Chapter 8.4, Problem 5E

(a)

To determine

To calculate: To sketch the curve represented by the parametric equations

Explanation of Solution

Given information: Parametric equations $x=t^2,y=t-2,2\le t\le 4$

Calculation:

Parametric equations are given as

  $x=t^2,y=t-2,2\le t\le 4$

Find the value of $x$ and $y$ for values of $t$

When $t=2$

  $\begin{array}{l}x={\left(2\right)}^2=4\\ y=2-2=0\end{array}$

When $t=3$

  $\begin{array}{l}x={\left(3\right)}^2=9\\ y=3-2=1\end{array}$

When $t=4$

  $\begin{array}{l}x={\left(4\right)}^2=16\\ y=4-2=2\end{array}$

Plotting the point on graph,

  

Conclusion:

Hence, Curve is sketched

(b)

To determine

To calculate: To find the rectangular coordinate equation for the curve

Answer to Problem 5E

The rectangular coordinate equation for the curve is $x={\left(y+2\right)}^2$

Explanation of Solution

Given information: Parametric equations $x=t^2,y=t-2,2\le t\le 4$

Calculation:

Parametric equations are given as

  $x=t^2,y=t-2,2\le t\le 4$

From equation $y=t-2$

  $t=y+2$

Substituting $t=y+2$ in $x=t^2$ ,

  $x={\left(y+2\right)}^2$

Conclusion:

Hence, rectangular coordinate equation for the curve is $x={\left(y+2\right)}^2$