Parametric Functions Modeling Planar Motion - AP Precalculus
Card 1 of 30
Find $y$ at $t = 2$ for $y = 3t - 5$.
Find $y$ at $t = 2$ for $y = 3t - 5$.
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$y = 1$. Substitute $t = 2$: $y = 3(2) - 5 = 1$.
$y = 1$. Substitute $t = 2$: $y = 3(2) - 5 = 1$.
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Convert $x = t + 1$, $y = t^2$ to a Cartesian equation.
Convert $x = t + 1$, $y = t^2$ to a Cartesian equation.
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$y = (x-1)^2$. From $x = t + 1$, get $t = x - 1$, substitute into $y = t^2$.
$y = (x-1)^2$. From $x = t + 1$, get $t = x - 1$, substitute into $y = t^2$.
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State the parametric form of a line through $(x_0, y_0)$ with slope $m$.
State the parametric form of a line through $(x_0, y_0)$ with slope $m$.
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$x = x_0 + t$, $y = y_0 + mt$. General form where $t$ acts as the parameter for direction.
$x = x_0 + t$, $y = y_0 + mt$. General form where $t$ acts as the parameter for direction.
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What is the path of $x = 2 \cos(t)$, $y = 3 \sin(t)$?
What is the path of $x = 2 \cos(t)$, $y = 3 \sin(t)$?
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An ellipse. Standard form of an ellipse with semi-axes 2 and 3.
An ellipse. Standard form of an ellipse with semi-axes 2 and 3.
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Find the point at $t = 1$ for $x = 3t$, $y = t^2 + 1$.
Find the point at $t = 1$ for $x = 3t$, $y = t^2 + 1$.
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$(3, 2)$. Substitute $t = 1$: $x = 3$, $y = 1 + 1 = 2$.
$(3, 2)$. Substitute $t = 1$: $x = 3$, $y = 1 + 1 = 2$.
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Convert $x = t^2$, $y = 2t$ to a Cartesian equation.
Convert $x = t^2$, $y = 2t$ to a Cartesian equation.
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$y^2 = 4x$. From $x = t^2$, get $t = \pm\sqrt{x}$, substitute into $y = 2t$.
$y^2 = 4x$. From $x = t^2$, get $t = \pm\sqrt{x}$, substitute into $y = 2t$.
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Find the slope of the line for $x = 2t + 1$, $y = 3t - 4$.
Find the slope of the line for $x = 2t + 1$, $y = 3t - 4$.
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Slope is $\frac{dy}{dx} = \frac{3}{2}$. Slope equals $\frac{dy/dt}{dx/dt} = \frac{3}{2}$.
Slope is $\frac{dy}{dx} = \frac{3}{2}$. Slope equals $\frac{dy/dt}{dx/dt} = \frac{3}{2}$.
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What type of curve is $x = a \cos(t)$, $y = a \sin(t)$?
What type of curve is $x = a \cos(t)$, $y = a \sin(t)$?
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A circle. A circle centered at origin with radius $a$.
A circle. A circle centered at origin with radius $a$.
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Convert $x = t + 1$, $y = t^2$ to a Cartesian equation.
Convert $x = t + 1$, $y = t^2$ to a Cartesian equation.
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$y = (x-1)^2$. From $x = t + 1$, get $t = x - 1$, substitute into $y = t^2$.
$y = (x-1)^2$. From $x = t + 1$, get $t = x - 1$, substitute into $y = t^2$.
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Find $x$ at $t = 2$ for $x = 4t + 3$.
Find $x$ at $t = 2$ for $x = 4t + 3$.
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$x = 11$. Substitute $t = 2$: $x = 4(2) + 3 = 11$.
$x = 11$. Substitute $t = 2$: $x = 4(2) + 3 = 11$.
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What is the range of $y = 2 \sin(t)$ for $0 \leq t \leq 2\pi$?
What is the range of $y = 2 \sin(t)$ for $0 \leq t \leq 2\pi$?
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$-2 \leq y \leq 2$. Sine function oscillates between -1 and 1, scaled by factor 2.
$-2 \leq y \leq 2$. Sine function oscillates between -1 and 1, scaled by factor 2.
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Find $y$ at $t = 3$ for $y = 2t - 1$.
Find $y$ at $t = 3$ for $y = 2t - 1$.
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$y = 5$. Substitute $t = 3$: $y = 2(3) - 1 = 5$.
$y = 5$. Substitute $t = 3$: $y = 2(3) - 1 = 5$.
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State parametric equations for the horizontal line $y = c$.
State parametric equations for the horizontal line $y = c$.
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$x = t$, $y = c$. Let $t$ vary while keeping $y$ constant at $c$.
$x = t$, $y = c$. Let $t$ vary while keeping $y$ constant at $c$.
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Find the initial point of $x = 2t + 1$, $y = 3t$ at $t = 0$.
Find the initial point of $x = 2t + 1$, $y = 3t$ at $t = 0$.
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$(1, 0)$. Substitute $t = 0$: $x = 1$, $y = 0$.
$(1, 0)$. Substitute $t = 0$: $x = 1$, $y = 0$.
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What is the meaning of $t$ in parametric equations?
What is the meaning of $t$ in parametric equations?
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A parameter, often representing time. Usually represents time or another independent variable.
A parameter, often representing time. Usually represents time or another independent variable.
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Convert $x = t$, $y = 2t + 3$ to Cartesian form.
Convert $x = t$, $y = 2t + 3$ to Cartesian form.
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$y = 2x + 3$. Since $x = t$, substitute directly into $y = 2t + 3$.
$y = 2x + 3$. Since $x = t$, substitute directly into $y = 2t + 3$.
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State the parametric equations for a circle with radius $r$.
State the parametric equations for a circle with radius $r$.
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$x = r , \cos(t)$, $y = r , \sin(t)$. Standard form using cosine for $x$ and sine for $y$ components.
$x = r , \cos(t)$, $y = r , \sin(t)$. Standard form using cosine for $x$ and sine for $y$ components.
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Identify the parameter in the equations $x = 3t$, $y = 2t + 1$.
Identify the parameter in the equations $x = 3t$, $y = 2t + 1$.
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The parameter is $t$. The independent variable that both $x$ and $y$ depend on.
The parameter is $t$. The independent variable that both $x$ and $y$ depend on.
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What is the path of $x = t$, $y = t^3$?
What is the path of $x = t$, $y = t^3$?
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A cubic curve. Third-degree polynomial relationship.
A cubic curve. Third-degree polynomial relationship.
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Find $x$ at $t = 0$ for $x = 7t - 2$.
Find $x$ at $t = 0$ for $x = 7t - 2$.
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$x = -2$. Substitute $t = 0$: $x = 7(0) - 2 = -2$.
$x = -2$. Substitute $t = 0$: $x = 7(0) - 2 = -2$.
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Convert $x = 6t$, $y = 2t + 3$ to a Cartesian equation.
Convert $x = 6t$, $y = 2t + 3$ to a Cartesian equation.
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$y = \frac{1}{3}x + 3$. From $x = 6t$, get $t = \frac{x}{6}$, substitute into $y$.
$y = \frac{1}{3}x + 3$. From $x = 6t$, get $t = \frac{x}{6}$, substitute into $y$.
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Convert $x = 2t + 3$, $y = 4t - 1$ to Cartesian form.
Convert $x = 2t + 3$, $y = 4t - 1$ to Cartesian form.
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$y = 2x - 7$. From $x = 2t + 3$, get $t = \frac{x-3}{2}$, substitute into $y$.
$y = 2x - 7$. From $x = 2t + 3$, get $t = \frac{x-3}{2}$, substitute into $y$.
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State the parametric form for a line parallel to $x$-axis.
State the parametric form for a line parallel to $x$-axis.
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$x = t$, $y = c$. Horizontal line where $y$ remains constant.
$x = t$, $y = c$. Horizontal line where $y$ remains constant.
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Find the range of $y = \sin(t)$ for $0 \leq t \leq 2\pi$.
Find the range of $y = \sin(t)$ for $0 \leq t \leq 2\pi$.
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$-1 \leq y \leq 1$. Standard range of the sine function.
$-1 \leq y \leq 1$. Standard range of the sine function.
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What motion does $x = 3 \cos(t)$, $y = 3 \sin(t)$ represent?
What motion does $x = 3 \cos(t)$, $y = 3 \sin(t)$ represent?
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Circular motion. Parametric equations for a circle with radius 3.
Circular motion. Parametric equations for a circle with radius 3.
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Convert $x = t^2 - 1$, $y = 2t$ to a Cartesian equation.
Convert $x = t^2 - 1$, $y = 2t$ to a Cartesian equation.
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$y^2 = 4(x + 1)$. From $y = 2t$, get $t = \frac{y}{2}$, substitute into $x$.
$y^2 = 4(x + 1)$. From $y = 2t$, get $t = \frac{y}{2}$, substitute into $x$.
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What is the result of $x = 2t$, $y = 3t$?
What is the result of $x = 2t$, $y = 3t$?
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A line through the origin. Linear relationship with slope $\frac{3}{2}$ passing through origin.
A line through the origin. Linear relationship with slope $\frac{3}{2}$ passing through origin.
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Convert $x = 4t$, $y = 5t + 1$ to a Cartesian equation.
Convert $x = 4t$, $y = 5t + 1$ to a Cartesian equation.
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$y = \frac{5}{4}x + 1$. From $x = 4t$, get $t = \frac{x}{4}$, substitute into $y$.
$y = \frac{5}{4}x + 1$. From $x = 4t$, get $t = \frac{x}{4}$, substitute into $y$.
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State the parametric equations for a line with slope $m$.
State the parametric equations for a line with slope $m$.
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$x = x_0 + at$, $y = y_0 + mt$. General form with direction vector $(a, m)$ and slope $\frac{m}{a}$.
$x = x_0 + at$, $y = y_0 + mt$. General form with direction vector $(a, m)$ and slope $\frac{m}{a}$.
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Find the coordinates at $t = 0$ for $x = t^2$, $y = 2t$.
Find the coordinates at $t = 0$ for $x = t^2$, $y = 2t$.
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$(0, 0)$. Both coordinates are zero when $t = 0$.
$(0, 0)$. Both coordinates are zero when $t = 0$.
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