Question

In problem graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility.

$x=2t-4,y=4t^2;-\infty <t<\infty .$

Short answer

The answer is$y=4\left(\frac{x+4}{2}\right).$

Step-by-step solution

Step 1. Given information.

$x=2t-4,y=4t^2$is given.

Step 2. Calculation.

Graph the curve defined by the parametric equations -

$t$	$x$	$y$	$(x,y)$
$-1$	$-6$	$4$	$(-6,4)$
$-2$	$-8$	$16$	$(-8,16)$
$-3$	$-10$	$36$	$(-10,36)$
$-4$	$-12$	$64$	$(-12,64)$

We find the rectangular equation by eliminating the parameter t from the parametric equations.

$$\begin{gathered} x=2t-4 \\ 2t=x+4 \\ t=\frac{x+4}{2} \end{gathered}$$

Put the value of t-

$$\begin{gathered} y=4t^2 \\ y=4{\left(\frac{x+4}{2}\right)}^2 \end{gathered}$$

Thus, the equation is$y=4\left(\frac{x+4}{2}\right).$