Answer
The required parabola is as shown below.
Work Step by Step
So, for the provided quadratic equation $f\left( x \right)=2x-{{x}^{2}}-2$ , when compared with the standard form of the quadratic equation $f\left( x \right)=a{{x}^{2}}+bx+c$ , one gets:
$\begin{align}
& a=-1, \\
& b=2, \\
& c=-2
\end{align}$
And use the steps shown below to determine the graph of the quadratic equation.
Step 1: Determine how the parabola opens:
Note that a, the coefficient of ${{x}^{2}}$ , is -1. If $a>0$ , the parabola opens in the downward direction. Also, if $\left| a \right|$ is small, the parabola opens more flatly than if $\left| a \right|$ is large.
Now, from the provided equation of the function, it is observed that the graph opens downwards as $a<0$.
Step 2: Calculate the vertex:
The x-coordinate can be calculated as:
$\begin{align}
& x=-\frac{b}{2a} \\
& =-\frac{-2}{2\times \left( -1 \right)} \\
& =1
\end{align}$
The y-coordinate can be calculated as:
$\begin{align}
& y=2\left( 1 \right)-{{\left( 1 \right)}^{2}}-2 \\
& =2-1-2 \\
& =-1
\end{align}$
So, the vertex is $\left( 1,-1 \right)$.
Step 3:
And above steps lead to the parabola that opens downwards and has a vertex at $\left( 1,-1 \right)$ and y-axis at $-2$.
So, the required parabola is as shown above.
