# stats_4_1_2

```4.1 Warm Up
Describe a monotonic function.
 Enter the given data in L1 and L2

Age
10
1
6
15
4
25
8
5
Number of
pets
4
1
2
6
1
10
3
2
Create a scatter plot for the data.
 Determine if a linear model is a good model.
If so what is the equation for the LSRL, If not
what would you do to transform the data.

AP Statistics, Section 4.1
1
Section 4.1
Exponential
Modeling
AP Statistics
Growth and Decay
Linear growth increases by a fixed amount
in each equal time period
 Exponential growth increase by a fixed
percentage of the previous total.

AP Statistics, Section 4.1
3
Linear Growth
X
Y
Difference
0
5
10-5=5
1
10
15-10=5
2
15
20-15=5
3
20
25-20=5
4
25
30-25=5
5
30
35-30=5
6
35
40-35=5
7
40
AP Statistics, Section 4.1
4
Linear Growth
3
3
3
4
4
4
X
Y
Difference
0
5
10-5=5
1
10
25-10=15
4
25
40-25=15
7
40
55-40=15
10
55
75-55=20
14
75
95-75=20
18
95
115-95=20
22
115
*Thus the original date is approximately linear.
Each
chunk has
the same
difference
5
Exponential Growth

is interesting, but what we really want is a
linear model
AP Statistics, Section 4.1
6
Exponential Growth
X
Y
Difference
0
1
2-1=1
1
2
4-2=2
2
4
8-4=4
3
8
16-8=8
4
16
32-16=16
5
32
64-32=32
6
64
128-64=64
7
128
AP Statistics, Section 4.1
7
Exponential Growth
X
Y
Ratio
0
1
2/1=2
1
2
4/2=2
2
4
8/4=2
3
8
16/8=2
4
16
32/16=2
5
32
64/32=2
6
64
128/64=2
7
128
AP Statistics, Section 4.1
8
Exponential Growth
X
Y
Ratio
0
1
2/1=2
1
2
8/2=4
3
8
32/8=4
5
32
128/32=4
7
128
512/128=4
9
512
4096/512=8
12
4096
32768/4096=8
15
32768
AP Statistics, Section 4.1
Same
ratio for x
values
with the
same gap
9
Exponential?
To check if a curve is truly exponential you
need to look at the ratio between a value
and the previous output.
 Skip non-consecutive values.

AP Statistics, Section 4.1
10
Year
Subscribers Ratios
Log(Sub)
1990
5,283
3.722880611
1993
16,009
4.204364205
1994
24,134
1.507527
4.382629308
1995
33,786
1.399934
4.528736778
1996
44,043
1.303587
4.643876893
1997
55,312
1.255864
4.742819362
1998
69,209
1.251247
4.840162574
1999
86,047
1.243292
4.934735733
AP Statistics, Section 4.1
11
100,000
90,000
80,000
70,000
60,000
50,000
40,000
30,000
20,000
10,000
1988 1990 1992 1994 1996 1998 2000
AP Statistics, Section 4.1
12
Year
Subscribers
Means 50.7%
growth in
Ratiossubscribers
Log(Sub)
that
year.
Log will undo
exponentiation
1990
5,283
3.722880611
1993
16,009
1994
24,134
1.507527
4.382629308
1995
33,786
1.399934
4.528736778
1996
44,043
1.303587
4.643876893
1997
55,312
1.255864
4.742819362
1998
69,209
1.251247
4.840162574
1999
86,047
1.243292
4.934735733
4.204364205
AP Statistics, Section 4.1
13
AP Statistics, Section 4.1
14
6
5
4
3
2
1
0
1988
1990
1992
1994
1996
AP Statistics, Section 4.1
1998
2000
15
Algebraic
Properties of
Logarithms.
16
AP Statistics, Section 4.1
Example

Enter in list 1
 0,1,2,3,4,5

Enter in list 2
 1,2,4,8,16,32

Enter in list 3
 Log(L2)

See results
 Graph

Calculate r for L1 and L2
 Graph

L1 and L2
L1 and L3
Calculate r for L1 and L3
17
Assignment

Exercises: 4.6, 4.8, 4.9, 4.11, 4.19, 4.21
AP Statistics, Section 4.1
18
```