Chapter 10.6, Problem 42E

To determine

Substitution of $x=t^2$ is not appropriate when trying to find a parametric representation of $y=x^2-5$

Explanation of Solution

Here the rectangular equation is,

  $y=x^2-5$

Now we convert the equation into parametric form,

  $\begin{array}{l}Let,\\ x=t\\ then,\\ y=t^2-5\end{array}$

Thus we get the parametric equation,

  $\begin{array}{l}x=t\\ \&\\ y=t^2-5\end{array}$

Now if we take,

  $x=t^2$

Then the parametric equation becomes,

  $\begin{array}{l}$ y={(t^2)}^2-5$y=t^4-5\end{array}$

Thus we can see that the parametric equation has become an equation of degree four. So it is not an appropriate while substituting into parametric form.