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. $$
\begin{aligned}
s(p) & =(p+7)^2-14.
\end{aligned}
$$ 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 $(-7,-14)$ and the equation for the axis of symmetry is $x = -7$.
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 $x$. $$
\begin{aligned}
s(0)& =(0+7)^2-14\\
&= 35\\
\textbf{y-intercept: (0,35)}
\end{aligned}
$$ Set $ y = 0$ and solve. $$
\begin{aligned}
(p+7)^2-14 & =0\\
(p+7)^2&= 14 \\
p+7& = \pm\sqrt{14}\\
p&= -7\pm \sqrt{14}.
\end{aligned}
$$ Find the two separate solutions: $$
\begin{array}{cl}
p= -7- \sqrt{14} &\text { or }\ p= -7+ \sqrt{14} \\
p= -10.74 & \text { or }\ p= -3.26\\
\textbf{x-intercepts: (-3.26,0), (-10.74,0)}
\end{array}
$$
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[-14, \infty) $.
