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. $$
\begin{aligned}
h(x) & =3(x+2)^2+12.
\end{aligned}
$$ 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 $(-2,12)$ and the equation for the axis of symmetry is $x = -2$.
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 $x$. $$
\begin{aligned}
& =3(0+2)^2+12\\
&= 24\\
&\textbf{y-intercept: (0,24)}.
\end{aligned}
$$ Set $ y = 0$ and solve. $$
\begin{aligned}
3(x+2)^2+12 & =0\\
3(x+2)^2&= -12
\end{aligned}
$$ 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: $[12, \infty) $.
The graph of the parabola is shown in the figure.
