AP PreCalculus Practice Quiz: Parametrization of Implicitly Defined Functions

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

Test your understanding with short quizzes. This quiz has 15 questions to check your progress.

Question 1 of 15

According to the provided content, what is a standard way to parametrize a function of the form y = f(x)?

A)(x(t), y(t)) = (f(t), t)B)(x(t), y(t)) = (t, f(t))C)(x(t), y(t)) = (t, t)D)(x(t), y(t)) = (f(t), f(t))

All Questions (15)

According to the provided content, what is a standard way to parametrize a function of the form y = f(x)?

A) (x(t), y(t)) = (f(t), t)

B) (x(t), y(t)) = (t, f(t))

C) (x(t), y(t)) = (t, t)

D) (x(t), y(t)) = (f(t), f(t))

Correct Answer: B

The content states that if f is a function of x, then y = f(x) can be parametrized as (x(t), y(t)) = (t, f(t)). This involves setting the independent variable x equal to the parameter t and then defining y in terms of t through the function f.

The standard parametrization for an ellipse, x(t) = h + a cos(t) and y(t) = k + b sin(t), uses which pair of trigonometric functions?

A) secant and tangent

B) sine and cosine

C) cosecant and cotangent

D) sine and tangent

Correct Answer: B

The provided content explicitly gives the parametrization for an ellipse using the trigonometric functions sine and cosine: x(t) = h + a cos(t) and y(t) = k + b sin(t).

Which of the following is a valid parametrization for the ellipse defined by the equation (x-2)²/25 + (y+1)²/9 = 1?

A) x(t) = 2 + 25 cos(t), y(t) = -1 + 9 sin(t)

B) x(t) = 2 + 5 cos(t), y(t) = -1 + 3 sin(t)

C) x(t) = 5 + 2 cos(t), y(t) = 3 - 1 sin(t)

D) x(t) = -2 + 5 cos(t), y(t) = 1 + 3 sin(t)

Correct Answer: B

The standard form for an ellipse is x(t) = h + a cos(t) and y(t) = k + b sin(t). For the given equation, the center (h, k) is (2, -1). The values for a and b are the square roots of the denominators, so a = √25 = 5 and b = √9 = 3. Substituting these values gives x(t) = 2 + 5 cos(t) and y(t) = -1 + 3 sin(t).

What is the fundamental condition that a parametrization (x(t), y(t)) must meet to correctly represent an implicitly defined function?

A) The derivatives dx/dt and dy/dt must be continuous.

B) The parameter t must be within the interval [0, 2π].

C) When substituted for x and y, the parametrization must satisfy the original equation for all values of t in the domain.

D) The functions x(t) and y(t) must be trigonometric.

Correct Answer: C

The provided content states: 'A parametrization (x(t), y(t)) for an implicitly defined function will, when x(t) and y(t) are substituted for x and y, respectively, satisfy the corresponding equation for every value of t in the domain.' This is the defining characteristic of a valid parametrization.

Which of the following is a possible parametrization for the hyperbola given by the equation (x-1)²/16 - (y-3)²/4 = 1?

A) x(t) = 1 + 16 sec(t), y(t) = 3 + 4 tan(t)

B) x(t) = 1 + 4 cos(t), y(t) = 3 + 2 sin(t)

C) x(t) = -1 + 4 sec(t), y(t) = -3 + 2 tan(t)

D) x(t) = 1 + 4 sec(t), y(t) = 3 + 2 tan(t)

Correct Answer: D

The content provides a standard parametrization for a hyperbola as x(t) = h + a sec(t) and y(t) = k + b tan(t). From the equation, the center (h, k) is (1, 3). The values for a and b are the square roots of the denominators, so a = √16 = 4 and b = √4 = 2. This leads to the parametrization x(t) = 1 + 4 sec(t) and y(t) = 3 + 2 tan(t).

A curve is defined by the parametrization x(t) = -3 + 5 sec(t) and y(t) = 2 + 4 tan(t). This parametrization represents which type of conic section?

A) A circle

B) An ellipse

C) A parabola

D) A hyperbola

Correct Answer: D

The content states that a hyperbola can be parametrized using trigonometric functions such as x(t) = h + a sec(t) and y(t) = k + b tan(t). The given parametrization fits this form, indicating it represents a hyperbola.

Using the standard method for parametrizing a function y = f(x), what is a correct parametrization for the parabola y = x² + 5?

A) x(t) = t² + 5, y(t) = t

B) x(t) = t, y(t) = t² + 5

C) x(t) = cos(t), y(t) = cos²(t) + 5

D) x(t) = t, y(t) = (t+5)²

Correct Answer: B

The content specifies that a function y = f(x) can be parametrized as (t, f(t)). For the function y = x² + 5, we let x = t. Substituting this into the function gives y = t² + 5. Therefore, the parametrization is x(t) = t, y(t) = t² + 5.

A curve is parametrized by x(t) = 4 + cos(t) and y(t) = -1 + 2 sin(t) for 0 ≤ t ≤ 2π. Which of the following implicit equations does this parametrization satisfy?

A) (x-4)²/1 + (y+1)²/4 = 1

B) (x+4)²/1 + (y-1)²/4 = 1

C) (x-4)²/1 - (y+1)²/4 = 1

D) y = 2x - 9

Correct Answer: A

The parametrization x(t) = 4 + cos(t) and y(t) = -1 + 2 sin(t) is for an ellipse with center (h, k) = (4, -1), a = 1, and b = 2. We can isolate the trigonometric functions: cos(t) = x - 4 and sin(t) = (y + 1)/2. Using the identity cos²(t) + sin²(t) = 1, we substitute to get (x - 4)² + ((y + 1)/2)² = 1, which simplifies to (x-4)²/1 + (y+1)²/4 = 1.

For the standard parametrization of an ellipse, x(t) = h + a cos(t) and y(t) = k + b sin(t), what is the typical domain for the parameter t to trace the entire curve exactly once?

A) 0 ≤ t ≤ π/2

B) 0 ≤ t ≤ π

C) 0 ≤ t ≤ 2π

D) -∞ < t < ∞

Correct Answer: C

The provided content explicitly states that for the parametrization of an ellipse, the domain for t is 0 ≤ t ≤ 2π. This range allows the sine and cosine functions to complete one full cycle, thus tracing the entire ellipse.

Consider the circle x² + y² = 16. Which of the following parametrizations satisfies this equation?

A) x(t) = 16 cos(t), y(t) = 16 sin(t)

B) x(t) = 4 sec(t), y(t) = 4 tan(t)

C) x(t) = 4 cos(t), y(t) = 4 sin(t)

D) x(t) = t, y(t) = 16 - t

Correct Answer: C

A circle is a special case of an ellipse where a = b. The equation x² + y² = 16 has a radius of √16 = 4. Using the ellipse formula with h=0, k=0, and a=b=4, we get x(t) = 4 cos(t) and y(t) = 4 sin(t). To verify, substitute these into the equation: (4 cos(t))² + (4 sin(t))² = 16cos²(t) + 16sin²(t) = 16(cos²(t) + sin²(t)) = 16(1) = 16. This satisfies the equation.

Which of the following is NOT a valid parametrization for the line y = 3x?

A) x(t) = t, y(t) = 3t

B) x(t) = 2t, y(t) = 6t

C) x(t) = t + 1, y(t) = 3t + 3

D) x(t) = t, y(t) = 3t + 1

Correct Answer: D

A valid parametrization must satisfy the original equation when substituted. We check if y(t) = 3 * x(t) for each option. For D, we check if (3t + 1) = 3 * (t). This simplifies to 3t + 1 = 3t, or 1 = 0, which is false. Therefore, this parametrization does not satisfy the equation y = 3x.

A curve is represented by x(t) = 7 + 3 cos(t) and y(t) = -2 + 3 sin(t). What geometric shape is described by this parametrization?

A) A hyperbola centered at (7, -2)

B) An ellipse with a horizontal major axis

C) A circle with radius 3 centered at (7, -2)

D) A parabola opening upwards

Correct Answer: C

This parametrization follows the form for an ellipse, x(t) = h + a cos(t) and y(t) = k + b sin(t). Here, h=7, k=-2, a=3, and b=3. Since a = b, the ellipse is a circle with center (7, -2) and radius 3.

A curve is parametrized by x(t) = 5 + sec(t) and y(t) = -2 + tan(t). Which of the following is the corresponding implicit equation?

A) (x-5)² + (y+2)² = 1

B) (x-5)² - (y+2)² = 1

C) (x+5)² - (y-2)² = 1

D) y = tan(x-5) - 2

Correct Answer: B

The parametrization uses secant and tangent, which suggests a hyperbola. We isolate the trigonometric functions: sec(t) = x - 5 and tan(t) = y + 2. Using the identity sec²(t) - tan²(t) = 1, we substitute to get (x - 5)² - (y + 2)² = 1.

In the parametrization of a hyperbola, x(t) = h + a sec(t) and y(t) = k + b tan(t), what do the parameters h and k determine?

A) The lengths of the axes

B) The location of the foci

C) The coordinates of the center of the hyperbola

D) The equations of the asymptotes

Correct Answer: C

Based on the provided formulas for both ellipses and hyperbolas, the parameters h and k represent the x and y coordinates of the center of the conic section, respectively. They cause a horizontal and vertical shift from the origin.

The parametrization x(t) = 3 cos(t) and y(t) = 5 sin(t) satisfies the equation x²/9 + y²/25 = 1. What would be the effect of changing the parametrization to x(t) = 3 cos(2t) and y(t) = 5 sin(2t) for 0 ≤ t ≤ 2π?

A) The shape of the curve changes from an ellipse to a hyperbola.

B) The center of the ellipse shifts to (2, 2).

C) The same ellipse is traced, but it is traced twice.

D) The parametrization is no longer valid as it does not satisfy the equation.

Correct Answer: C

Substituting x(t) = 3 cos(2t) and y(t) = 5 sin(2t) into the equation gives (3cos(2t))²/9 + (5sin(2t))²/25 = (9cos²(2t))/9 + (25sin²(2t))/25 = cos²(2t) + sin²(2t) = 1. The equation is still satisfied, so the curve is the same ellipse. However, because of the '2t', the parameter t only needs to go from 0 to π to complete one full trace. Since the domain is 0 ≤ t ≤ 2π, the ellipse will be traced twice.