Download 8.7 Exponential and Power Function Models 8.8 Logistics Model

8.7 Exponential and Power
Function Models
8.8 Logistics Model
Writing an Exponential Function
You can write an exponential function, if you have two points
on the graph of the exponential graph.
Procedure:
1. Use the two points to write two equations by substituting
into the equation, y = abx .
2. Take the first equation, solve for a.
3. Substitute the results of the first equation into the second
equation. Solve for b.
4. Write the equation.
Example
Write an exponential function of the form y = abx whose graph
passes through the points (1, 4) and (3, 16).
Solution
4  ab1
4
a
b
4
2a
2
The equation is y = 2(2)x .
16  ab3
4 3
16  b
b
16  4b2
16
 b2
4
4  b2
2b
Finding an Exponential Model
for a Set of Data
x
0
1
2
3
4
y
14.7 13.5 12.9 12.4 11.9
5
6
7
11.4
10.9 10.4 10.0 9.6
The data appears to be an exponential decay.
Procedure with the TI83 calculator.
STAT EDIT - Put the x values in L1.
Put the y values in L2.
STAT CALC – select ExpReg - ENTER
2nd LIST – selected L1 , L2 - ENTER
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9
Result
y = a*b^x
a = 14.29665308
b = .9558971055
So, the exponential model is y = 14.3(0.956)x .
Writing Power Function
If we have two points on the graph, we can find the
equation using the same procedure we used for exponential
functions.
Procedure:
1. Use the two points to write two equations by substituting
into the equation, y = axb.
2. Take the first equation, solve for a.
3. Substitute the results of the first equation into the second
equation. Solve for b.
4. Write the equation.
Power Function Example
• A power function, y = axb passes through the following points:
and (3, 36). Find the equation of the power function
y  ax b
y  ax b
16  a (2)b
Solve for a.
16
a
b
2
36  a (3)b
Substitute for a.
16 b
36  b  3
2
Solve for b.
3
36  16   
2
36  3 
 
16  2 
9 3
 
4 2
b
b
9 3
 
4 2
2
Substitute for b
in the first equation,
b
32  3 
 
2
2 2
(2, 16)
and solve for a.
b
3 3
   
2 2
b2
b
16
a
b
2
16
a
2
2
4a
b
The power function is
y  4 x2
Logistic Growth Functions
The problem with exponential and power growth
functions that both models grow without bounds. Many
growths in real life have a limit to the growth based on
various constraints, such as available resources.
The logistic model is one model that grows to a limit.
This model is used frequently to model the spread of
epidemics such as the Ebola virus or SARS.
The Graph of the Logistic
Growth Function
10
8
6
4
2
-10 -8
-6
-4
-2
-2
-4
-6
-8
-10
2
4
6
8
10
Characteristics
The form of the equation:
c
y
1  ae rx
•The horizontal lines y = 0 and y = c are asymptotes.
•The y-intercepts is
c
1 a
•The domain is all real numbers, and the range is 1 < y <c.
•The graph increases from left to right.
•The point of maximum growth is  ln a , c 

 r

2
•To the left of the maximum growth the rate of growth is
increasing. To the right of the maximum growth the rate of
growth is decreasing.
Example
You planted a seedling and kept track of its height h (in
centimeters) over time t (in weeks). Use the data to find a
model that gives h as a function of t.
t
0
1
2
3
4
5
6
7
h
5
12
26
39
51
88
94
103 112
Procedure with the TI83 calculator.
STAT EDIT - Put the x values in L1.
Put the y values in L2.
STAT CALC – select Logistic - ENTER
2nd LIST – selected L1 , L2 - ENTER
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Result
y = c/(1 + ae^(-bx))
a = 18.18103216
b = .7317420617
c = 116.8290157
117
So, the logistic function is h 
1  18e 0.73t
Graph of the Example
h
117
1  18e 0.73t
200
180
160
140
120
100
80
60
40
20
-10
-8
-6
-4
-2
2
-20
-40
4
6
8
10
12
14