Answer
$g(x)=-x^2-2.5x+1$
Work Step by Step
The $y$-intercept of the parabola on the right is $(0,1)$, so an equation is
$g(x)=ax^2+bx+1$……….$(1)$
Since the points $(-2,2)$ and $(1,-2.5)$ are on the parabola,
we will substitute in equation $(1)$ the values $-2$ for $x$ and $2$ for $y$ as well as $1$ for $x$ and $-2.5$ for $y$ to obtain two equations with two unknowns $a$ and $b$.
$$\begin{cases}
4a-2b+1=2\\
a+b+1=-2.5
\end{cases}$$
Rewrite the equations:
$$\begin{cases}
4a-2b=1\\
a+b=-3.5
\end{cases}$$
AMultiply the second equation by $2$ and add it to the first to determine $a$:
$4a-2b+2a+2b=1-7$
$6a=-6$
$a=-1$
Calculate $b$:
$b=-3.5-a=-3.5-(-1)=-2.5$
The function is:
$g(x)=-x^2-2.5x+1$