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