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. $$
\begin{aligned}
m(p) &=6\left(p+\frac{5}{6}\right)^2+\frac{101}{6}\\
& = 6(p+0.833)^2+16.833
\end{aligned}
$$ Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant, $a= 6$ is positive.
Step 2: Determine the vertex and the equation for the axis of symmetry. The vertex is $(-0.833,16.833)$ and the equation for the axis of symmetry is $x = -0.833$.
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 $x$.
$$
\begin{aligned}
m(0)& =6\left(0+\frac{5}{6}\right)^2+\frac{101}{6}\\
&= 21\\
&\text{y-intercept: (0,21)}.
\end{aligned}
$$ Set $ y = 0$ and solve. $$
\begin{aligned}
6\left(p+\frac{5}{6}\right)^2+\frac{101}{6}& =0\\
\left(p+\frac{5}{6}\right)^2&= -\frac{101}{36}.
\end{aligned}
$$ The equation has no solution, therefore there are no $x$-intercepts.
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[16.833, \infty) $.
