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. $$
\begin{aligned}
f(x)=(x-5)^2-16.
\end{aligned}
$$ Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant, $a= 1$ is positive.
Step 2: Determine the vertex and the equation for the axis of symmetry. The vertex is $(5,-16)$ and the equation for the axis of symmetry is $x = 5$.
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 $x$.
$$
\begin{aligned}
f(0)& =(x-5)^2-16= 9\\
&\textbf{y-intercept: (0,9)}.
\end{aligned} $$
Set $ y = 0$ and solve. $$
\begin{aligned}
(x-5)^2-16& =0\\
\left(m-18\right)^2&= 16 \\
m-5& = \pm\sqrt{16}\\
m&= 5\pm 4.
\end{aligned}
$$ Find the two separate solutions:
$$
\begin{aligned}
x&=5+ 4= 9 \\
\textbf{or}\\
m&= 5-4 = 1.
\end{aligned} $$ $x$-intercepts: $(1,0), (9 , 0)$.
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[-16, \infty) $.
