Chapter 8.4, Problem 7E

(a)

To determine

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

Explanation of Solution

Given information: Parametric equations $x=\sqrt{t},y=1-t$

Calculation:

Parametric equations are given as

  $x=\sqrt{t},y=1-t$

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

When $t=0$

  $\begin{array}{l}x=\sqrt{0}=0\\ y=1-0=1\end{array}$

When $t=1$

  $\begin{array}{l}x=\sqrt{1}=1\\ y=1-1=0\end{array}$

When $t=4$

  $\begin{array}{l}x=\sqrt{4}=2\\ y=1-4=-3\end{array}$

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 7E

The rectangular coordinate equation for the curve is $y=1-x^2$

Explanation of Solution

Given information: Parametric equations $x=\sqrt{t},y=1-t$

Calculation:

Parametric equations are given as

  $x=\sqrt{t},y=1-t$

From equation $y=1-t$

  $t=1-y$

Substituting $t=1-y$ in $x=\sqrt{t}$ ,

  $\begin{array}{l}x=\sqrt{1-y}\\ x^2=1-y\\ y=1-x^2\end{array}$

Conclusion:

Hence, rectangular coordinate equation for the curve is $y=1-x^2$