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.

$x=3t^2,y=t+1;-\infty <t<\infty .$

Short answer

The answer is$x=3{\left(y-1\right)}^2.$

Step-by-step solution

Step 1. Given information.

$x=3t^2,y=t+1$is given.

Step 2. Calculation.

Graph the curve defined by the parametric equations -

$t$	$x$	$y$	$(x,y)$
$0$	$0$	$1$	$(0,1)$
$1$	$3$	$2$	$(3,2)$
$2$	$12$	$3$	$(12,3)$
$3$	$27$	$4$	$(27,4)$

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

$$\begin{gathered} y=t+1 \\ t=y-1 \end{gathered}$$

Put the value of t-

$$\begin{gathered} x=3t^2 \\ x=3{(y-1)}^2. \end{gathered}$$

Thus the equation is$x=3{\left(y-1\right)}^2.$