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. $$
h(x)=-0.5(x+3)^2+32.
$$ Step 1: We first determine the direction in which the function opens. The given function opens downward because the constant $a= -0.5$ is negative.
Step 2:Determine the vertex and the equation for the axis of symmetry. The vertex is $(-3,32)$ and the equation for the axis of symmetry is $x = -3$.
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 $x$.
$$
\begin{aligned}
h(0) & =-0.5(0+3)^2+32 \\
& =-0.5(9 )+32 \\
& =27.5
\end{aligned}
$$ $y$-intercept: $(0,27.5)$.
Set $ y = 0$ and solve.
$$
\begin{aligned}
-0.5(x+3)^2+32 & =0 \\
(x+3)^2-64 & =0 \\
(x+3)^2 & =64 \\
(x+3)^2 & = \pm \sqrt{64} \\
(x+3) & = \pm 8 \\
x & =-3 \pm 8
\end{aligned}
$$Find the two separate solutions: $$
\begin{aligned}
x & =-3-8 \\
& =-11 \\
x & =-3+8 \\
& =5
\end{aligned}
$$ $x$-intercept: $(-11,0), (5, 0)$.
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $(-\infty, 32] $.
