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Notes: Logistic Functions I. Use, Function, and General Graph If Notes: Logistic Functions I. Use, Function, and General Graph If growth begins slowly, then increases rapidly and eventually levels off, the data often can be model by an “S-curve”, or a logistic function. The logistic function describes certain kinds of growth. These functions, like exponential functions, grow quickly at first, but because of restrictions that place limits on the size of the underlying population, eventually grow more slowly and then level off. For real numbers a, b, and c, the function is a logistic function. If a >0, a logistic function increases when b >0 and decreases when b < 0. Logistic Growth Function Equation: x-Intercept: none y-intercept: Horizontal Asymptote: The number, , is called the limiting value or the upper limit of the function because the graph of a logistic growth function will have a horizontal asymptote at y = c. As is clear from the graph above, the characteristic S-shape in the graph of a logistic function shows that initial exponential growth is followed by a period in which growth slows and then levels off, approaching (but never attaining) a maximum upper limit. Logistic functions are good models of biological population growth in species which have grown so large that they are near to saturating their ecosystems, or of the spread of information within societies. They are also common in marketing, where they chart the sales of new products over time; in a different context, they can also describe demand curves: the decline of demand for a product as a function of increasing price can be modeled by a logistic function, as in the figure below. II. Example Problems with logistics Ex 1: The percents of live births to unmarried mothers for selected years 1970-2003 are show in the table below: Year Percent Year Percent 1970 10.7 1990 28.0 1975 14.3 1995 32.2 1980 18.4 2000 33.2 1985 22.0 2003 34.6 a. Find a logistic function that models the data, with the percent and the number of years from 1960. Graphing calculator STAT>Edit: Enter the data in the table. Take note of your column names (L1, L2 ), etc. STAT>Calc>Logistics: make sure the column names are entered/selected. Press enter. Give it a moment to calculate. It will give you something like the following: y=c(1+ae^(-bx)) a=7.2156 b=-.0929 c=39.606 This tells you that the logistics function is: 39.606 푦 = 1 + 7.2156푒−.0929푥 I typically round to 4 decimal places. b. At what point will this data “level off”? ‘c’ is the limiting value. This tells us that the data will level off approximately 78 years after 1960- in 2038. c. What percent of live births to unmarried mothers should we expect this year (2013)? Use your table function. Tips: go to TBLSET, set TblStart to 53 (2013 is 53 years after 1960), set Indpnt to Auto, then go to table. The table tells you that in 2013, about 38% of live births will be to unmarried mothers. Ex 2: Find the logistic regression for the population of Florida since 1800. What is the “limit to growth?” Year Population (millions 1900 0.5 1910 0.8 1920 1 1930 1.5 1940 1.9 1950 2.8 1960 5 1970 6.8 1980 9.7 1990 12.9 2000 16 Credits: Thank you to the following websites: http://www.cs.xu.edu/math/math120/01f/logistic.pdf http://homepages.ius.edu/MEHRINGE/M122/Notes%20Summer%2010/Chapter%205/Section %205.7%20Logisitics%20Functions.htm http://www.pleasanton.k12.ca.us/avhsweb/dipaola/powpts/section3-2Bnotes.pdf .
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