SOLUTION: Please help. Find the equation of the following in the form y=ab^x. Be sure to check the x-increments when finding b. (?) I was given these tables: a) x:1,2,3,4 y:3,6,12,2

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Please help. Find the equation of the following in the form y=ab^x. Be sure to check the x-increments when finding b. (?) I was given these tables: a) x:1,2,3,4 y:3,6,12,2      Log On


   

Question 276907: Please help.
Find the equation of the following in the form y=ab^x. Be sure to check the x-increments when finding b. (?)
I was given these tables:
a) x:1,2,3,4
y:3,6,12,24
b) x:4,8,12,16
y:2,2/3,2/9,2/27
c)x:1,2,3,4
y:0.75,2.25,6.75,20.25
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
y+=+ab%5Ex
I'll do one problem (the one I think is the hardest) and from that you should be able to do the others.
b) (4, 2) (8, 2/3) (12, 2/9) (16, 2/27)
The coordinates of these points make the equation true. So
2+=+ab%5E4
2%2F3+=+ab%5E8
2%2F9+=+ab%5E12
2%2F27+=+ab%5E16
To find "a" and "b" we just pick any two of these equations (it doesn't matter which) and solve that system. I'll pick the first two and I'll use the Substitution method:
2+=+ab%5E4
2%2F3+=+ab%5E8
If I take the first equation and divide both sides by b%5E4 we get:
2%2Fb%5E4+=+a
Substituting this into the second equation we get:
2%2F3+=+%282%2Fb%5E4%29b%5E8
which simplifies to:
2%2F3+=+2b%5E4
Dividing both sides by 2 we get:
1%2F3+=+b%5E4
Raising both sides to the 1/4 power we get:
%281%2F3%29%5E1%2F4+=+b
Substituting this into 2%2Fb%5E4+=+a we get:
2%2F%28%281%2F3%29%5E%281%2F4%29%29%5E4+=+a
which simplifies as follows:
2%2F%281%2F3%29+=+a
6+=+a
Now that we know a and b we can write the equation:
y+=+6%28%281%2F3%29%5E%281%2F4%29%29%5Ex
which can be written as
y+=+6%28%281%2F3%29%5E%28%281%2F4%29x%29%29
or
y+=+6%28root%284%2C+1%2F3%29%29%5Ex
To check, we can try the other two points in this equation.

The other two are much easier. Just
  1. set up two equations using the given points
  2. solve the system for a and b
  3. Write the equation using "x", "y" and the numbers you found for a and b
  4. Simplify the equation, if possible
  5. (Optional) Check your equation with one or both of the other points.