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. $$
\begin{aligned}
k(m) & =(m+2.5)^2-18.25.
\end{aligned}
$$ 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 $(-2.5,-18.25)$ and the equation for the axis of symmetry is $x = -2.5$.
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 $x$. $$
\begin{aligned}
k(0)& =(0+2.5)^2-18.25\\
&= -12\\
\textbf{y-intercept: (0,-12)}.
\end{aligned}
$$ Set $ y = 0$ and solve. $$
\begin{aligned}
(m+2.5)^2-18.25 & =0\\
(m+2.5)^2&= 18.25 \\
m+2.5& = \pm\sqrt{18.25}\\
m&= -2.5\pm \sqrt{18.25}.
\end{aligned}
$$ Find the two separate solutions: $$
\begin{array}{cl}
m = -2.5 +\sqrt{18.25} &\text { or } m = -2.5 -\sqrt{18.25} \\
m= 1.77 & \text { or }\ m= -6.77\\
\textbf{x-intercept: (-6.77,0), (1.77,0)}.
\end{array}
$$ Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[-18.25, \infty) $.
