How to find the formula of a quadratic function knowing that it contains the points...

Step 1
${\displaystyle y=a{x}^2+bx+c}$
Find a, b, and c.
Write that this graph passes at the 3 points: ${\displaystyle (-2,2);(0,1)}$ and ${\displaystyle (1,-2.5)}$.
${\displaystyle 2=4a-2b+c}$
${\displaystyle 1=c}$
$-2.5=a+b+c$
Solve this system of linear equations.
First ${\displaystyle c=1\to 1=4a-2b\to 4a-2b=1}$
${\displaystyle -3.5=a+b\to 2a+2b=-7}$
Step 2
${\displaystyle 6a=-6\to a=-1}$
${\displaystyle a+b=-3.5\to b=-3.5+1=-2.5}$
Finally the quadratic function is:
$y=-x^2-2.5\times x+1$
Check:
${\displaystyle x=-2\to y=-4+5+1=2}$ Correct
${\displaystyle x=1\to y=-1-2.5+1=2.5}$ Correct