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. $$
\begin{aligned}
f(x)=(x+6)^2-11.
\end{aligned}
$$ Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant, $a= 1$ is positive.
Step 2:Determine the vertex and the equation for the axis of symmetry. The vertex is $(-6,-11)$ 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 $x$. $$
\begin{aligned}
f(0) & =(0+6)^2-11 \\
& =36-11 \\
& =25.
\end{aligned}
$$ $y$-intercept: $(0,25)$
Set $ y = 0$ and solve.$$
\begin{aligned}
(x+6)^2-11 & =0 \\
(x+6)^2 & =11 \\
x+6 & = \pm \sqrt{11} \\
x & =-6 \pm \sqrt{11}.
\end{aligned}
$$ Find the two separate solutions: $$
\begin{aligned}
x & =-6+\sqrt{11}\approx =-2.68\\
x & =-6-\sqrt{11}\approx =-9.32.
\end{aligned}
$$ $x$-intercepts: $(-9.32,0), (-2.68 , 0)$.
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers
Range: $[-11, \infty) $.
