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. $$
\begin{aligned}
h(x)=2(x+2)^2+5.
\end{aligned}
$$ Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant, $a= 2$ is positive.
Step 2: Determine the vertex and the equation for the axis of symmetry. The vertex is $(-2,5)$ 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 $x$. $$
\begin{aligned}
h(0)= & 2(0+2)^2+5 \\
= & 2(4)+5 \\
= & 13.
\end{aligned}
$$ $y$-intercept: $(0,13)$
Set $ y = 0$ and solve. $$
\begin{aligned}
2(x+2)^2+5 & =0 \\
2(x+2) & =-5.
\end{aligned}
$$ 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: $[5, \infty) $.
