Question

How do you find the exponential function that contains both of these points (2,12.6) and (5, 42.525)?

Answer 1

Suppose `f(x) = k*a^x` for some `k in RR` and `a > 0`, `a != 1`

Then `a = root(3)(42.525/12.6) = root(3)(3.375) = 1.5`

and `k = 12.6/(a^2) = 12.6/2.25 = 5.6`

So `f(x) = 5.6(1.5)^x`

Explanation:

Suppose `f(x) = k*a^x` for some `k in RR` and `a > 0`, `a != 1`

Then `f(5)/f(2) = (k*a^5)/(k*a^2) = a^3`

So `a = root(3)(f(5)/f(2)) = root(3)(42.525/12.6) = root(3)(3.375) = 1.5`

`f(2) = k*a^2`, so `k = f(2)/(a^2) = 12.6/(1.5^2) = 12.6 / 2.25 = 5.6`

So `f(x) = 5.6(1.5)^x`