Expert: Steve Holleran - 6/19/2007

Question
Steve, the question reads "Evaluate the functions for the values of x given as 1, 2, 4, 8, and 16. Describe the differences in the rate at which each function changes with increasing values of x."  

I have solved the values of x and have come up with the following table of values:

x    y

1   12
2   20
4   42
8   110
16   342

Can you please help with finding the differences in the rate at which each function changes with increasing values of x?  

Answer
Hi Pam,

First of all, thank you for all the nice comments you have made.  It is truly appreciated.

I'm going to give you what I hope is the answer you're looking for, but I'm a little unsure about the directions, so I may have missed the mark... I hope not.

If you think of the x and y values as ordered pairs, we have

(1, 12)   (2, 20)   (4, 42)  (8, 110)  (16, 342)

Now, rate of change of a function is defined to be

  (change in y) / (change in x)

so, for example, the rate of change for the first two ordered pairs would be :

    (20 - 12) / (2 - 1)  = 8

    
Doing this for the other ordered pairs, in succession, we get:

(2, 20 ) to (4, 42)  = 42 - 20 / 4 - 2 = 11

(4, 42) to (8, 110)  = 110 - 42 / 8 - 4 = 17

(8, 110 ) to (16, 342) = 342 - 110 / 16 - 8 = 29


So now look at the rates of change:

8 --> 11 --> 17 --> 29 --> ?

From 8 to 11 is an increase of 3

From 11 to 17 is an increase of 6

Form 17 to 29 is an increase of 12

This tells us that the rate is doubling each time.  
So, for example, the next increase should be 24, which means the next rate of increase should be 29 + 24 = 53,  and so on.

I hope this is the information you needed.
Please let me know if its not.

Steve Holleran