Precalculus, Chapter 1, Review Exercises, Section Review Exercises, Problem 12

Find the intercepts of the equation $9x^2 - y^2 = 9$ and test for symmetry with respect to the $x$-axis, the $y$-axis and the origin.

$x$-intercepts


$
\begin{equation}
\begin{aligned}

9x^2 - y^2 =& 9
&& \text{Given equation}
\\
9x^2 - (0)^2 =& 9
&& \text{To find the $x$-intercept, we let $y = 0$ and solve for $x$}
\\
9x^2 =& 9
&&
\\
x^2 =& 1
&&
\\
x =& \pm 1
&&

\end{aligned}
\end{equation}
$


The $x$-intercepts are $(-1,0)$ and $(1,0)$

$y$-intercepts


$
\begin{equation}
\begin{aligned}

9x^2 - y^2 =& 9
&& \text{Given equation}
\\
9(0)^2 - y^2 =& 9
&& \text{To find the $y$-intercept, we let $x = 0$ and solve for $y$}
\\
-y^2 =& 9
&&
\\
y^2 =& -9
&&
\\
y =& \sqrt{-9}
&&

\end{aligned}
\end{equation}
$


Theres is no real solutions for $y$.

Test for symmetry

$x$-axis


$
\begin{equation}
\begin{aligned}

9x^2 - y^2 =& 9
&& \text{Given equation}
\\
9x^2 - (-y)^2 =& 9
&& \text{To test for $x$-axis symmetry, replace $y$ by $-y$ and see if the equation is still the same}
\\
9x^2 - y^2 =& 9
&&

\end{aligned}
\end{equation}
$


The equation is still the same so it is symmetric to the $x$-axis.

$y$-axis


$
\begin{equation}
\begin{aligned}

9x^2 - y^2 =& 9
&& \text{Given equation}
\\
9(-x)^2 - y^2 =& 9
&& \text{To test for $y$-axis symmetry, replace$ x$ by $-x$ and see if the equation is still the same}
\\
9x^2 - y^2 =& 9
&&

\end{aligned}
\end{equation}
$


The equation is still the same so it is symmetric to the $y$-axis.

Origin


$
\begin{equation}
\begin{aligned}

9x^2 - y^2 =& 9
&& \text{Given equation}
\\
9(-x)^2 - (-y)^2 =& 9
&& \text{To test for origin symmetry, replace both $x$ by $-x$ and y by $-y$ and see if the equation is still the same}
\\
9x^2 - y^2 =& 9
&&


\end{aligned}
\end{equation}
$


The equation is still the same so it is symmetric to the origin.