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. $$
\begin{aligned}
k(p) &=5(p-5)^2-105.
\end{aligned}
$$ Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant $a= 5$ is positive.
Step 2: Determine the vertex and the equation for the axis of symmetry. The vertex is $(5,-105)$ and the equation for the axis of symmetry is $x = 5$.
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 $x$. $$
\begin{aligned}
k(0)& =5(0-5)^2-105\\
&= 20\\
&\text{y-intercept: (0,20)}.
\end{aligned}
$$ Set $ y = 0$ and solve. $$
\begin{aligned}
5(p-5)^2-105& =0\\
(p-5)^2&= \frac{105}{5} \\
p-5& = \pm\sqrt{21}\\
p&= 5\pm \sqrt{21}.
\end{aligned}
$$ Find the two separate solutions:
$$
\begin{aligned}
p&= 5+ \sqrt{21}\\
&\approx 9.58 \\
\textbf{or}\\
p&= 5- \sqrt{21}\\
& \approx 0.42
\end{aligned}
$$ $x$-intercepts: $(0.42,0), (9.58, 0)$.
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[-105, \infty) $.
