Chapter 1.4, Problem 35E

To determine

Observe the graph of the equation as $c$ varies.

Answer to Problem 35E

The size of the loop will increase.

Explanation of Solution

Given information:

What happens to the graph of the equation $y^2=c{x}^3+x^2$ as $c$ varies?

Calculation:

Consider the equation,

  $y^2=c{x}^3+x^2$ , where $c$ is a real number

The sketch of this equation for $c=-5,-2,2,5$ is shown below:

  

From the figure, we observe that the loop of the graph is to the right of the origin when $c<0$ .

And the loop of the graph is to the left of the origin when $c>0$ .

Hence, as the value of $c$ is near to $0$ on the real line, the size of the loop will increase.