How do I find the curve for an exponential function in the form y=ab^x when given two points on the line?

Question

For example I am given the points (1,6) and (3,24). How do I find the equation for this exponential function?

Answer 1

You have a function and a few points - you can plug them in and solve for the constants a and b. If you plug in your first set of points, you have 6 = ab^1. Likewise for the second set of points - 24 = ab^3. Two equations, two unknowns - let's combine them:

6 = ab
a = 6/b

24 = ab^3
24 = (6/b)b^3
24 = 6b^2
4 = b^2
b = 2

Then, since a = 6/b:

a = 6/b
a = 6/2
a = 3

So your exponential becomes y = 3*2^x. This checks out if you plug in the points you have:

(1) 6 = 3*2^1
(2) 24 = 3*2^3