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. $$
\begin{aligned}
g(t) & = (t-4)^2-36.
\end{aligned}
$$ 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 $(4,-36)$ and the equation for the axis of symmetry is $x = -36$.
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 $x$. $$
\begin{aligned}
g(0)& =(0-4)^2-36\\
&= -20\\
\textbf{y-intercept: (0,-20)}
\end{aligned}
$$ Set $ y = 0$ and solve. $$
\begin{aligned}
(t-4)^2-36 & =0\\
(t-4)^2&= 36 \\
t-4& = \pm\sqrt{36}\\
t&= 4\pm 6
\end{aligned}
$$
Find the two separate solutions:
$$
\begin{array}{cl}
t= 4+ 6 &\text { or }\ t= 4-6 \\
t= 10 & \text { or }\ t=-2\\
\textbf{x-intercept: (-2,0), (10,0)}
\end{array}
$$ Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[-36, \infty) $.
