Quadratic Rule from a Table of Values

How do you find a quadratic rule from a table of values?

For example,

	1	2	3	4
	0	6	14	24

Find the first difference

First Difference

6-0=6

14-6=8

24-14=10

Find the second difference (if the second difference is a constant, then it is quadratic)

Second Difference

8-6=2

10-8=2

The general equation of a quadratic is

The second difference is

Hence our equation is now

The value is the vertical intercept (). We can back track in the table

	0	1	2	3	4
		0	6	14	24

As the first differences are 6, 8, 10, the one between 0 and 1 must be 4

Our equation is now .

We can now use any other point to find the value.

Let’s use the point

The function is

Let’s try another one

	3	4	5	6
	7	17	31	49

First differences

First difference

10

14

18

Second difference

Second Difference

4

4

Hence , therefore

The equation is now

Instead of back tracking, this time I am going to use two points and simultaneous equations.

Using points and

   

(1)  

   

(2)  

Equation 2 – Equation 1

Substitute into equation 1

Hence the equation is