When we get into equations that have different degrees, like quadratics and cubics, we've got a whole new ball of wax. The teacher really needs to be particular about the type of equation he/she wants to use here because not every equation works unless you get into a numerical analysis concept called the "divided-difference table."

Let's take a look at the following table:

x

y

1

1

2

4

3

9

4

16

 5

25

It's obvious to just about eveyone that the equation that will give us this table is y = x^2. We are able to eye-ball this table and see what the necessary equation is, but let's go through the steps we went through at the beginning of this unit.

The common difference in the x column is 1, but the common difference in the y column is 3, then 5, then 7, then 9. How can this be?

If we get bold and take the common difference of our common differences, we see that this new common difference is 2.

Our students have decided a long time ago that this equation will start up something like:

y = x^2

The issue, once again, is what goes in the box. It appears that the number that should go in the box is a 2. But 2 will not go in the box and give us the table above. OK, OK, must be some kind of mistake. Let's try another one. Given the table below, find the values that will make the equation give our table's values.

x

y

1

6

2

15

3

30

4

51

 5

78

Your students may be in a state of panic by now. Just go through the same steps we did before. The commond difference in the x column is again 1. The common differences for the y column are 9, 15, 21, and 27. That gives us the second common difference on the right to be 6. So from what we have now, our equation should look something like:

y = 6x^2 + .

When we go back and check the first set of values, the number in the box is a 0, but that clearly will not work for the second set of values - not even close.

So we are obviously doing something incorrectly. Now, teachers, how fast you want to reveal this to your students is your call. You can go really slowly and get them to figure out the problem, but you can also go ahead and tell them that when we go to the "second level" we must divide the coefficient of the x term by 2.

 

 

 

return to previous page