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. $$
\begin{aligned}
g(x) &=0.5\left(x+7\right)^2-54.5.
\end{aligned}
$$ Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant $a= 0.5$ is positive.
Step 2: Determine the vertex and the equation for the axis of symmetry. The vertex is $(-7,-54.5)$ and the equation for the axis of symmetry is $x = -7$.
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 $x$. $$
\begin{aligned}
g(0)& =0.5\left(0+7\right)^2-54.5\\
&= -30\\
&\text{y-intercept: (0,-30)}.
\end{aligned}
$$ Set $ y = 0$ and solve. $$
\begin{aligned}
0.5\left(x+7\right)^2-54.5 & =0\\
\left(x+7\right)^2&= \frac{54.5}{0.5} \\
x+7& = \pm\sqrt{109}\\
x&= -7\pm \sqrt{109}.
\end{aligned}
$$ Find the two separate solutions: $$
\begin{aligned}
x&= -7+ \sqrt{109}\\
&\approx 3.44 \\
\textbf{or}\\
x&= -7- \sqrt{109}\\
& \approx -17.44.
\end{aligned}
$$ $x$-intercept: $(-17.44,0), (3.44 , 0)$.
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[-54.5, \infty) $.
