8.8 Logistic Growth Functions

8.8 Logistic Growth Functions P. 517 Hello, my name is Super Power Hero.

General formLogistic 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:

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

Lets look at Example #5 p.519 • We’ll use the calculator to model a Logistic Growth Function.

Assignment

8.8 Logistic Growth Functions