# What is the equation of a parabola that passes through (-2,2), (0,1), and (1, -2.5)?

May 8, 2018

See explanation below

#### Explanation:

A general parabola is like $a {x}^{2} + b x + c = f \left(x\right)$
We need to "force" that this parabola passes thru these points. How do we do?. If parabola passes through these points, their coordinates acomplishes the parabola expresion. It say

If $P \left({x}_{0} , {y}_{0}\right)$ is a parabola point, then $a {x}_{0}^{2} + b {x}_{0} + c = {y}_{0}$

Apply this to our case. We have

1.- $a {\left(- 2\right)}^{2} + b \left(- 2\right) + c = 2$
2.- a·0+b·0+c=1
3.- a·1^2+b·1+c=-2.5

From 2. $c = 1$
From 3 $a + b + 1 = - 2.5$ multiply by 2 this equation and add to 3
From 1 $4 a - 2 b + 1 = 2$

$2 a + 2 b + 2 = - 5$
$4 a - 2 b + 1 = 2$

$6 a + 3 = - 3$, then $a = - 1$

Now from 3...$- 1 + b + 1 = - 2.5$ give $b = - 2.5$

The the parabola is $- {x}^{2} - 2.5 x + 1 = f \left(x\right)$