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. $$
\begin{aligned}
m(d) &= 4(d+3)^2-66.
\end{aligned}
$$
Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant $a= 4$ is positive.
Step 2: Determine the vertex and the equation for the axis of symmetry. The vertex is $(-3,-66)$ and the equation for the axis of symmetry is $x = -3$.
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 $x$. $$
\begin{aligned}
m(0)& =4(0+3)^2-66\\
&= -30\\
&\textbf{y-intercept: (0,-30)}.
\end{aligned}
$$ Set $ y = 0$ and solve. $$
\begin{aligned}
4(d+3)^2-66 & =0\\
(d+3)^2&= \frac{66}{4} \\
d+3& = \pm\sqrt{16.5}\\
d&= -3\pm \sqrt{16.5}.
\end{aligned}
$$ Find the two separate solutions: $$
\begin{aligned}
d&= -3+ \sqrt{16.5}\approx 1.062 \\
\textbf{or}\\
d&= -3-\sqrt{16.5}\approx = -7.062 \\
&\textbf{x-intercepts: (-7.062,0), (1.062, 0)}.
\end{aligned}
$$ Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[-66, \infty) $.
