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?

1 Answer
May 30, 2017

Answer:

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]"#