Write the following parametric equation in the form y=f(x) . \begin{alignat}{3} x = t^2...

Question:

Write the following parametric equation in the form {eq}y=f(x) {/eq}.

{eq}\begin{alignat}{3} x &=&& t^2 +2, \\ y&=&& t^2-4. \end{alignat} {/eq}.

Parametric Equations:

Parametric equations area set of equations contains an independent variable. We are given two parametric equations and {eq}x= p(t) , y= q(t) {/eq} we need to write the Cartesian equation {eq}y=f(x) {/eq} of the given curves by eliminating the parameter {eq}t. {/eq}

We can equate the parameter and use algebraic operations to solve this kind of problems.

Answer and Explanation:

We are given:

{eq}\begin{alignat}{3} x &=&& t^2 +2, \\ y&=&& t^2-4. \end{alignat} {/eq}


Isolate t from the first equation {eq}x =t+2\Rightarrow t^2 =x-2 \Rightarrow t =\sqrt{ x-2 } {/eq}

Isolate t from the second equation: {eq}y=t^2-4\Rightarrow t^2 =y +4 \Rightarrow t =\sqrt{ y+4 } \\ {/eq}


Now equating both values of {eq}t: {/eq}

{eq}\Rightarrow \sqrt{ y+4 } = x-2 {/eq}

{eq}\Rightarrow y+4= (x-2)^2 {/eq}

{eq}\Rightarrow y= (x-2)^2-4 {/eq}


Hence the Cartesian equation of the curve is {eq}{\boxed{ y= (x-2)^2-4.}} {/eq}


Learn more about this topic:

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Evaluating Parametric Equations: Process & Examples

from Precalculus: High School

Chapter 24 / Lesson 3
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