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. $$
\begin{aligned}
g(x)=(x-3)^2-9.
\end{aligned}
$$ 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 $(3,-9)$ 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 $x$. $$
\begin{aligned}
g(0)& =(0-3)^2-9= 0\\
&\textbf{y-intercept: (0,0)}.
\end{aligned}
$$ Set $ y = 0$ and solve. $$
\begin{aligned}
(x-3)^2-9& =0\\
\left(x-3\right)^2&= 9 \\
x-3& = \pm\sqrt{9}\\
x&= 3\pm 3.
\end{aligned}
$$ Find the two separate solutions:
$$
\begin{aligned}
x&=3+3= 6 \\
\textbf{or}\\
x&= 3-3 = 0.
\end{aligned}
$$ $x$-intercepts: $(6,0), (0 , 0)$.
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[-9, \infty) $.
