Answer
The graph is shown below:
Work Step by Step
Let us consider the parabola defined by the quadratic equation $y=-0.25{{x}^{2}}+40x$.
The coefficient of ${{x}^{2}}$ is $-0.25$.
So, the value of $a$ is negative and the parabola would be opening downwards.
Compare the equation with the standard form of the quadratic equation, which is
$f\left( x \right)=a{{x}^{2}}+bx+c$.
So,
$\begin{align}
& a=-0.25 \\
& b=40 \\
& c=0
\end{align}$
The general form of the parabola vertex is $\left( -\frac{b}{2a},f\left( -\frac{b}{2a} \right) \right)$.
Putting in the values of $a$ and $b$ to find the $x\text{-}$ coordinate of the vertex, we get:
$\begin{align}
& x=-\frac{b}{2a} \\
& =-\frac{40}{2\left( -0.25 \right)} \\
& =\frac{40}{\left( \frac{5}{10} \right)} \\
& =80
\end{align}$
Putting in the value of $x$ in the equation to find the $y\text{-}$ coordinate of the vertex, we get:
$\begin{align}
& f\left( x \right)=-0.25{{x}^{2}}+40x \\
& =-0.25{{\left( 80 \right)}^{2}}+40\left( 80 \right) \\
& =-1600+3200 \\
& =1600
\end{align}$
So, the vertex of the parabola is $\left( -\frac{b}{2a},f\left( -\frac{b}{2a} \right) \right)=\left( 80,1600 \right)$.
Use a graphing utility to plot the graph as follows:
Step 1: Write the function
Step 2: Set the window $x:\left( -160,256,32 \right)$ and $y:\left( -1920,2160,240 \right)$.
Step 3: Plot the graph.
