Chapter 1.7, Problem 43E

To determine

To find: The family of the curve determined by the parametric given parametric equations. Also, determine the change in shape as the $c$ increases and then graph the several members of the family.

Answer to Problem 43E

The graph is in cusp form for $c<0$ and when the value of $c$ decreases the size of the cusp increase. The graph for $c>0$ is in the shape of loop and the size of the loop increases as the value of $c$ decreases. The graph for the curves is shown in Figure 1 and Figure 2

Explanation of Solution

Given:

The given parametric equations is,

  $\begin{array}{l}x=t^2\\ y=t^3-ct\end{array}$

Calculation:

Consider the given parametric equations is,

  $\begin{array}{l}x=t^2\\ y=t^3-ct\end{array}$

Then,

  $t=\pm \sqrt{x}$

Then,

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

The sketch for the curve for $c<0$ is shown in Figure 1

  

Figurer 1

The graph is in cusp form and when the value of $c$ decreases the size of the cusp increase.

The sketch for the curve $c>0$ is shown in Figure 2

  

Figure 2

The graph for $c>0$ is in the shape of loop and the size of the loop increases as the value of $c$ decreases.