Chapter 10.7, Problem 13AYU

In Problems 7-26, 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.

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

To determine

To find:

a. The rectangular equation of the curve.

Answer to Problem 13AYU

$x=y+8,t\ge 0$

Explanation of Solution

Given:

$x=3t², y=t+1;-\infty < t<\infty$

Calculation:

$x=3t², y=t+1;-\infty < t<\infty$

$x=3\left(y-1\right)²$

$x=3y²-6y+3, -\infty <y-1<\infty$

To determine

To graph:

b. $x=3t², y=t+1;-\infty < t<\infty$, verify with the graph of rectangular equation of the curve.

Answer to Problem 13AYU

Explanation of Solution

Given:

$x=3t², y=t+1;-\infty <t<\infty$

Graph:

The graph of $x=3t², y=t+1;-\infty <t<\infty$ is plotted.

The graph of $x=3y²-6y+3, -\infty <y-1<\infty$ is plotted.

Graph 1 represents the graph of $x=3t², y=t+1;-\infty <t<\infty$, plotted in parametric mode and the orientation is shown.

The rectangular equation of the curve is calculated. Graph 2 represents the curve of the rectangular equation $x=3y²-6y+3, -\infty <y-1<\infty$.

Both the equation represents the same graph and orientation. Hence it is verified.