Answer
The graph is shown below.
Work Step by Step
Let us consider the parabola defined by the provided quadratic equation $y=-4{{x}^{2}}+20x+160$.
The coefficient of ${{x}^{2}}$ is $-4$.
Therefore, 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,
$\begin{align}
& a=-4 \\
& b=20 \\
& c=160
\end{align}$
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:
$\begin{align}
& x=-\frac{b}{2a} \\
& =-\frac{20}{2\left( -4 \right)} \\
& =-\frac{20}{\left( -8 \right)} \\
& =2.5
\end{align}$
Putting in the value of $x$ in the equation to find the $y\text{-}$ coordinate of the vertex, we get:
$\begin{align}
& f\left( x \right)=-4{{x}^{2}}+20x+160 \\
& f\left( 2.5 \right)=-4{{\left( 2.5 \right)}^{2}}+20\left( 2.5 \right)+160 \\
& =-25+210 \\
& =185
\end{align}$
So, the vertex of the parabola is $\left( -\frac{b}{2a},f\left( -\frac{b}{2a} \right) \right)=\left( 2.5,185 \right)$.
Use a graphing utility to plot the graph as follows:
Step 1: Write the function.
Step 2: Set the window $x:\left( -24,40,4 \right)$ and $y:\left( -210,300,30 \right)$.
Step 3: Plot the graph.
