The points (2,9) and (4, 65) lie on the graph of f(x)=2^(ax)+b. What are the values of a and b?

Question

If someone could show me the working behind this I would be very grateful!!!

Answer 1

If we plug in your values, we get two equations:

9=2^(2a)+b

65=2^(4a)+b

 

The second equation can be written as 65=(2^(2a))^2 +b

and the first as 9-b=2^(2a)

 

Now we can substitute the second into the first and get 65=(9-b)^2 +b

Expand: 65=81-18b+b^2+b

0=b^2-17b+16

0=(b-16)(b-1)

So b is either 1 or 16.

 

Lets try b=1 first. Plug it back into an equation and get:

9=2^(2a)+1

 

8=2^(2a)

 

3=2a

 

a=3/2

 

And if b=16, we get:

9=2^(2a)+16

-7=2^(2a)

But unfortunately, we can't multiply 2s together until we get a negative number. So this isn't a solution.

 

So a=3/2, b=1