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. $$
g(x)=\frac{3}{4}(x+8)^2-27.
$$ Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant $a= \frac{3}{4}$ is positive.
Step 2: Determine the vertex and the equation for the axis of symmetry. The vertex is $(-8,-27)$ and the equation for the axis of symmetry is $x = -8$.
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 $x$. $$
\begin{aligned}
g(0) & =\frac{3}{4}(0+8)^2-27 \\
& =\frac{3}{4}(64)-\frac{27 \cdot 4}{4} \\
& =3(16)-27 \\
& =21
\end{aligned}
$$ $y$-intercept: $(0,21)$.
Set $ y = 0$ and solve. $$
\begin{aligned}
\frac{3}{4}(x+8)^2-27 & =0 \\
\frac{3}{4}(x+8)^2 & =27 \\
(x+8)^2 & =27 \cdot \frac{4}{3} \\
(x+8)^2 & =36 \\
x+8 & = \pm \sqrt{36} \\
x & =-8 \pm 6.
\end{aligned}
$$ Find the two separate solutions: $$
\begin{aligned}
x & =-8-6 \\
& =-14 \\
x & =-8+6 \\
& =-2.
\end{aligned}
$$ $x$-intercepts: $(-14,0), (-2, 0)$.
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[-27, \infty) $.
The graph of the parabola is shown in the figure below.
