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.