Question

How do you write the quadratic function in standard form given points (-6, 46), (2,14), (4, 56)?

Answer 1

`f(x)=2.5x^2+6x-8`

Explanation:

If we are required to find a quadratic in standard form
`color(white)("XXX")f(x)=ax^2+bx+c`
with solutions:
`color(white)("XXX")f(-6)=46`
`color(white)("XXX")f(2)=14`
`color(white)("XXX")f(4)=56`
(or, in the form provided in the question, for the points `(-6,46), (2,14), (4,56)`
then
`color(white)("XXX")a(-6)^2+b(-6)+c=46`
`color(white)("XXX")a(2)^2+b(2)+c=14`
`color(white)("XXX")a(4)^2+b(4)+c=14`
or simplified as a set of equations:
`color(white)("XXX")36a-6b+c=46`
`color(white)("XXXx")4a+2b+c=14`
`color(white)("XXX")16a+4b+c=56`

This set can be solved using common operations.
For demonstration purposes, I will include a solution method (perhaps less common) using Crammer's Rule:

Given a solution for `a=2.5, b=6,` and `c=-8` which allows us to write the required quadratic as
`color(white)("XXX")f(x)=2.5x^2+6x-8`