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. $$
\begin{aligned}
f(t) &= 2(t-4)^2-44.
\end{aligned}
$$ Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant $a= 2$ is positive.
Step 2: Determine the vertex and the equation for the axis of symmetry. The vertex is $(4,-44)$ and the equation for the axis of symmetry is $x = 4$.
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 $x$. $$
\begin{aligned}
f(0)& =2(0-4)^2-44\\
&= -12\\
&\text{y-intercept: (0,-12)}.
\end{aligned}
$$ Set $ y = 0$ and solve. $$
\begin{aligned}
2(t-4)^2-44 & =0\\
(t-4)^2&= \frac{44}{2} \\
t-4& = \pm\sqrt{22}\\
t&= 4\pm \sqrt{22}.
\end{aligned}
$$ Find the two separate solutions: $$
\begin{aligned}
t&= 4+ \sqrt{22}
&\approx 8.69 \\
\textbf{or}\\
t&= 4- \sqrt{22}\\
& \approx -0.69.\
\end{aligned}
$$ $x$-intercepts: $(-0.69,0), (8.69, 0)$.
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[-44, \infty) $.
