AP PreCalculus Practice Quiz: Parametrically Defined Circles and Lines

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

Test your understanding with short quizzes. This quiz has 10 questions to check your progress.

Question 1 of 10

Which of the following parametric functions models a complete counterclockwise revolution around the unit circle, starting and ending at the point (1, 0)?

A)x(t) = cos(t), y(t) = sin(t) for 0 ≤ t ≤ 2πB)x(t) = sin(t), y(t) = cos(t) for 0 ≤ t ≤ 2πC)x(t) = t, y(t) = 1 - t for 0 ≤ t ≤ 1D)x(t) = cos(t), y(t) = sin(t) for 0 ≤ t ≤ π

All Questions (10)

Which of the following parametric functions models a complete counterclockwise revolution around the unit circle, starting and ending at the point (1, 0)?

A) x(t) = cos(t), y(t) = sin(t) for 0 ≤ t ≤ 2π

B) x(t) = sin(t), y(t) = cos(t) for 0 ≤ t ≤ 2π

C) x(t) = t, y(t) = 1 - t for 0 ≤ t ≤ 1

D) x(t) = cos(t), y(t) = sin(t) for 0 ≤ t ≤ π

Correct Answer: A

The provided content explicitly states that a complete counterclockwise revolution around the unit circle starting and ending at (1, 0) is modeled by (x(t), y(t)) = (cos t, sin t) with the domain 0 ≤ t ≤ 2π.

A particle's motion is described by the parametric function (x(t), y(t)) = (cos t, sin t) with a domain of 0 ≤ t ≤ π. What path does the particle trace?

A) A full circle, traversed once.

B) A semicircle, starting at (1, 0) and ending at (-1, 0).

C) A full circle, traversed twice.

D) A straight line segment from (1, 0) to (-1, 0).

Correct Answer: B

The standard parametrization for a full counterclockwise circle uses the domain 0 ≤ t ≤ 2π. Using a domain of 0 ≤ t ≤ π covers only half of the revolution, starting at t=0 (point (1,0)) and ending at t=π (point (-1,0)), thus tracing a semicircle.

Based on the principles of transforming the standard unit circle parametrization, which set of equations models a circular path centered at the origin with a radius of 7?

A) x(t) = 7 + cos(t), y(t) = 7 + sin(t)

B) x(t) = cos(7t), y(t) = sin(7t)

C) x(t) = 7cos(t), y(t) = 7sin(t)

D) x(t) = cos(t), y(t) = sin(t) for 0 ≤ t ≤ 14π

Correct Answer: C

The content states that transformations of (cos t, sin t) can model any circular path. To change the radius from 1 (for the unit circle) to 7, the x and y components are multiplied by 7. Adding 7 would shift the center, and multiplying t by 7 would change the speed of revolution.

Which set of parametric equations represents a unit circle (radius 1) centered at the point (-2, 4)?

A) x(t) = -2cos(t), y(t) = 4sin(t)

B) x(t) = cos(t) - 2, y(t) = sin(t) + 4

C) x(t) = cos(t - 2), y(t) = sin(t + 4)

D) x(t) = 4cos(t), y(t) = -2sin(t)

Correct Answer: B

Transformations of the basic unit circle (cos t, sin t) can model any circular path. To shift the center from the origin (0,0) to a new center (h, k), we add h to the x-component and k to the y-component. For a center at (-2, 4), the equations become x(t) = cos(t) - 2 and y(t) = sin(t) + 4.

A particle moves along a linear path starting at point (5, 2) and ending at point (1, 10) as the parameter t goes from 0 to 1. Which parametrization models this motion?

A) x(t) = 5 + t, y(t) = 2 + 10t

B) x(t) = 5 - 4t, y(t) = 2 + 8t

C) x(t) = 1 + 4t, y(t) = 10 - 8t

D) x(t) = 5 + 4t, y(t) = 2 - 8t

Correct Answer: B

A linear path from (x₁, y₁) to (x₂, y₂) can be parametrized using an initial position and rates of change. The initial position is (5, 2). The total change in x is 1 - 5 = -4, and the total change in y is 10 - 2 = 8. For t from 0 to 1, the parametrization is x(t) = x₁ + (Δx)t = 5 - 4t and y(t) = y₁ + (Δy)t = 2 + 8t.

A particle's motion is described by the equations x(t) = 1 + 5t and y(t) = 3 - 2t. According to the provided content, what type of path is being modeled?

A) A circular path centered at (1, 3).

B) A circular path with a radius of 5.

C) A linear path starting at (1, 3).

D) A linear path ending at (5, -2).

Correct Answer: C

The content describes that a linear path can be parametrized using an initial position and rates of change. The form x(t) = x₁ + at, y(t) = y₁ + bt represents a line. Here, (x₁, y₁) is the initial position (at t=0), which is (1, 3). The values 5 and -2 are the rates of change for x and y.

Which set of parametric equations could model the motion of a particle along a circular path with a radius of 3, centered at (8, -1)?

A) x(t) = 8 + 3cos(t), y(t) = -1 + 3sin(t)

B) x(t) = 3 + 8cos(t), y(t) = 3 - sin(t)

C) x(t) = 8cos(t), y(t) = -sin(t)

D) x(t) = 3cos(t) - 8, y(t) = 3sin(t) + 1

Correct Answer: A

This requires combining two transformations on the base unit circle (cos t, sin t). To change the radius to 3, we multiply by 3, resulting in (3cos t, 3sin t). To shift the center to (8, -1), we add 8 to the x-component and -1 to the y-component, giving x(t) = 8 + 3cos(t) and y(t) = -1 + 3sin(t).

The parametrization (x(t), y(t)) = (cos t, -sin t) for 0 ≤ t ≤ 2π traces the unit circle. What is the direction of motion and the starting point?

A) Counterclockwise, starting at (1, 0).

B) Clockwise, starting at (1, 0).

C) Counterclockwise, starting at (-1, 0).

D) Clockwise, starting at (0, -1).

Correct Answer: B

This is a transformation of the standard counterclockwise path (cos t, sin t). At t=0, the position is (cos(0), -sin(0)) = (1, 0). At a slightly later time, t=π/2, the position is (cos(π/2), -sin(π/2)) = (0, -1). The path moves from (1, 0) to (0, -1), which is a clockwise direction.

Based on the provided content, a primary structural difference between parametric equations for circles and lines is that:

A) Circular paths require a parameter domain of 2π, while linear paths do not.

B) Linear paths have constant rates of change for x and y, while circular paths have rates of change that vary.

C) Circular paths are modeled using trigonometric functions, while linear paths can be modeled using linear functions of the parameter.

D) Only linear paths can be defined by a starting and ending point.

Correct Answer: C

The content explicitly models circles using (cos t, sin t) and their transformations, which are trigonometric. It models lines using an initial position and rates of change, resulting in equations of the form x = x₁ + at, y = y₁ + bt, which are linear functions of the parameter t.

For the linear path from (x₁, y₁) to (x₂, y₂), the content mentions using 'rates of change for x with respect to t and y with respect to t'. In the specific parametrization x(t) = x₁ + (x₂ - x₁)t, y(t) = y₁ + (y₂ - y₁)t for 0 ≤ t ≤ 1, what do the terms (x₂ - x₁) and (y₂ - y₁) represent?

A) The initial position of the particle.

B) The constant rates of change for x and y, respectively.

C) The midpoint of the line segment.

D) The length of the line segment.

Correct Answer: B

In this standard form for a line segment, the initial position at t=0 is (x₁, y₁). The coefficients of t, which are (x₂ - x₁) and (y₂ - y₁), represent the total change in each coordinate over the interval t=[0,1]. These are the constant rates of change for x and y with respect to the parameter t.