Chapter 10.7, Problem 11AYU

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.

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

To determine

To find:

a. The rectangular equation of the curve.

Answer to Problem 11AYU

$y=x-8,-\infty <\sqrt{x-4}<\infty$

Explanation of Solution

Given:

$x=\surd 2t,y=4t;t\ge 0$

Calculation:

$x=t²+4,y=t²-4;-\infty <t<\infty$

$x=y+4-4,-\infty <t<\infty$

$y=x-8,-\infty <\sqrt{x-4}<\infty$

To determine

To graph:

b. $x=t²+4,y=t²-4;-\infty <t<\infty$, verify with the graph of rectangular equation of the curve.

Answer to Problem 11AYU

Explanation of Solution

Given:

$x=t²+4,y=t²-4;-\infty <t<\infty$

Graph:

The graph of $x=t²+4,y=t²-4;-\infty <t<\infty$ is plotted.

The graph of $y=x-8,-\infty <\sqrt{x-4}<\infty$ is plotted.

Graph 1 represents the graph of $x=t²+4,y=t²-4;-\infty <t<\infty$ plotted in parametric mode and the orientation is shown. . For $\text{t }<0$ the movement is towards the left and for $\text{t }>0$ the movement is towards right.

The rectangular equation of the curve is calculated. Graph 2 represents the curve of the rectangular equation $y=x-8,-\infty <\sqrt{x-4}<\infty$.

Both the equation represents the same graph and orientation. Hence it is verified. For $\surd \left(x-4\right)<0$ the movement is towards the left and for $\surd \left(x-4\right)>0$ the movement is towards right.