# Find a set of parametric equations for the graphs of rectangular equation y = 2x^2 that satisfies...

## Question:

Find a set of parametric equations for the graphs of rectangular equation {eq}y = 2x^2 {/eq} that satisfies the condition {eq}t = 3 {/eq} at the point {eq}(3,18) {/eq}.

## Parametric Equations:

Parametric equations are useful in some calculus concepts such as line integrals.

The explicit function {eq}y=f(x) {/eq} is transformed into a set of parametric equations by substituting {eq}t {/eq} for {eq}x {/eq} so as to get {eq}x=t {/eq} and {eq}y=f(t) {/eq}.

## Answer and Explanation:

Note that by letting {eq}x=t {/eq} we get {eq}t=3 {/eq} at {eq}x=3 {/eq} so the indicated condition is satisfied.

Substituting {eq}t {/eq} for {eq}x {/eq} gives us the parametric equation for {eq}y {/eq}:

{eq}\begin{align*} \displaystyle y& = 2x^2\\ y& = 2t^2\\ \end{align*} {/eq}

This also satisfies the condition as {eq}t=3 {/eq} when {eq}y = 18 {/eq}: {eq}y(3) = 2(3)^2 \implies y(3) = 18 {/eq}.

Thus, the set of parametric equations we're looking for is {eq}x=t {/eq} and {eq}y = 2t^2 {/eq}.