Question

What is the quadratic function of a graph if the points are (-2, -4) (-1, -2.5) (0, 0) (1, 3.5) (2, 8)?

Answer 1

It can be guessed since we deal with easy numbers, but let's approach this problem theoretically.

The general equation of a quadratic function looks like this:
`y=ax^2+bx+c`
There are three unknown coefficients here - `a`, `b` and `c`

Let's use three points out of 5 given to determine these three coefficients.

Point `(-2,-4)` results in equation
`-4=a(-2)^2+b(-2)+c`
or, simplifying this,
(1) `-4=4a-2b+c`

Point `(-1,-2.5)` results in equation
`-2.5=a(-1)^2+b(-1)+c`
or, simplifying this,
(2) `-2.5=a-b+c`

Point `(0,0)` results in equation
`0=a(0)^2+b(0)+c`
or, simplifying this,
(3) `0=c`

Equations (1), (2) and (3) constitute a system of 3 linear equations with three unknown variables. Let's solve it by substitution.

Step 1.
Substitute `c=0` from equation (3) into equations (1) and (2):
(1) `-4=4a-2b`
(2) `-2.5=a-b`

Step2.
Simplify the equation (1) by dividing left and right sides by 2:
(1) `-2=2a-b`
Solve it for `b`:
`b=2a+2`

Step 3.
Substitute an expression for `b` into equation (2):
(2) `-2.5=a-2a-2`
or
`a=0.5`

Step 4.
Find the value of `b`:
`b=2*0.5+2=3`

Together with previously determined `c=0`, we have an equation:
`y=0.5x^2+3x`

All we have to do is to check that two other points specified in the problem lie on this graph.

Point `(1, 3.5)`:
`0.5*1^2+3*1=0.5+3=3.5` (check!)
Point `(2,8)`:
`0.5*2^2+3*2=0.5*4+6=2+6=8` (check!)

So, the answer to this problem is
`y=0.5x^2+3x`