College Algebra, Chapter 2, Review Exercises, Section Review Exercises, Problem 30

a.) Test the equation $\displaystyle y = \frac{4}{x}$ for symmetry with respect to the $x$-axis, $y$-axis and the origin.
Testing the symmetry with respect to $y$-axis, set $x = -x$

$
\begin{equation}
\begin{aligned}
y &= \frac{4}{(-x)}\\
\\
y &= \frac{-4}{x}
\end{aligned}
\end{equation}
$

Since the new equations is not equal to the original equation, then $\displaystyle y = \frac{4}{x}$ is not symmetric to $y$-axis.



Testing the symmetry with respect to $x-$axis, set $y = -y$

$
\begin{equation}
\begin{aligned}
(-y) &= \frac{4}{x}\\
\\
-y &= \frac{4}{x}\\
\\
y &= -\frac{4}{x}
\end{aligned}
\end{equation}
$

Since the new equation is not equal to the original equation. Then $\displaystyle y = \frac{4}{x}$ is not symmetric to the $x$-axis.



Testing the symmetry with respect to origin. Set, $x = -x$ and $y = -y$

$
\begin{equation}
\begin{aligned}
(-y) &= \frac{4}{(-x)}\\
\\
-y &= -\frac{4}{x}\\
\\
y &= \frac{4}{x}
\end{aligned}
\end{equation}
$

Since the new equation is equal to the original equation. Then $\displaystyle y = \frac{4}{x}$ is symmetric to the origin.

b.) Find the $x$ and $y$-intercepts of the equation.
To find for $y$-intercept, set $x = 0$,
$\displaystyle y = \frac{4}{0}$ , $y$-intercept does not exist.

To find for $x$-intercept, set $y = 0$
$\displaystyle 0 = \frac{4}{x}$, $x$-intercept does not exist.