# How do you find a quadratic model?

Mar 22, 2015

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 $\left({p}_{1} , {q}_{1}\right) , \left({p}_{2} , {q}_{2}\right) , \left({p}_{3} , {q}_{3}\right)$ which satisfy the quadratic model.

A quadratic can be expressed as:
$a {x}^{2} + b x + c$

With $3$ points we can write $3$ equations with $a , b , c$ as variables:
$a {\left({p}_{1}\right)}^{2} + b \left({p}_{1}\right) + c = {q}_{1}$
$a {\left({p}_{2}\right)}^{2} + b \left({p}_{2}\right) + c = {q}_{2}$
$a {\left({p}_{3}\right)}^{2} + b \left({p}_{3}\right) + c = {q}_{3}$
which can then be solved for $a , b ,$ and $c$.

For example if the points are:
$\left(1 , - 1\right) , \left(4 , - 1\right) ,$ and $\left(5 , 3\right)$
the equations are
$1 \cdot a + 1 \cdot b + c = - 1$
$16 \cdot a + 4 \cdot b + c = - 1$
$25 \cdot a + 5 \cdot b + c = 3$
which can be solved for
$a = 1$, $b = - 5$, and $c = 3$
${x}^{2} - 5 x + 3$