Chapter 1, Problem 12RE

To determine

To list: the intercepts and to test for symmetry

Answer to Problem 12RE

The $x$ intercept are 1 and -1 and no $y$ -intercepts.

The graph of the equation is symmetric with respect to the $x$ -axis, $y$ -axis and the origin.

Explanation of Solution

Given:

  $9x^2-y^2=9$

Calculation:

In order to find the $x$ -intercept, substitute 0 for $y$ in the given equation and evaluate.

  $\begin{array}{r}\hfill 9x^2-0^2=9\\ \hfill 9x^2=9\end{array}$

Solve for $x$

  $x^2=1$

  $x=\pm 1$

The $x$ intercept are 1 and -1

Now, substitute 0 for $x$ and evaluate to find the $y$ -intercept.

  $9{(0)}^2-y^2=9$

  $-y^2=9$

Solve for $y$

  $y=\sqrt{-9}$

Since the value of $y$ is imaginary, the given equation has no $y$ -intercepts.

We can test for the symmetry.

  $x-$ Axis: In order to test for symmetry with respect to the $x$ -axis, replace $y$ by $-y$ .

  $9x^2-{(-y)}^2=99x^2-y^2=9$

Since $9x^2-{(-y)}^2=9$ is equivalent to $9x^2-y^2=9$ , so the graph of the equation is symmetric with respect to the $x$ -axis.

  $y$ -Axis: Replace $x$ by $-x,$ to test for symmetry with respect to the $y$ -axis

  $9{(-x)}^2-y^2=99x^2-y^2=9$

Since $9{(-x)}^2-y^2=9$ is equivalent to $9x^2-y^2=9,$ the graph of the equation is symmetric with respect to the $y$ -axis.

Origin: To test for symmetry with respect to the origin, replace $x$ by $-x,$ and $y$ by $-y$ .

  $9{(-x)}^2-{(-y)}^2=99x^2-y^2=9$

Since $9{(-x)}^2-{(-y)}^2=9$ is equivalent to $9x^2-y^2=9$ , the graph of the equation issymmetric with respect to the origin.

Conclusion:

Therefore, the $x$ intercept are 1 and -1 and no $y$ -intercepts.

The graph of the equation is symmetric with respect to the $x$ -axis, $y$ -axis and the origin.