Intro: Today we’re going to look at a useful family of equations. It models many real-world phenomenon.

Logistic Growth Function:
(a, c, and r are all constants)

Example: Evaluate for each value of x given.

a. f(-3) b. f(0) c. f(2) d. f(4)

Solution: a. 0.0275 b. 10 c. 85.8
d. 99.7
Let’s look at the graph of the function:

Graphs of Logistic Growth Functions

• Asymptotes: Horizontal lines: y = 0 and y = c
  
• y-intercept: ^c/_{(1 + a)}
  
• Domain: All real numbers
  
• Range: 0 < y < c
  
• The graph is increasing from left to right
  
  

• To the left of the inflection point the rate of increase is increasing
  
• To the right of the inflection point the rate of increase is decreasing
  

Solving Logistic Growth Equations

Example: Solve
Solution:
50 = (40)(1 + 10e^-3x)
⁵/₄ = 1 + 10e^-3x (divided out 40, reduced)
¹/₄ = 10e^-3x (subtracted one)
¹/₄₀ = e^-3x (divided by 10)
ln (¹/₄₀) = -3x (natural logged both sides)
-3.689 = -3x (Calculated ln ¹/₄₀)
x = 1.23 (divided by -3)

Your turn: Solve
Solution: x = ln 5 = 1.61

Word Problems (they save lives)

A colony of bacteria B. dendroides is growing in a petri dish. The colony’s area A can be modeled by: where t is the time in days and the area is in square centimeters.
a. Sketch the model
b. What is the maximum area of the colony?
c. Where does the colony’s rate of growth change?
Solution:


c.
After 2.5 days, the rate of growth changes.