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.

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

Short answer

The final answer is$y-x+8=0$.

Step-by-step solution

Step 1. Given information.

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

Step 2. Calculation.

Graph the curve defined by the parametric equations -

$t$	$x$	y	$(x,y)$
-1	$5$	$-3$	$(5,-3)$
-2	$8$	$0$	$(8,0)$
1	$5$	$-3$	$(5,-3)$
2	$8$	$0$	$(8,0)$

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

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

Put the value of t-

role="math" localid="1648376651280" $$\begin{gathered} y=t^2-4 \\ y=x-4-4 \\ y=x-8 \\ y-x+8=0 \end{gathered}$$

Thus, the equation is$y-x+8=0$.