Precalculus, Chapter 1, Review Exercises, Section Review Exercises, Problem 10

Find the intercepts of the equation $y = 5x$ and test for symmetry with respect to the $x$-axis, the $y$-axis and the origin.

$x$-intercepts


$\begin{equation} \begin{aligned} y =& 5x && \text{Given equation} \\ 0 =& 5x && \text{To find the $x$-intercept, we let $y = 0$ and solve for $x$} \\ \frac{0}{5} =& \frac{5x}{5} && \\ x =& 0 && \end{aligned} \end{equation}$


The $x$-intercept is $(0,0)$

$y$-intercepts


$\begin{equation} \begin{aligned} y =& 5x && \text{Given equation} \\ y =& 5(0) && \text{To find the $y$-intercept, we let $x = 0$ and solve for $y$} \\ y =& 0 && \end{aligned} \end{equation}$


The $y$-intercept is $(0,0)$

Test for symmetry

$x$-axis


$\begin{equation} \begin{aligned} y =& 5x && \text{Given equation} \\ -y =& 5x && \text{To test for $x$-axis symmetry, replace $y$ by $-y$ and see if the equation is still the same} \end{aligned} \end{equation}$


The equation changes so the equation is not symmetric to $x$-axis.

$y$-axis


$\begin{equation} \begin{aligned} y =& 5x && \text{Given equation} \\ y =& 5(-x) && \text{To test for $y$-axis symmetry, replace$ x$ by $-x$ and see if the equation is still the same} \\ y =& -5x && \end{aligned} \end{equation}$


The equation changes so the equation is not symmetric to $y$-axis.

Origin


$\begin{equation} \begin{aligned} y =& 5x && \text{Given equation} \\ -y =& 5(-x) && \text{To test for origin symmetry, replace both $x$ by $-x$ and y by $-y$ and see if the equation is still the same} \\ -y =& -5x && \\ y =& 5x && \end{aligned} \end{equation}$


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