Answer
See graph
Work Step by Step
The function that we want to graph is shown below. Follow the steps outline to graph the function. $$
\begin{aligned}
d(m) &= 0.25\left(m-18\right)^2-68.
\end{aligned}
$$ Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant $a= 0.25$ is positive.
Step 2: Determine the vertex and the equation for the axis of symmetry. The vertex is $(18,-68)$ and the equation for the axis of symmetry is $x = 18$.
Step 3: Find the $y$ and $x$ intercepts. To find the $y$ intercept, set $x= 0$ to find the value of $y$. Similarly, to find the $x$ intercept, set $ y= 0$ to find the values of $x$. $$
\begin{aligned}
d(0)& =0.25\left(0-18\right)^2-68\\
&= 13\\
&\text{y-intercept: (0,13)}.
\end{aligned}
$$ Set $ d = 0$ and solve. $$
\begin{aligned}
0.25\left(m-18\right)^2-68& =0\\
\left(m-18\right)^2&= \frac{68}{0.25} \\
m-18& = \pm\sqrt{272}\\
m&= 18\pm \sqrt{272}
\end{aligned}
$$ Find the two separate solutions: $$
\begin{aligned}
m&=18+ \sqrt{272}\\
&= 34.49 \\
\textbf{or}\\
m&= 18-\sqrt{272}\\
& = 1.51
\end{aligned}
$$ $x$-intercept: $(1.51,0), (34.49 , 0)$.
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[-68, \infty) $.
