1. Logistic Growth Functions HW-attached worksheet Graph Logistic Functions Determine Key Features of Logistic Functions Solve equations involving Logistic Functions.

2. Warm Up 4 = 12 (3)3x 1 3 = 3 3𝑥 ; 𝑙𝑛 1 3 =𝑙𝑛 3 3𝑥 ; 𝑙𝑛 1 3 =3𝑥𝑙𝑛3
Solve each of the following exponential equations:
4 = 12 (3)3x
1 3 = 3 3𝑥 ; 𝑙𝑛 1 3 =𝑙𝑛 3 3𝑥 ; 𝑙𝑛 1 3 =3𝑥𝑙𝑛3
𝑙𝑛 𝑙𝑛3 =x; −1 3 = x
5 = 10(4)x+1
1 2 = 4 𝑥+1 ; 𝑙𝑛 1 2 =𝑙𝑛 4 𝑥+1 ;
𝑙𝑛 1 2 = 𝑥+1 𝑙𝑛4= 𝑙𝑛 1 2 𝑙𝑛4 =x+1; −3 2 = x

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

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

5. Graph on your calculator:

6. Graph on your calculator:

7. Graph on your calculator:

8. 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.

9. Concave down
Concave up
Inflection point

10. 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

11. 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)

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

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

14. Your turn! Solve: .46 ≈ x 30 = 10(1+5e-2x) 30 = 10 + 50e-2x
ln.4 = -2xlne

15. Using 20th-century U.S. census data, the population of Ohio can be modeled by where p is the population in millions and t is the number of years since Based on this model. a.) What was the population of New York in 1850? b.) What will New York state’s population be in 2015? c.) What is New York’s maximum sustainable population (limit to growth)?