Write an exponential function

Question

Given the two points (2, 27) and (4, 243), write an exponential equation. I only know how to do this when the two points are right next to eachother. Help! 🙏

Answer 1

Write the exponential equation as y = abⁿ

27 = ab²

243 = ab⁴

243/27 = ab⁴/(ab²)

9 = b²

3 = b

27 = ab²

27 = a(3²)

27 = 9a

a = 3

y = 3*3ⁿ or y = 3ⁿ⁺¹

Answer 2

Kate

A general exponential function can be written as f(x)=f₀e^kx where f₀ and k are constants.

We were given two points to fit the exponential x=2 f(2)=27 and x=4 f9)4=243

so now the equation becomes two relationships from which we can determine k and f₀

The equations are:

27=f₀e^2k and 243= f₀e^4k form which we can solve for k by taking the ratio 243/27 = e^2k so k=1/2(ln(243/27)=1.1 and f₀=27/e^2.2 = 3

The required function f(x)=3e^1.1x

Regards

Jim