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. $$
g(x)=4(x+3.5)^2-16.
$$ Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant $a= 4$ is positive.
Step 2: Determine the vertex and the equation for the axis of symmetry. The vertex is $(-3.5,-16)$ and the equation for the axis of symmetry is $x = -3.5$.
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 $x$. $$
\begin{aligned}
g(0) & =4(0+3.5)^2-16 \\
& =4(12.25)-16 \\
& =33.
\end{aligned}
$$ $y$-intercept: $(0,33)$.
Set $ y = 0$ and solve. $$
\begin{aligned}
4(x+3.5)^2-16 & =0 \\
(x+3.5)^2 & =\frac{16}{4} \\
(x+3.5)^2 & =4 \\
x+3.5 & = \pm \sqrt{4} \\
x+3.5 & = \pm 2 \\
x & =-3.5 \pm 2.
\end{aligned}
$$ Find the two separate solutions: $$
\begin{aligned}
x & =-3.5-2 \\
& =-5.5 \\
x & =-3.5+2 \\
& =-1.5
\end{aligned}
$$ $x$-intercepts: $(-5.5,0), (-1.5, 0)$.
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[-16, \infty) $.
The graph of the parabola is shown in the figure below.
