Question

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

Answer 1

See explanation below

Explanation:

A general parabola is like `ax^2+bx+c=f(x)`
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(x_0,y_0)` is a parabola point, then `ax_0^2+bx_0+c=y_0`

Apply this to our case. We have

1.- `a(-2)^2+b(-2)+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 `4a-2b+1=2`

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

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

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

The the parabola is `-x^2-2.5x+1=f(x)`