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AP PreCalculus Practice Quiz: The Tangent Function

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

Test your understanding with short quizzes. This quiz has 16 questions to check your progress.

Question 1 of 16

The tangent function, f(θ) = tan θ, is defined as which of the following ratios?

All Questions (16)

The tangent function, f(θ) = tan θ, is defined as which of the following ratios?

A) sin θ / cos θ

B) cos θ / sin θ

C) 1 / sin θ

D) 1 / cos θ

Correct Answer: A

Based on the provided content, 'The tangent function is the ratio sin θ / cos θ, where cos θ ≠ 0.'

On the unit circle, what geometric property does the value of tan θ represent?

A) The length of the arc from (1, 0) to the terminal point

B) The y-coordinate of the terminal point

C) The x-coordinate of the terminal point

D) The slope of the terminal ray

Correct Answer: D

The provided text states, 'The tangent function, f(θ) = tan θ, gives the slope of the terminal ray on the unit circle.'

What is the period of the basic tangent function, f(θ) = tan θ?

A) π/2

B) π

C) 2π

D) 4π

Correct Answer: B

The content states, '...the tangent function has a period of π.'

According to the provided text, why is the period of the tangent function π?

A) Because sin(θ) and cos(θ) both have a period of 2π.

B) Because the slope values of the terminal ray repeat every one-half revolution of the circle.

C) Because the function has asymptotes at θ = π/2 + kπ.

D) Because the range of the tangent function is all real numbers.

Correct Answer: B

The text explicitly states, 'Because the slope values of the terminal ray repeat every one-half revolution of the circle, the tangent function has a period of π.'

The graph of f(θ) = tan θ has vertical asymptotes at values of θ where which condition is met?

A) sin θ = 0

B) cos θ = 0

C) tan θ = 1

D) tan θ = 0

Correct Answer: B

The content explains that the tangent function has 'periodic asymptotic behavior at input values... because cos θ = 0 at those values.' This is due to the definition tan θ = sin θ / cos θ, where division by zero is undefined.

The tangent function demonstrates periodic asymptotic behavior at which set of input values, for any integer k?

A) θ = kπ

B) θ = 2kπ

C) θ = π/2 + kπ

D) θ = π/2 + 2kπ

Correct Answer: C

The content specifies that asymptotes occur at 'input values θ = π/2 + kπ, for integer values of k.'

In the function y = a tan(b(θ+c)) + d, which parameter causes a vertical dilation of the graph of y = tan θ?

A) a

B) b

C) c

D) d

Correct Answer: A

The content states that the graph of the transformed function is a 'vertical dilation of the graph of y = tan θ by a factor of |a|.'

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

A) 4π

B) π

C) π/2

D) π/4

Correct Answer: D

For a function of the form y = a tan(b(θ+c)) + d, the period is |π/b|. In this case, b=4, so the period is |π/4| = π/4.

What is the phase shift of the function f(θ) = tan(θ + π/3)?

A) π/3 units to the right

B) π/3 units to the left

C) π/3 units up

D) π/3 units down

Correct Answer: B

For the form y = a tan(b(θ+c)) + d, the phase shift is -c. Here, c = π/3, so the shift is -π/3, which is π/3 units to the left.

How does the graph of g(θ) = tan(θ) - 2 relate to the graph of f(θ) = tan(θ)?

A) It is shifted 2 units to the right.

B) It is shifted 2 units to the left.

C) It is shifted 2 units up.

D) It is shifted 2 units down.

Correct Answer: D

The parameter d in y = a tan(b(θ+c)) + d represents a vertical shift. Here, d = -2, so the graph is shifted 2 units down.

What is the period of the function y = -3 tan( (π/4)θ )?

A) π/4

B) 4

C) 4π

D) π²/4

Correct Answer: B

The period is given by the formula |π/b|. In this function, b = π/4. Therefore, the period is |π / (π/4)| = |π * (4/π)| = 4.

Consider the function g(θ) = 2 tan(θ - π/2) + 5. What transformation does the value d=5 represent?

A) A phase shift of 5 units to the right.

B) A vertical dilation by a factor of 5.

C) A change in period to 5π.

D) A vertical shift of the line containing the points of inflection by 5 units.

Correct Answer: D

The content states that d represents 'a vertical shift of the line containing the points of inflection of the graph of y = tan θ by d units.' Here, d=5.

The function f(θ) = tan θ is undefined for θ = -π/2. This is because at this value:

A) sin θ = 0

B) sin θ = -1

C) cos θ = 0

D) cos θ = 1

Correct Answer: C

The tangent function is the ratio sin θ / cos θ. It is undefined when the denominator, cos θ, is equal to 0. At θ = -π/2, the value of cos θ is 0.

What is the phase shift of the function y = 4 tan(2θ + π)?

A) π units to the left

B) π units to the right

C) π/2 units to the left

D) π/2 units to the right

Correct Answer: C

First, the function must be in the form y = a tan(b(θ+c)) + d. We factor out b=2: y = 4 tan(2(θ + π/2)). In this form, c = π/2. The phase shift is -c, which is -π/2, or π/2 units to the left.

Which statement best describes the relationship between the unit circle and the periodic asymptotic behavior of the tangent function?

A) The asymptotes occur when the terminal ray on the unit circle is horizontal, as the slope is zero.

B) The asymptotes occur when the terminal ray on the unit circle is vertical, as the slope is undefined.

C) The asymptotes occur when the terminal ray completes a full revolution, as the slope returns to its starting value.

D) The asymptotes occur when the sine and cosine values are equal, resulting in a slope of 1.

Correct Answer: B

The tangent function represents the slope of the terminal ray. Asymptotes occur where the function is undefined. This happens when cos θ = 0, which corresponds to points (0, 1) and (0, -1) on the unit circle. At these points, the terminal ray is vertical, and a vertical line has an undefined slope.

For the function g(θ) = a tan(b(θ+c)) + d, which parameters directly affect the horizontal locations of the vertical asymptotes compared to the parent function f(θ) = tan θ?

A) a and d

B) b and c

C) a and c

D) b and d

Correct Answer: B

The vertical asymptotes of tan θ occur at θ = π/2 + kπ. In the transformed function, the input is b(θ+c). The asymptotes will occur when b(θ+c) = π/2 + kπ. Solving for θ involves b (which changes the period, or horizontal spacing, of the asymptotes) and c (which shifts them horizontally). The parameters a (vertical stretch) and d (vertical shift) do not affect the horizontal position of the asymptotes.