Question

How do you find a quadratic model?

Answer 1

How you establish a quadratic model depends upon what information you have available.
Probably the easiest way to find a quadratic model is if you are given `3` points `(p_1,q_1), (p_2,q_2), (p_3,q_3)` which satisfy the quadratic model.

A quadratic can be expressed as:
`ax^2 + bx + c`

With `3` points we can write `3` equations with `a, b, c` as variables:
`a(p_1)^2 + b(p_1) + c = q_1`
`a(p_2)^2 + b(p_2) +c = q_2`
`a(p_3)^2 + b(p_3) + c = q_3`
which can then be solved for `a,b,` and `c`.

For example if the points are:
`(1,-1), (4,-1),` and `(5,3)`
the equations are
`1*a + 1*b + c = -1`
`16*a + 4*b + c = -1`
`25*a + 5*b + c = 3`
which can be solved for
`a=1`, `b=-5`, and `c=3`
for a quadratic model
`x^2 - 5x + 3`