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.$$
\begin{aligned}
g(x)=-2(x-4)^2+18.
\end{aligned}
$$ 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 $(4,18)$ and the equation for the axis of symmetry is $x = 4$.
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 $x$. $$
\begin{aligned}
g(0) & =-2(0-4)^2+18 \\
& =-2(16)+18 \\
& =-14.
\end{aligned}
$$ $y$-intercept: $(0,-14)$
Set $ y = 0$ and solve. $$
\begin{aligned}
-2(x-4)^2+18 & =0 \\
-2(x-4)^2 & =-18 \\
(x-4)^2 & =\frac{-18}{-2} \\
(x-4)^2 & =9 \\
x-4 & = \pm \sqrt{9} \\
x & =4 \pm 3.
\end{aligned}
$$ Find the two separate solutions: $$
\begin{aligned}
x & =4+3 \\
& =7
\end{aligned}
$$ $$
\begin{aligned}
x & =4-3 \\
& =1
\end{aligned}
$$ $x$-intercepts: $(1,0), (7 , 0)$
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers
Range: $(-\infty, 18] $.
