Q. 69 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 69, (a) find the intercepts of each equation, (b) test each equation for symmetry with respect to the $x$-axis, the $y$-axis, and the origin, and (c) graph each equation by hand by plotting points. Be sure to label the intercepts on the graph and use any symmetry to assist in drawing the graph. Verify your results using a graphing utility.

$y=x^2-5$

Step-by-Step Solution

Answer

The $x$-intercepts are $-\sqrt{5}$, and $\sqrt{5}$ and the $y$-intercept is $-5$. The equation is symmetric with respect to the $y$-axis.

The graph is shown below:

Step 1: Given information

The given equation is $y=x^2-5$.

Step 2: Determine the x -intercept.

Substitute $0$ for $y$ into the given function and factor it.

Apply the Zero-product property and solve for $x$.

$\begin{array}{rcl}{x}^2-5& =& 0\\ \left(x-\sqrt{5}\right)\left(x+\sqrt{5}\right)& =& 0\\ x& =& \sqrt{5}, -\sqrt{5}\end{array}$

Step 3: Determine the y intercept.

Substitute $0$ for $x$ into the given function and then solve for $y$.

$\begin{array}{rcl}y& =& 0^2-5\\ & =& -5\end{array}$

Step 4: Determine the symmetry.

Replace $y$ with $-y$ in the given function and simplify to check the symmetry about the $x$-axis.

$\begin{array}{rcl}-y& =& x^2-5\\ y& =& -x^2+5\end{array}$

As the obtained function is not equivalent to the given function, there is no symmetry about the $x$ axis.

Replace $x$ with $-x$ in the given function and simplify to check the symmetry about the $y$-axis.

$\begin{array}{rcl}y& =& {\left(-x\right)}^2-5\\ & =& x^2-5\end{array}$

As the obtained function is equivalent to the given function, there is a symmetry about the $y$-axis.

Replace $y$ with $-y$ and $x$ with $-x$ in the given function and simplify to check the symmetry about the origin.

$\begin{array}{rcl}-y& =& {\left(-x\right)}^2-5\\ y& =& -x^2+5\end{array}$

As the obtained function is not equivalent to the given function, there is no symmetry about the origin.

Step 5: Graph the equation

Substitute different values of $x$ into the given function to find the corresponding values of $y$. Represent the data in tabular form.

$x$	$y$	$\left(x,y\right)$
$-3$	${\left(-3\right)}^2-5=4$	$\left(-3,4\right)$
$-2$	${\left(-2\right)}^2-5=-1$	$\left(-2,-1\right)$
$-1$	${\left(-1\right)}^2-5=-4$	$\left(-1,-4\right)$
$0$	${\left(0\right)}^2-5=-5$	$\left(0,-5\right)$
$1$	${\left(-1\right)}^2-5=-4$	$\left(1,-4\right)$
$2$	${\left(2\right)}^2-5=-1$	$\left(2,-1\right)$
$3$	${\left(3\right)}^2-5=4$	$\left(3,4\right)$

The graph is shown below:

Step 6: Verify the graph using graphing utility.

By using the graphing utility, the graph is shown below:

Step 7: Write the conclusion

The $x$-intercept are $\left(\sqrt{5},0\right)$ and $\left(-\sqrt{5},0\right)$.

The $y$-intercept is $\left(0,-5\right)$.

The graph is symmetric with respect to $y$-axis.

The graph is shown below: