# Parametric Equations

A rectangular equation, or an equation in rectangular form is an equation composed of variables like $x$ and $y$ which can be graphed on a regular Cartesian plane. For example $y=4x+3$ is a rectangular equation.

A curve in the plane is said to be parameterized if the set of coordinates on the curve, $\left(x,y\right)$ , are represented as functions of a variable $t$ .

$\begin{array}{l}x=f\left(t\right)\\ y=g\left(t\right)\end{array}$

These equations may or may not be graphed on Cartesian plane.

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Example 1:
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Find a set of parametric equations for the equation $y={x}^{2}+5$ .

Solution:

Assign any one of the variable equal to $t$ . (say $x$ = $t$ ).

Then, the given equation can be rewritten as $y={t}^{2}+5$ .

Therefore, a set of parametric equations is $x$ = $t$ and $y={t}^{2}+5$ .

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Example 2:
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Eliminate the parameter and find the corresponding rectangular equation.

$\begin{array}{l}x=t+5\\ y={t}^{2}\end{array}$

Solution:

Rewrite the equation $x=t+5$ as $t$ in terms of $x$ .

$\begin{array}{l}t+5=x\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t=x-5\end{array}$

Now, replace $t$ by ( $x-5$ ) in the equation $y={t}^{2}$ .

$y={\left(x-5\right)}^{2}$

Therefore, the corresponding rectangular equation is $y={\left(x-5\right)}^{2}$ .

There is another type of equations called polar equations which need to be graphed on a polar plane .