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. $$
g(x)=\frac{1}{5}(x+7)^2+10.
$$ Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant $a= \frac{1}{5}$ is positive.
Step 2: Determine the vertex and the equation for the axis of symmetry. The vertex is $(-7,10)$ and the equation for the axis of symmetry is $x = -7$.
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 $x$. $$
\begin{aligned}
g(0) & =\frac{1}{5}(0+7)^2+10 \\
& =\frac{49}{5}+10 \cdot\frac{5}{5} \\
& =\frac{99}{5} \\
& =19.8
\end{aligned}
$$ $y$-intercept: $(0,19.8)$.
Set $ y = 0$ and solve. $$
\begin{aligned}
\frac{1}{5}(x+7)^2+10&=0 \\
(x+7)^2+50&=0 \\
(x+7)^2&=-50
\end{aligned}
$$ 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: $[10, \infty) $.
The graph of the parabola is shown in the figure.
