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.$$
f(x)=-2(x+6)^2+20.
$$ Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant $a= -2$ is negative.
Step 2: Determine the vertex and the equation for the axis of symmetry. The vertex is $(-6,20)$ and the equation for the axis of symmetry is $x = -6$.
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 $x$. $$
\begin{aligned}
f(0) & =-2(0+6)^2+20 \\
& =-2(36)+20 \\
& =-52.
\end{aligned}
$$ $y$-intercept: $(0,-52)$.
Set $ y = 0$ and solve. $$
\begin{aligned}
-2(x+6)^2+20 & =0 \\
-2(x+6)^2 & =-20 \\
(x+6)^2 & =\frac{-20}{-2} \\
(x+6)^2 & =10 \\
x+6 & = \pm \sqrt{10} \\
x & =-6 \pm \sqrt{10}.
\end{aligned}
$$ Find the two separate solutions:
$$
\begin{aligned}
x & =-6-\sqrt{10} \\
& =-9.16 \\
x & =-6+\sqrt{10} \\
& =-2.84
\end{aligned}
$$ $x$-intercept: $(-9.16,0), (-2.84, 0)$.
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $(-\infty, 20] $.
