Question

If the first point is (-2, 45/4) and the second point is (1, 10/3), how do I find the formula for exponential growth?

Answer 1

Any exponential growth/decay will fit the general form:

`y = Ae^(lambdax)" [1]"`

Use the two points to write two equations and then solve for `lambda` and A.

Explanation:

Substitute the point `(-2,45/4)` into equation [1]:

`45/4 = Ae^(-2lambda)" [2]"`

Substitute the point `(1,10/3)` into equation [1]:

`10/3 = Ae^(lambda)" [3]"`

We can make A disappear by dividing equation [2] by equation [3]:

`(45/4)/(10/3) = (Ae^(-2lambda))/(Ae^(lambda))`

`(45/4)(3/10) = e^(-3lambda)`

`135/40 = 27/8 = e^(-3lambda)`

`ln(27/8) = -3lambda`

`-1/3ln(27/8) = lambda`

`lambda = ln(2/3)`

Substitute the value of `lambda` into equation [1]:

`y = Ae^(ln(2/3)x)`

The exponential function and the natural logarithm are inverses, therefore, they cancel:

`y = A(2/3)^x" [4]"`

Use the point, `(1,10/3)`, to find the value of A:

`10/3= A(2/3)`

`A = 10/3(3/2)`

`A = 5`

Substitute into equation [4]:

`y = 5(2/3)^x" [5]"`