What type of power function is represented in the table?

Question

What type of power function is represented in the table?

 

xl –2 –1 0 1 2 3
f(x)l 12 3 0 3 12 27

 

A linear

B quadratic

C cubic

D quartic

Answer 1

Since the x values are separated by 1, it's easy to use the method of taking differences to figure this out.

 

First take the differences between the successive values of f(x):

 

3 - 12 = -9

0 - 3 = -3

3 - 0 = 3

12 - 3 = 9

27 - 12 = 15

 

If the difference between each value of f(x) was the same for every x, then the equation would be a first degree polynomial, or a linear function.

 

So now take the differences of the first set of differences:

 

(-3) - (-9) = 6

3 - (-3) = 6

9 - 3 = 6

15 = 9 = 6

 

Since the second set of differences are the same, that means that the function is a second degree polynomial, which is quadratic.

 

In other words, the number of differences you have to take until you get a constant series is equal to the degree of the polynomial.

 

HOWEVER:

 

- If the increments between the values of x were not all 1, then this would NOT work.  You would need an adjustment to the procedure.

- Not all functions are polynomials.  You will only reach a constant set of differences for a polynomial equation.

 

Play around with this.  Make up polynomial equations of different degrees and make tables of values using successive values of x.  You'll see how it works!

 

 

Jon