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. $$
f(x)=\frac{1}{3}(x-5)^2-12.
$$
Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant $a= \frac{1}{3}$ is positive.
Step 2:Determine the vertex and the equation for the axis of symmetry. The vertex is $(5,-12)$ and the equation for the axis of symmetry is $x = 5$
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 $x$. $$
\begin{aligned}
f(0) & =\frac{1}{3}(0-5)^2-12 \\
& =\frac{1}{3}(25)-12 \\
& =\frac{25-3\cdot12}{3} \\
& =\frac{25-36}{3} \\
& =-\frac{11}{3} \\
& \approx -3.67.
\end{aligned}
$$ $y$-intercept: $(0,-3.67)$.
Set $ y = 0$ and solve. $$
\begin{aligned}
\frac{1}{3}(x-5)^2-12 & =0 \\
\frac{1}{3}(x-5)^2 & =12 \\
(x-5)^2 & =12(3) \\
(x-5)^2 & =36 \\
x-5 & = \pm \sqrt{36} \\
x & =5 \pm 6.
\end{aligned}
$$ Find the two separate solutions: $$
\begin{aligned}
x & =5-6 \\
& =-1 \\
x & =5+6 \\
& =11.
\end{aligned}
$$ $x$-intercepts: $(-1,0), (11, 0)$.
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[-12, \infty) $.
