Answer
See image
Work Step by Step
Write $f$ in the form $f(x)=a(x-h)^{2}+k$:
$\displaystyle \begin{aligned}f(x)&=-2x^{2}+6x+2\displaystyle \\&=-2(x^{2}-3x)+2\\&=-2(x^{2}-3x+\displaystyle \frac{9}{4})+2+2\cdot\frac{9}{4}\\&=-2\displaystyle (x-1.5)^{2}+6.5\end{aligned}$
Let $f_{1}(x)=x^{2}$.
Then, $f(x)=-2f_{1}(x-1.5)+6.5$.
The graph is obtained from $f_{1}(x)$ by
- reflecting about the x-axis $\quad(x,y)\rightarrow(x,-y),$
- stretching by factor 2 $\quad (x,-y)\rightarrow(x,-2y)$
- shifting to the right by $1.5$ units,$\quad (x,-2y)\rightarrow (x+1.5,-2y)$
- and then up by 6.5 units $\quad (x+1.5,-2y+6.5)$
Point by point,
$\left[\begin{array}{lll}
(x,y) & \rightarrow & (x+1.5,-2y+6.5)\\
& & \\
(0,0) & \rightarrow & (1.5,6.5)\\
(-1,1) & \rightarrow & (0.5,4.5)\\
(1,1) & \rightarrow & (2.5,4.5)\\
(-2,4) & \rightarrow & (-0.5,-1.5)\\
(2,4) & \rightarrow & (3.5,-1.5)
\end{array}\right]$
