7.6: Logistic Functions

Explanation:

Logistic functions are characterized by when something grows rapidly but then tapers off because it reaches a maximum or limit. The logistic funciton is a distorted form of the exponential function. A graph of a logistic function is concave up AND concave down. There is a point of inflection between the two stages. There are asymptotes above and below the graph. The high asymptote is labeled "c" and is in the equation for a logistic function. The general forms for a logistic function's equation are:

y=c/(1+a(b^-x)) and y=c/(1+a(e^bx))

The graphs of a logistic function look like this:












when b is greater than 1 ...................................when b is between 0 and 1

To find the value of the point of inflection, you use y=c/2. The higher asymptote is where the graph will stop, and you know that the point of inflection is exactly between the two asymptotes. If you divide c by 2, you'll get the y-value of the point of inflection.

What kind of situations would you use the logistic function for? Populations mostly. Because the logistic functions have a maximum and a minimum, the situations that would most likely be represented with a logistic graph are ones with limits.

Because the growth of a logistic function is NOT consistent, we cannot assign any given pattern to a set of numerical information. If we are given the data table and the asymptote (or carrying capacity) OR "c", then we can determine the equation. "C" is the higher asymptote of the logistic equation, so in most problems they will give you the maximum of whatever the problem is about.

Sample Problem:

A version of The Grove is constructed in Ohio. Because nobody in Ohio has ever been to something like The Grove, all the people rush in to see it and shop there. The numbers of people visiting the outdoor mall increase rapidly at first. At a certain point, however, the numbers level off because there is no longer a rush to go see it. The maximum number of shoppers in one week is 396,000. Given the data table below, find the particular equation for this logistic function using the second and last data points.

x(weeks).............................y(# of shoppers in thousands)

1............................................................71
10........................................................182
15........................................................239
21........................................................281
30........................................................321
50........................................................370
74.........................................................394

How to solve it:
First plug in the numbers of your two points. The two points are (10, 182,000) and
(74, 394,000). "C" is the maximum number of shoppers (396,000)

y=c/1+ab^-x
182,000=396,000/1+ab^-10
394,000=396,000/1+ab^-74

Then multiply each side by the denominator to get rid of the fraction.

182,000+182,000ab^-10=396,000
394,000+394,000ab^-74=396,000

Then subtract the value without variables from both sides.

182,000ab^-10=214,000
394,000ab^-74=2,000

Now divide the top equation by the bottom equation. The a's will cancel out, and you can combine the b's together. We know that when a value with an exponent is divided by another value with an exponent, we can combine them by subtracting the bottom exponent from the top exponent. We have to be extra careful with logistic functions however, because both exponents will be negative! So, -10 - (-74) = 64.

.46193b^64=107

Now isolate b by dividing both sides by .46193 ...

b^64=231.6374

Now get the b by itself by taking both sides to the 1/64 power...

b=1.088

To get the rest of the equation, all you have to do is plug b back into one of the original equations and solve for a.

182,000=396,000/1+a(1.088)^-10...multiply by the denominator

182,000+182,000a(1.088)^-10=396,000...take 1.088 to the -10 power

182,000+182,000a(.4271)=396,000...multiply 182,000 times .4271

182,000+77,726.67a=396,000...subtract 182,000 from 396,000

77,726.67a=214,000...divide both sides by 77,726.67

a=2.753! (if you got a couple answers a little different than mine, that's because I used the whole decimal, but I wrote them on the blog as if I'd rounded them.)

Your equation to fit this data table is: y=396,000/1+(2.753)(1.088)^-x!!!

Other problems that you may come across will usually ask you to find the x or y value that corresponds to a given x or y value (which is not in the table) that they give you. For this kind of problem, all you have to do plug this number in to your equation.

Other web source:

If you need more help, http://www.wmueller.com/precalculus/families/1_80.html is helpful. It explains logistic functions really well.

Landry, you're up next! Good luck!

Ok this is so cool. Stare at the point in the center of the circles for around 15 seconds. Then move your head forward. NEAT!