8.8 Logistic Growth Functions

P. 517

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General formLogistic Growth Functions

• a, c, r are positive real constants

• y = rxae

c1

Evaluating

• f(x) =

• f(-3) =

• f(0) =

xe 291

100

3291

100 e

≈ .0275

0291

100 e

= 100/10 = 10

Graph on your calculator:

xey

1

1

Graph on your calculator:

xey

251

10

Graph on your calculator:

xey

2101

5

• 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.

Increasing growth rate

Decreasing growth rate

Point of maximumGrowth where the graphSwitches from growthTo 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

a

c

1

rxae

cy

1

, 2

,ln

c

r

a

Graph

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

xey

5.21

6

(0,2)

Your turn! Graph:

xey

251

3

•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

40101

503

xe

40101

503

xe

Your turn!

• Solve:

• .46 ≈ x

1051

302

xe

Lets look at Example #5 p.519

• We’ll use the calculator to model a Logistic Growth Function.

Assignment