SOLUTION: Solve for the logistic function with initial condition = 14, limit to growth = 42 and passing through (1, 28) Thank you!

Algebra ->  Rational-functions -> SOLUTION: Solve for the logistic function with initial condition = 14, limit to growth = 42 and passing through (1, 28) Thank you!      Log On


   

Question 1127178: Solve for the logistic function with initial condition = 14, limit to growth = 42 and passing through (1, 28)
Thank you!
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


The standard form of a logistic function is

f%28x%29+=+c%2F%281%2Bab%5Ex%29

The "ab^x" in the denominator is a decreasing exponential function; as x gets very large that exponential goes to zero, so the denominator goes to 1. That means the numerator c is the limiting value of the function.

So in this example we know the function is of the form

f%28x%29+=+42%2F%281%2Bab%5Ex%29

The initial value of the function is when x is 0. When x is 0, the denominator of the function is just 1+a. Use the given initial value to find the value of a.

f%280%29+=+42%2F%281%2Bab%5E0%29+=+42%2F%281%2Ba%29+=+14
1%2Ba+=+3
a+=+2

Now we know the function is of the form

f%28x%29+=+42%2F%281%2B2b%5Ex%29

Now, to finish finding the logistic function, find the value of b by using the given data point.

f%281%29+=+42%2F%281%2B2b%5E1%29+=+28
42%2F%281%2B2b%29+=+28
42+=+28%2B56b
14+=+56b
b+=+1%2F4

The logistic function for this problem is

f%28x%29+=+42%2F%281%2B2%281%2F4%29%5Ex%29

A graph, including horizontal lines at the initial, intermediate, and limiting values of 14, 28, and 42:

graph%28400%2C400%2C-1%2C5%2C-10%2C50%2C42%2F%281%2B2%28.25%5Ex%29%29%2C14%2C28%2C42%29