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. $$
f(x)=\frac{2}{5}(x-6)^2+\frac{1}{5}.
$$ Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant $a= \frac{2}{5}$ is positive.
Step 2: Determine the vertex and the equation for the axis of symmetry. The vertex is $(6,0.2)$ and the equation for the axis of symmetry is $x = 6$.
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 $x$. $$
\begin{aligned}
f(0) & =\frac{2}{5}(0-6)^2+\frac{1}{5} \\
& =\frac{2(36)+1}{5} \\
& =\frac{73}{5} \\
& =14.6
\end{aligned}
$$ $y$-intercept: $(0,14.6)$.
Set $ y = 0$ and solve. $$
\begin{aligned}
& \frac{2}{5}(x-6)^2+\frac{1}{5}=0 \\
& 2(x-6)^2+1=0 \\
& (x-6)^2=-\frac{1}{2}
\end{aligned}
$$ There is no $x$ intercept because we can't take the square root of a negative number.
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[1/5, \infty) $.
The graph of the parabola is shown in the figure.
