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. $$
\begin{aligned}
g(x)=3(x-7)^2+10.
\end{aligned}
$$ Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant, $a= 3$ 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 $x$. $$
\begin{aligned}
g(0) & =3(0-7)^2+10 \\
& =3(49)+10 \\
& =157.
\end{aligned}
$$ $y$-intercept: $(0,157)$
Set $ y = 0$ and solve. $$
\begin{aligned}
3(x-7)^2+10 & =0 \\
3(x-7)^2 & =-10.
\end{aligned}
$$ There is no x intercept. 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) $.
