10.6 Parametric Equations

Parameterize the curve given by $x = y^{3} - 2 y .$

Given $x \left(t\right) = t^{2} + 1$ and $y \left(t\right) = 2 + t ,$ eliminate the parameter, and write the parametric equations as a Cartesian equation.

Given the equations below, eliminate the parameter and write as a rectangular equation for $y$ as a function of $x .$

Eliminate the parameter and write as a Cartesian equation: $x \left(t\right) = e^{- t}$ and $y \left(t\right) = 3 e^{t}$

Eliminate the parameter and write as a Cartesian equation: $x \left(t\right) = \sqrt{t} + 2$ and $y \left(t\right) = \log \left(t\right) .$

Eliminate the parameter and write as a rectangular equation.

Eliminate the parameter from the given pair of parametric equations and write as a Cartesian equation: $x \left(t\right) = 2 \cos t$ and $y \left(t\right) = 3 \sin t .$

Use two different methods to find the Cartesian equation equivalent to the given set of parametric equations.

Write the given parametric equations as a Cartesian equation: $x \left(t\right) = t^{3}$ and $y \left(t\right) = t^{6} .$

What is a system of parametric equations?

Explain how to eliminate a parameter given a set of parametric equations.

What is a benefit of using parametric equations?

Eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

$\left\{\right. \begin{matrix}x \left(t\right) = 5 - t \\ y \left(t\right) = 8 - 2 t\end{matrix}$

Eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

$\left\{\right. \begin{matrix}x \left(t\right) = 2 t + 1 \\ y \left(t\right) = 3 \sqrt{t}\end{matrix}$

Eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

$\left\{\right. \begin{matrix}x \left(t\right) = 2 e^{t} \\ y \left(t\right) = 1 - 5 t\end{matrix}$

Eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

$\begin{cases} x \left(t\right) = 4 \text{log} \left(t\right) \\ y \left(t\right) = 3 + 2 t \end{cases}$

Eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

$\left\{\right. \begin{matrix}x \left(t\right) = t^{3} - t \\ y \left(t\right) = 2 t\end{matrix}$

Eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

$\left\{\right. \begin{matrix}x \left(t\right) = e^{2 t} \\ y \left(t\right) = e^{6 t}\end{matrix}$

Eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

$\begin{cases} x \left(t\right) = 4 \text{cos} t \\ y \left(t\right) = 5 \sin t \end{cases}$

Eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

$\begin{cases} x \left(t\right) = 2 \text{cos}^{2} t \\ y \left(t\right) = - \sin t \end{cases}$

Eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

$\begin{cases} x \left(t\right) = t - 1 \\ y \left(t\right) = t^{2} \end{cases}$

Eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

$\begin{cases} x \left(t\right) = 2 t - 1 \\ y \left(t\right) = t^{3} - 2 \end{cases}$

Rewrite the parametric equation as a Cartesian equation by building an $x - y$ table.

$\begin{cases} x \left(t\right) = 4 - t \\ y \left(t\right) = 3 t + 2 \end{cases}$

Rewrite the parametric equation as a Cartesian equation by building an $x - y$ table.

$\begin{cases} x \left(t\right) = 4 t - 1 \\ y \left(t\right) = 4 t + 2 \end{cases}$

Parameterize (write parametric equations for) each Cartesian equation by setting $x \left(t\right) = t$ or by setting $y \left(t\right) = t .$

$y \left(x\right) = 2 \sin x + 1$

Parameterize (write parametric equations for) each Cartesian equation by setting $x \left(t\right) = t$ or by setting $y \left(t\right) = t .$

$x \left(y\right) = \sqrt{y} + 2 y$

Parameterize (write parametric equations for) each Cartesian equation by using $x \left(t\right) = a \cos t$ and $y \left(t\right) = b \sin t .$ Identify the curve.

$\frac{x^{2}}{16} + \frac{y^{2}}{36} = 1$

Parameterize (write parametric equations for) each Cartesian equation by using $x \left(t\right) = a \cos t$ and $y \left(t\right) = b \sin t .$ Identify the curve.

$x^{2} + y^{2} = 10$

Parameterize (write parametric equations for) each Cartesian equation by using $x \left(t\right) = a \cos t$ and $y \left(t\right) = b \sin t .$ Identify the curve.

Parameterize the line from $\left(−1 , 0\right)$ to $\left(3 , −2\right)$ so that the line is at $\left(−1 , 0\right)$ at $t = 0 ,$ and at $\left(3 , −2\right)$ at $t = 1.$

Parameterize (write parametric equations for) each Cartesian equation by using $x \left(t\right) = a \cos t$ and $y \left(t\right) = b \sin t .$ Identify the curve.

Parameterize the line from $\left(4 , 1\right)$ to $\left(6 , −2\right)$ so that the line is at $\left(4 , 1\right)$ at $t = 0 ,$ and at $\left(6 , −2\right)$ at $t = 1.$

Use the table feature in the graphing calculator to determine whether the graphs intersect.

$\left\{\right. \begin{matrix}x_{1} \left(t\right) = t^{2} \\ y_{1} \left(t\right) = 2 t - 1\end{matrix} \textrm{ }\text{and}\textrm{ } \left\{\right. \begin{matrix}x_{2} \left(t\right) = - t + 6 \\ y_{2} \left(t\right) = t + 1\end{matrix}$

Use a graphing calculator to complete the table of values for each set of parametric equations.

$\left\{\right. \begin{matrix}x_{1} \left(t\right) = t^{2} - 4 \\ y_{1} \left(t\right) = 2 t^{2} - 1\end{matrix}$

Find two different sets of parametric equations for $y = \left(x + 1\right)^{2} .$

Find two different sets of parametric equations for $y = x^{2} - 4 x + 4.$