8.8 Logistic Growth Functions

Objectives/Assignment Evaluate and graph logistic growth functions Use logistic growth functions Assignment: 17-43 odd

General form Logistic Growth Functions a, c, r are positive real constants y =

Evaluating f(x) = f(-3) = f(0) = ≈ .0275 = 100/10 = 10

Graph on your calculator:

Graph on your calculator:

Graph on your calculator:

From these graphs you can see that a logistic growth function has an upper bound of y=c. Logistic growth functions are used to model real-life quantities whose growth levels off because the rate of growth changes – from an increasing growth rate to a decreasing growth rate.

Decreasing growth rate Increasing growth rate Point of maximum Growth where the graph Switches from growth To decrease.

The graphs of The horizontal lines y=0 & y=c are asymptotes The y intercept is (0, ) The Domain is all reals and the Range is 0<y<c The graph is increasing from left to right To the left of it’s point of maximum growth, the rate of increase is increasing. To the right of it’s point of maximum growth, the rate of increase is decreasing

Graph Asy: y=0, y=6 Y-int: 6/(1+2)=6/3=2 Max growth: (ln2/.5 , 6/2) = (1.4 , 3) (0,2)

Your turn! Graph: Asy: y=0 & y=3 Y-int: (0,1/2) Max growth: (.8, 1.5)

Solving Logistic Growth Functions Solve: 50 = 40(1+10e-3x) 50 = 40 + 400e-3x 10 = 400e-3x .025 = e-3x ln.025 = -3x 1.23 ≈ x

Your turn! Solve: .46 ≈ x