Exponential Functions

Question

Find a formula for f(x), an exponential function such that f(−8) = 200 and f(30) = 580

Answer 1

Erin,

For exponential functions, a fixed difference in domain translates into a fixed ratio of range.

Difference given in domain = 38 (30 - (-8)). Ratio of range = 580/200 = 2.9, ln(ratio of range) = 1.06471074

So for function f(x) = a * exp(kx)

take ln each side:

ln (f(x)) = kx

1.06471074=38k

k = 0.0280187037

Then plug back in to solve for a (the preexponential term): 200 = a exp (0.0280187037 * (-8)) = a exp(-0.2241496296) = 0.799195542113 .

a= 200 / 0.799195542113

a=250.251646

Check: is 250.251646 * exp(0.0280187037 * 30) = 580? Yes, = 580.0000017723705.