Question

Find a quadratic model for the set of values: (-2, -20), (0, -4), (4, -20)? Show your work.

Answer 1

`y=-2x^2 +4x -4`

Explanation:

A quadratic model can be expressed as;

`y=ax^2 + bx + c`

We have three unknown coefficients, `a`, `b`, and `c`, as well as two variables, `x` and `y`. Since we are given three points, lets plug those `x` and `y` values in and see what we get.

`-20 = a(-2)^2 +b(-2) + c`
`-4 = a(0)^2 + b(0) +c`
`-20 = a(4)^2 + b(4) +c`

Simplifying the expressions, we have;

`-20 = 4a - 2b + c`
`c=-4`
`-20 = 16a + 4b + c`

Conveniently, one of the middle expression has given us the value of one of the unknown constants, `c=-4`. We can subtract one of the remaining equations from the other to find an equation in terms of only `a` and `b`, the remaining unknown coefficients.

`(-20 = 16a +4b + c)`
`-(-20 = 4a + -2b + c)/(0 = 12a +6b + 0)`

In order to avoid dealing with fractions, lets solve for `b` in terms of `a`.

`6b = -12a`
`b = -2a`

Now we can plug this in for `b` in one of our remaining equations and solve for `a`.

`-20 = 4a -2(-2a) -4`
`-16 = 8a`
`a=-2`

Working backwards and solving for `b` then, we get;

`b = 4`

Now we have all of our coefficients, and we can write our model.

`y=-2x^2 +4x -4`