Answer
The graph is shown below:
Work Step by Step
Let us consider the parabola defined by the quadratic equation $f\left( x \right)=5{{x}^{2}}+40x+600$
The coefficient of ${{x}^{2}}$ is $5$.
Therefore, the value of $a$ is positive and the parabola opens in the upward direction.
Compare the equation with the standard form of the quadratic equation $f\left( x \right)=a{{x}^{2}}+bx+c$.
So,
$\begin{align}
& a=5 \\
& b=40 \\
& c=600
\end{align}$
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:
$\begin{align}
& x=-\frac{b}{2a} \\
& =-\frac{40}{2\left( 5 \right)} \\
& =-\frac{40}{\left( 10 \right)} \\
& =-4
\end{align}$
Putting in the value of $x$ in the equation to find the $y\text{-}$ coordinate of the vertex, we get:
$\begin{align}
& f\left( x \right)=5{{x}^{2}}+40x+600 \\
& f\left( -4 \right)=5{{\left( -4 \right)}^{2}}+40\left( -4 \right)+600 \\
& =80-160+600 \\
& =520
\end{align}$
So, the vertex of the parabola is $\left( -\frac{b}{2a},f\left( -\frac{b}{2a} \right) \right)=\left( -4,520 \right)$.
Use a graphing utility to plot the graph:
Step 1: Write the function.
Step 2: Set the window $x:\left( -15,90,15 \right)$ and $y:\left( 0,5000,500 \right)$.
Step 3: Plot the graph.
