AP PreCalculus Flashcards: Parametrization of Implicitly Defined Functions
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 11 cards to help you master important concepts.
How can a function of the form $y = f(x)$ be parametrized?
A simple and direct parametrization is to let $x=t$, which results in the parametric equations $(x(t), y(t)) = (t, f(t))$.
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How can a function of the form $y = f(x)$ be parametrized?
A simple and direct parametrization is to let $x=t$, which results in the parametric equations $(x(t), y(t)) = (t, f(t))$.
Parametrize the function $y = x^3 - 2x$.
Using the simplest method for a function of x, the parametrization is $(x(t), y(t)) = (t, t^3 - 2t)$.
Identify the conic section represented by the parametric equations $x(t) = -2 + 4 \sec t$ and $y(t) = 1 + 3 \\tan t$.
This represents a hyperbola, because the parametrization follows the form $x = h + a \sec t$ and $y = k + b \\tan t$.
What does it mean to represent a curve parametrically?
It means to define the coordinates of points on the curve, (x, y), as functions of a single independent variable, called a parameter (e.g., t).
What is a common parametrization for a hyperbola centered at $(h, k)$?
A common parametrization uses the trigonometric functions secant and tangent: $x(t) = h + a \sec t$ and $y(t) = k + b \\tan t$.
What is the standard parametrization for an ellipse centered at $(h, k)$?
The standard parametrization uses trigonometric functions: $x(t) = h + a \\cos t$ and $y(t) = k + b \\sin t$.
Why is the combination of cosine and sine effective for parametrizing an ellipse?
This combination is effective because it relies on the Pythagorean identity $\\cos^2 t + \\sin^2 t = 1$, which mirrors the structure of the standard ellipse equation.
What is the typical range for the parameter $t$ needed to trace an entire ellipse exactly once?
The typical range is $0 \le t \le 2\\pi$, which completes one full cycle of the sine and cosine functions.
Identify the conic section represented by the parametric equations $x(t) = 3 + 5 \\cos t$ and $y(t) = -1 + 2 \\sin t$.
This represents an ellipse, because the parametrization follows the form $x = h + a \\cos t$ and $y = k + b \\sin t$.
Why is the combination of secant and tangent effective for parametrizing a hyperbola?
This combination is effective because it relies on the Pythagorean identity $\sec^2 t - \\tan^2 t = 1$, which mirrors the structure of the standard hyperbola equation.
How do you verify that a parametrization $(x(t), y(t))$ correctly represents an implicitly defined function?
You must substitute the expressions for $x(t)$ and $y(t)$ into the implicit equation; the equation must hold true for every value of $t$ in the domain.