Parametric Functions - AP Precalculus
Card 1 of 30
What is a parametric equation?
What is a parametric equation?
Tap to reveal answer
An equation that expresses variables as functions of a parameter. Both $x$ and $y$ depend on the same parameter.
An equation that expresses variables as functions of a parameter. Both $x$ and $y$ depend on the same parameter.
← Didn't Know|Knew It →
What is the parametric form of a line segment from $(2,1)$ to $(5,4)$?
What is the parametric form of a line segment from $(2,1)$ to $(5,4)$?
Tap to reveal answer
$x = 2 + 3t$, $y = 1 + 3t$, $0 \leq t \leq 1$. Direction vector $(3,3)$ from start to end point.
$x = 2 + 3t$, $y = 1 + 3t$, $0 \leq t \leq 1$. Direction vector $(3,3)$ from start to end point.
← Didn't Know|Knew It →
Convert $x = 1 + 2t$, $y = 3 + 4t$ to a Cartesian equation.
Convert $x = 1 + 2t$, $y = 3 + 4t$ to a Cartesian equation.
Tap to reveal answer
$y = 2x + 1$. From $x = 1 + 2t$ get $t = \frac{x-1}{2}$, substitute.
$y = 2x + 1$. From $x = 1 + 2t$ get $t = \frac{x-1}{2}$, substitute.
← Didn't Know|Knew It →
State the parametric equations for a circle centered at the origin.
State the parametric equations for a circle centered at the origin.
Tap to reveal answer
$x = r , \cos(t)$, $y = r , \sin(t)$. Standard form using trigonometric functions with radius $r$.
$x = r , \cos(t)$, $y = r , \sin(t)$. Standard form using trigonometric functions with radius $r$.
← Didn't Know|Knew It →
What is the parametric form of a line segment from $(2,1)$ to $(5,4)$?
What is the parametric form of a line segment from $(2,1)$ to $(5,4)$?
Tap to reveal answer
$x = 2 + 3t$, $y = 1 + 3t$, $0 \leq t \leq 1$. Direction vector $(3,3)$ from start to end point.
$x = 2 + 3t$, $y = 1 + 3t$, $0 \leq t \leq 1$. Direction vector $(3,3)$ from start to end point.
← Didn't Know|Knew It →
What is the parametric equation for a vertical line $x = c$?
What is the parametric equation for a vertical line $x = c$?
Tap to reveal answer
$x = c$, $y = t$. Parameter varies while $x$ remains constant.
$x = c$, $y = t$. Parameter varies while $x$ remains constant.
← Didn't Know|Knew It →
Convert $x = 2t$, $y = t^2$ to a Cartesian equation.
Convert $x = 2t$, $y = t^2$ to a Cartesian equation.
Tap to reveal answer
$y = \frac{x^2}{4}$. Eliminate parameter: $t = \frac{x}{2}$, so $y = \left(\frac{x}{2}\right)^2$.
$y = \frac{x^2}{4}$. Eliminate parameter: $t = \frac{x}{2}$, so $y = \left(\frac{x}{2}\right)^2$.
← Didn't Know|Knew It →
Find the Cartesian equation from $x = 2 \cos(t)$, $y = 3 \sin(t)$.
Find the Cartesian equation from $x = 2 \cos(t)$, $y = 3 \sin(t)$.
Tap to reveal answer
$\frac{x^2}{4} + \frac{y^2}{9} = 1$. Use identity $\cos^2(t) + \sin^2(t) = 1$.
$\frac{x^2}{4} + \frac{y^2}{9} = 1$. Use identity $\cos^2(t) + \sin^2(t) = 1$.
← Didn't Know|Knew It →
Convert $x = 5 \sin(t)$, $y = 5 \cos(t)$ to a Cartesian equation.
Convert $x = 5 \sin(t)$, $y = 5 \cos(t)$ to a Cartesian equation.
Tap to reveal answer
$x^2 + y^2 = 25$. Apply Pythagorean identity to eliminate parameter.
$x^2 + y^2 = 25$. Apply Pythagorean identity to eliminate parameter.
← Didn't Know|Knew It →
Convert $x = 5t - 2$, $y = 3t + 1$ to a Cartesian equation.
Convert $x = 5t - 2$, $y = 3t + 1$ to a Cartesian equation.
Tap to reveal answer
$y = \frac{3}{5}(x + 2) - 1$. From $x = 5t - 2$ get $t = \frac{x+2}{5}$, substitute.
$y = \frac{3}{5}(x + 2) - 1$. From $x = 5t - 2$ get $t = \frac{x+2}{5}$, substitute.
← Didn't Know|Knew It →
What is the parametric form for a parabola $y = ax^2$?
What is the parametric form for a parabola $y = ax^2$?
Tap to reveal answer
$x = t$, $y = at^2$. General parabola form with coefficient $a$.
$x = t$, $y = at^2$. General parabola form with coefficient $a$.
← Didn't Know|Knew It →
What are the parametric equations for a line parallel to $y = 2x + 3$?
What are the parametric equations for a line parallel to $y = 2x + 3$?
Tap to reveal answer
$x = t$, $y = 2t + c$. Same slope but different $y$-intercept constant.
$x = t$, $y = 2t + c$. Same slope but different $y$-intercept constant.
← Didn't Know|Knew It →
Convert $x = 4t$, $y = 9 - t^2$ to a Cartesian equation.
Convert $x = 4t$, $y = 9 - t^2$ to a Cartesian equation.
Tap to reveal answer
$y = 9 - \left(\frac{x}{4}\right)^2$. Substitute $t = \frac{x}{4}$ into the $y$ equation.
$y = 9 - \left(\frac{x}{4}\right)^2$. Substitute $t = \frac{x}{4}$ into the $y$ equation.
← Didn't Know|Knew It →
Convert $x = 3 \sin(t)$, $y = 4 \cos(t)$ to a Cartesian equation.
Convert $x = 3 \sin(t)$, $y = 4 \cos(t)$ to a Cartesian equation.
Tap to reveal answer
$\frac{x^2}{9} + \frac{y^2}{16} = 1$. Ellipse using sine for $x$ and cosine for $y$.
$\frac{x^2}{9} + \frac{y^2}{16} = 1$. Ellipse using sine for $x$ and cosine for $y$.
← Didn't Know|Knew It →
What is the parametric form for a circle with radius $r$?
What is the parametric form for a circle with radius $r$?
Tap to reveal answer
$x = r \cos(t)$, $y = r \sin(t)$. General circle equation with specified radius.
$x = r \cos(t)$, $y = r \sin(t)$. General circle equation with specified radius.
← Didn't Know|Knew It →
Convert $x = 2t$, $y = t^2$ to a Cartesian equation.
Convert $x = 2t$, $y = t^2$ to a Cartesian equation.
Tap to reveal answer
$y = \frac{x^2}{4}$. Eliminate parameter: $t = \frac{x}{2}$, so $y = \left(\frac{x}{2}\right)^2$.
$y = \frac{x^2}{4}$. Eliminate parameter: $t = \frac{x}{2}$, so $y = \left(\frac{x}{2}\right)^2$.
← Didn't Know|Knew It →
What is the parametric equation for a vertical line $x = c$?
What is the parametric equation for a vertical line $x = c$?
Tap to reveal answer
$x = c$, $y = t$. Parameter varies while $x$ remains constant.
$x = c$, $y = t$. Parameter varies while $x$ remains constant.
← Didn't Know|Knew It →
Determine the Cartesian equation from $x = 4 \cos(t)$, $y = 5 \sin(t)$.
Determine the Cartesian equation from $x = 4 \cos(t)$, $y = 5 \sin(t)$.
Tap to reveal answer
$\frac{x^2}{16} + \frac{y^2}{25} = 1$. Ellipse with semi-axes 4 and 5.
$\frac{x^2}{16} + \frac{y^2}{25} = 1$. Ellipse with semi-axes 4 and 5.
← Didn't Know|Knew It →
What is the parametric equation for a line with slope $m$?
What is the parametric equation for a line with slope $m$?
Tap to reveal answer
$x = t$, $y = mt + c$. Standard form with slope $m$ and parameter $t$.
$x = t$, $y = mt + c$. Standard form with slope $m$ and parameter $t$.
← Didn't Know|Knew It →
Identify parametric equations for a horizontal line $y = c$.
Identify parametric equations for a horizontal line $y = c$.
Tap to reveal answer
$x = t$, $y = c$. Parameter varies while $y$ remains constant.
$x = t$, $y = c$. Parameter varies while $y$ remains constant.
← Didn't Know|Knew It →
What do $a$ and $b$ represent in the ellipse parametric equations $x = a \cos(t)$, $y = b \sin(t)$?
What do $a$ and $b$ represent in the ellipse parametric equations $x = a \cos(t)$, $y = b \sin(t)$?
Tap to reveal answer
The semi-major and semi-minor axes. The lengths of the ellipse's major and minor axes.
The semi-major and semi-minor axes. The lengths of the ellipse's major and minor axes.
← Didn't Know|Knew It →
Convert $x = 2 \cos(t)$, $y = 2 \sin(t)$ to a Cartesian equation.
Convert $x = 2 \cos(t)$, $y = 2 \sin(t)$ to a Cartesian equation.
Tap to reveal answer
$x^2 + y^2 = 4$. Circle with radius 2 centered at origin.
$x^2 + y^2 = 4$. Circle with radius 2 centered at origin.
← Didn't Know|Knew It →
What is the purpose of parametric equations in mathematics?
What is the purpose of parametric equations in mathematics?
Tap to reveal answer
To describe geometric figures and motions. Enables modeling of complex paths and trajectories.
To describe geometric figures and motions. Enables modeling of complex paths and trajectories.
← Didn't Know|Knew It →
Identify parametric equations for a parabola $y = x^2$.
Identify parametric equations for a parabola $y = x^2$.
Tap to reveal answer
$x = t$, $y = t^2$. Simplest parametrization using $t$ as $x$-coordinate.
$x = t$, $y = t^2$. Simplest parametrization using $t$ as $x$-coordinate.
← Didn't Know|Knew It →
What is the parametric equation for a straight line through $(x_1, y_1)$, $(x_2, y_2)$?
What is the parametric equation for a straight line through $(x_1, y_1)$, $(x_2, y_2)$?
Tap to reveal answer
$x = x_1 + (x_2 - x_1)t$, $y = y_1 + (y_2 - y_1)t$. Linear interpolation between two given points.
$x = x_1 + (x_2 - x_1)t$, $y = y_1 + (y_2 - y_1)t$. Linear interpolation between two given points.
← Didn't Know|Knew It →
Find the Cartesian equation from $x = 1 + t$, $y = 2t - 1$.
Find the Cartesian equation from $x = 1 + t$, $y = 2t - 1$.
Tap to reveal answer
$y = 2x - 3$. From $x = 1 + t$, get $t = x - 1$, substitute.
$y = 2x - 3$. From $x = 1 + t$, get $t = x - 1$, substitute.
← Didn't Know|Knew It →
What is the general form of parametric equations for a line?
What is the general form of parametric equations for a line?
Tap to reveal answer
$x = x_0 + at$, $y = y_0 + bt$. Point $(x_0, y_0)$ with direction vector $(a,b)$.
$x = x_0 + at$, $y = y_0 + bt$. Point $(x_0, y_0)$ with direction vector $(a,b)$.
← Didn't Know|Knew It →
Define what a parameter is in terms of parametric equations.
Define what a parameter is in terms of parametric equations.
Tap to reveal answer
An independent variable that defines a set of equations. The controlling variable in parametric representation.
An independent variable that defines a set of equations. The controlling variable in parametric representation.
← Didn't Know|Knew It →
Convert $x = \cos(t)$, $y = \sin(t)$ to a Cartesian equation.
Convert $x = \cos(t)$, $y = \sin(t)$ to a Cartesian equation.
Tap to reveal answer
$x^2 + y^2 = 1$. Unit circle using fundamental trigonometric identity.
$x^2 + y^2 = 1$. Unit circle using fundamental trigonometric identity.
← Didn't Know|Knew It →
Which parametric equations describe the line segment from $(1,2)$ to $(4,8)$?
Which parametric equations describe the line segment from $(1,2)$ to $(4,8)$?
Tap to reveal answer
$x = 1 + 3t$, $y = 2 + 6t$, $0 \leq t \leq 1$. Direction vector $(3,6)$ with parameter range $[0,1]$.
$x = 1 + 3t$, $y = 2 + 6t$, $0 \leq t \leq 1$. Direction vector $(3,6)$ with parameter range $[0,1]$.
← Didn't Know|Knew It →