AP Precalculus Flashcards: Parametric Functions

What this deck covers

This deck focuses on Parametric Functions, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Determine the Cartesian equation from $x = 4 \cos(t)$, $y = 5 \sin(t)$.

Answer: $\frac{x^2}{16} + \frac{y^2}{25} = 1$. Ellipse with semi-axes 4 and 5.

Flashcard 2: What is the parameter in parametric equations?

Answer: A variable, often $t$, that both $x$ and $y$ are functions of. The independent variable that controls both coordinates.

Flashcard 3: Convert $x = 1 + 2t$, $y = 3 + 4t$ to a Cartesian equation.

Answer: $y = 2x + 1$. From $x = 1 + 2t$ get $t = \frac{x-1}{2}$, substitute.

Flashcard 4: Convert $x = \cos(t)$, $y = \sin(t)$ to a Cartesian equation.

Answer: $x^2 + y^2 = 1$. Unit circle using fundamental trigonometric identity.

Flashcard 5: What is the purpose of parametric equations in mathematics?

Answer: To describe geometric figures and motions. Enables modeling of complex paths and trajectories.

Flashcard 6: Convert $x = 5t - 2$, $y = 3t + 1$ to a Cartesian equation.

Answer: $y = \frac{3}{5}(x + 2) - 1$. From $x = 5t - 2$ get $t = \frac{x+2}{5}$, substitute.

Flashcard 7: What is a parametric equation?

Answer: An equation that expresses variables as functions of a parameter. Both $x$ and $y$ depend on the same parameter.

Flashcard 8: Identify parametric equations for a parabola $y = x^2$.

Answer: $x = t$, $y = t^2$. Simplest parametrization using $t$ as $x$-coordinate.

Flashcard 9: Identify parametric equations for a horizontal line $y = c$.

Answer: $x = t$, $y = c$. Parameter varies while $y$ remains constant.

Flashcard 10: Define what a parameter is in terms of parametric equations.

Answer: An independent variable that defines a set of equations. The controlling variable in parametric representation.

Flashcard 11: What is the parametric equation for a line with slope $m$?

Answer: $x = t$, $y = mt + c$. Standard form with slope $m$ and parameter $t$.

Flashcard 12: Convert the parametric equations $x = t + 1$, $y = 2t$ to Cartesian form.

Answer: $y = 2(x - 1)$. Solve for $t$ from first equation, substitute into second.

Flashcard 13: What is the parametric equation for a vertical line $x = c$?

Answer: $x = c$, $y = t$. Parameter varies while $x$ remains constant.

Flashcard 14: Which parametric equations describe the line segment from $(1,2)$ to $(4,8)$?

Answer: $x = 1 + 3t$, $y = 2 + 6t$, $0 \leq t \leq 1$. Direction vector $(3,6)$ with parameter range $[0,1]$.

Flashcard 15: What is the parametric form for a parabola $y = ax^2$?

Answer: $x = t$, $y = at^2$. General parabola form with coefficient $a$.

Flashcard 16: Convert $x = 4t$, $y = 9 - t^2$ to a Cartesian equation.

Answer: $y = 9 - \left(\frac{x}{4}\right)^2$. Substitute $t = \frac{x}{4}$ into the $y$ equation.

Flashcard 17: What is the role of $t$ in parametric equations?

Answer: It is the independent variable or parameter. Controls the position along the curve as it varies.

Flashcard 18: Identify the parametric form for the line $y = 3x + 2$.

Answer: $x = t$, $y = 3t + 2$. Set parameter $t = x$ for simplest form.

Flashcard 19: State the parametric equations for a circle centered at the origin.

Answer: $x = r \, \cos(t)$, $y = r \, \sin(t)$. Standard form using trigonometric functions with radius $r$.

Flashcard 20: Convert $x = 3t$, $y = 4t$ to a Cartesian equation.

Answer: $y = \frac{4}{3}x$. Eliminate $t$ by solving $x = 3t$ gives $t = \frac{x}{3}$.

Flashcard 21: Find the Cartesian equation from $x = 1 + t$, $y = 2t - 1$.

Answer: $y = 2x - 3$. From $x = 1 + t$, get $t = x - 1$, substitute.

Flashcard 22: Find the Cartesian equation from $x = 2 \cos(t)$, $y = 3 \sin(t)$.

Answer: $\frac{x^2}{4} + \frac{y^2}{9} = 1$. Use identity $\cos^2(t) + \sin^2(t) = 1$.

Flashcard 23: Convert $x = 3 \sin(t)$, $y = 4 \cos(t)$ to a Cartesian equation.

Answer: $\frac{x^2}{9} + \frac{y^2}{16} = 1$. Ellipse using sine for $x$ and cosine for $y$.

Flashcard 24: What do $a$ and $b$ represent in the ellipse parametric equations $x = a \cos(t)$, $y = b \sin(t)$?

Answer: The semi-major and semi-minor axes. The lengths of the ellipse's major and minor axes.

Flashcard 25: What is an advantage of parametric equations?

Answer: They allow the description of curves not functions. Can represent curves that fail the vertical line test.

Flashcard 26: What is the parametric form of a line segment from $(2,1)$ to $(5,4)$?

Answer: $x = 2 + 3t$, $y = 1 + 3t$, $0 \leq t \leq 1$. Direction vector $(3,3)$ from start to end point.

Flashcard 27: What are the parametric equations for a line parallel to $y = 2x + 3$?

Answer: $x = t$, $y = 2t + c$. Same slope but different $y$-intercept constant.

Flashcard 28: Convert $x = 2 \cos(t)$, $y = 2 \sin(t)$ to a Cartesian equation.

Answer: $x^2 + y^2 = 4$. Circle with radius 2 centered at origin.

Flashcard 29: Convert $x = 2t$, $y = t^2$ to a Cartesian equation.

Answer: $y = \frac{x^2}{4}$. Eliminate parameter: $t = \frac{x}{2}$, so $y = \left(\frac{x}{2}\right)^2$.

Flashcard 30: What is the parametric form for a circle with radius $r$?

Answer: $x = r \cos(t)$, $y = r \sin(t)$. General circle equation with specified radius.

Flashcard 31: What is the general form of parametric equations for a line?

Answer: $x = x_0 + at$, $y = y_0 + bt$. Point $(x_0, y_0)$ with direction vector $(a,b)$.

Flashcard 32: Convert $x = 5 \sin(t)$, $y = 5 \cos(t)$ to a Cartesian equation.

Answer: $x^2 + y^2 = 25$. Apply Pythagorean identity to eliminate parameter.

Flashcard 33: What is the parametric equation for a straight line through $(x_1, y_1)$, $(x_2, y_2)$?

Answer: $x = x_1 + (x_2 - x_1)t$, $y = y_1 + (y_2 - y_1)t$. Linear interpolation between two given points.

Flashcard 34: State the parametric equations for an ellipse centered at the origin.

Answer: $x = a \, \cos(t)$, $y = b \, \sin(t)$. Uses different coefficients for horizontal and vertical radii.