Q. 58 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 58, list the intercepts and test for symmetry.

$y=\frac{x^2-4}{2x}$

Step-by-Step Solution

Answer

There are no $y$-intercepts and the $x$-intercepts are $(-2,0)$ and $(2,0)$. The graph is symmetric with respect to the origin only.

Step 1: Given information

The given equation is $y=\frac{x^2-4}{2x}$.

Step 2: Determine the x -intercept

For $x$-intercept, substitute 0 for $y$ and then solve for $x$.

$\begin{array}{rcl}0& =& \frac{x^2-4}{2x}\\ x^2-4& =& 0\\ \left(x-2\right)\left(x+2\right)& =& 0\\ x& =& -2,2\end{array}$

Step 3: Determine the y -intercept.

For $y$-intercept, substitute 0 for $x$ and then solve for $y$.

$\begin{array}{rcl}y& =& \frac{0^2-4}{2\left(0\right)}\\ & =& \mathrm{Undefined}\end{array}$

Step 4: Determine the symmetry

Substitute $y$ for $-y$ in the equation to check for symmetry with respect to $x$ axis:

$\begin{array}{rcl}-y& =& \frac{x^2-4}{2x}\\ y& =& \frac{-x^2+4}{2x}\end{array}$

Notice that the right side of the equation after substitution is not equal to the right side of the initial equation.

Since the equations are not equivalent, the graph is not symmetric with respect to the $x$-axis.

Substitute $x$ with $-x$ in the equation to check for the symmetry with respect to $y$-axis:

$\begin{array}{rcl}y& =& \frac{{\left(-x\right)}^2-4}{2\left(-x\right)}\\ & =& \frac{-x^2+4}{2x}\end{array}$

Notice that the right side of the equation after substitution is not equal to the right side of the initial equation.

Since the equations are not equivalent, the equation is not symmetric with respect to the $y$-axis.

Substitute $x$ with $-x$ and $y$ with $-y$ in the equation to check for the symmetry with respect to the origin:

$\begin{array}{rcl}-y& =& \frac{{\left(-x\right)}^2-4}{2\left(-x\right)}\\ y& =& \frac{x^2-4}{2x}\end{array}$

Notice that the right side of the equation after substitution is equal to the right side of the initial equation.

Since the equations are equivalent, the equation is symmetric with respect to the origin.

Step 5: Write the conclusion

There is no $y$-intercept.

The $x$-intercept are $\left(2,0\right)$ and $\left(-2,0\right)$.

The given equation is symmetric about the origin.