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.