Name:___________________
Group Members:___________________
Non-linear Models
We will be looking at 4 types of non-linear functions/models. Linear functions are listed
here for completeness. Please draw examples of each type of function in the boxes.
Linear Function 𝑦 = 𝑚𝑥 + 𝑏
Linear functions arise when
there is a constant rate of change.
Quadratic Function 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
The graph of a quadratic function
is called a parabola.
Exponential Function 𝑦 = 𝐴𝑒 𝑘𝑥
Exponential functions arise when
the growth rate is proportional to how
much there already is (like money).
Power Function 𝑓(𝑥) = 𝑎𝑥 𝑘
Power functions typically arise
when two things tend to grow together,
but not necessarily in a linear manner.
Power functions are different from
exponential functions because the
variable is in the base. Power functions
are often written as roots.
𝐴
Logistic Function 𝑓(𝑥) = 1+𝐵𝑒 −𝑘𝑥
Logistic functions are often used
when there is a natural limit to something.
Modeling with Data
For each set of data/graph, explain what type of model best fits the data.
1. The weight of a child in kilograms t months after birth.
weight (kg)
0
1
2.5
4
11
12
18
23
3.12
4.11
5.16
6.44
10
11.34
12.81
14.29
16
14
12
Weight (kg)
t
10
8
6
4
2
0
0
5
10
15
20
25
Months after birth
Best Type of Function to Use: ______________________
Best-fit Function:_________________________________
2. The population of the world t years after 1800 (according to Wikipedia).
population
(billions)
4
127
160
174
187
199
212 (est.)
1
2
3
4
5
6
7
8
World Population (billions)
year
7
6
5
4
3
2
1
0
0
50
100
150
Year after 1800
Best Type of Function to Use: ______________________
Best-fit Function:_________________________________
200
250
3. A wolf population t years after wolves are reintroduced into a park.
300
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
Population
20
46
59
73
93
129
144
174
189
220
228
237
246
234
238
243
250
Wolf Population
t
200
150
100
50
0
0
5
10
15
20
25
30
35
Years after Reintroduction
Best Type of Function to Use: ______________________
Best-fit Function:_________________________________
4. The height in inches & weight in pounds of some people with BMIs between 18 & 24.
Height
(in)
30
45
70
65
69
51
73
64
42
61
18
78
90
80
70
Height (in)
Weight
(lb)
27
68
158
118
153
85
166
130
60
117
10
202
60
50
40
30
20
10
0
0
50
Best Type of Function to Use: ______________________
Best-fit Function:_________________________________
100
150
Weight (lbs)
200
250
5. The Gross Domestic Product of the Democratic Republic of Congo (formerly Zaire) t
years after 1990.
GDP
($billions)
3
5
7
9
11
13
15
17
19
10.7
5.6
6.1
4.7
4.7
5.7
7.1
10.0
10.6
12
Gross Domestic Product
(billions of US$)
year
10
8
6
4
2
0
0
5
10
15
20
Year after 1990
Best Type of Function to Use: ______________________
Best-fit Function:_________________________________
6. The percentage of workers who were in a union in a given year1 vs the inflation
adjusted average U.S. salary for that year.2 (Year included for reference.)
Real Wages
302.52
311.3
312.94
331.39
309.61
298.87
276.23
270.32
258.34
259.97
265.22
275.62
279.94
30
% of Workers Who Were Unionized
Year
1964
1967
1970
1973
1976
1979
1985
1988
1991
1994
1997
2000
2003
% of
Unionized
Wage &
Salary
Workers
28.3
27.8
27.4
23.5
21.6
23.4
17.4
16.2
15.5
14.9
13.6
12.9
12.4
25
20
15
10
5
0
240
260
280
300
Inflation Adjusted Weekly Earnings (1982 dollars)
Best Type of Function to Use: ______________________
What conclusions can you draw from this data:
1
2
320
http://digitalcommons.ilr.cornell.edu/cgi/viewcontent.cgi?article=1176&context=key_workplace
http://www.workinglife.org/wiki/Wages+and+Benefits%3A+Real+Wages+%281964-2004%29
340