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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 $y = 4 x + 3$ is a rectangular equation.

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

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

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

Example 1:

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 $y = t^{2} + 5$ .

Example 2:

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 $x$ .

$\begin{array}{l} t + 5 = x \\ t = x - 5 \end{array}$

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

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

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

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