occurs across its midline (y=d). This specific detail is an extension of the provided content about 'a' representing a vertical dilation.