8-3 Growing, Growing, Growing - Concepts and Explanations
Exponential Growth
An exponential pattern of change can often be recognized in a verbal description of a situation or in the pattern of change in a table of (x, y) values.
The exponential growth in rewards for good-work days in the example can be represented in a graph. The increasing rate of growth is reflected in the upward curve of the plotted points.
Example
![Example graph of exponential growth](https://connectedmath.msu.edu/sites/_connectedMath/assets/File/Families/Images/8_3-exponential-growth.png)
Suppose a reward is offered for days of good work. At the start, 1¢ is put in a party fund. On the first good-work day, 2. is added; on the second good-work day, 4. is added; and on each succeeding good-work day, the reward is doubled. How much money is added on the eighth good-work day?
Good-Work Day | Reward (cents) |
---|---|
0 (start) | 1 |
1 | 2 |
2 | 4 |
3 | 8 |
4 | 16 |
5 | 32 |
6 | 64 |
7 | 128 |
8 |
Growth Factor
A constant factor can be obtained by dividing each successive y-value by the previous y-value. This ratio is called the growth factor of the pattern.
Example
For each good-work day, the reward doubles. You multiply the previous award by 2 to get the new reward. This constant factor can also be obtained by dividing successive y-values: 2/1 = 2, 4/2 = 2, etc
Exponential Equation
Examining the growth pattern leads to a generalization that can be expressed as an equation.
An exponential growth pattern y = a(b)x may increase slowly at first but grows at an increasing rate because its growth is multiplicative. The growth factor is b.
Example
Day | Calculation | Reward (cents) |
---|---|---|
0 | 1 | 1 |
1 | 1 x 2 = 21 | 2 |
2 | 1 x 2 x 2 = 22 | 4 |
3 | 1 x 2 x 2 x 2 = 23 | 8 |
: | : | : |
6 | 1 x 2 x 2 x 2 x 2 x 2 x 2 = 26 | 64 |
: | : | : |
n | 1 x 2 x 2 x ... x 2 = 2n | 2n |
On the nth day, the reward R will be R = 1 x 2n. Because the independent variable in this pattern appears as an exponent, the growth pattern is called exponential. The growth factor is the base 2. The exponent n tells the number of times the 2 is a factor.
Exponential Decay
Exponential models describe patterns in which the value decreases. Decay factors result in decreasing relationships because they are less than 1.
Example
![Graph of Exponential Decay](https://connectedmath.msu.edu/sites/_connectedMath/assets/File/Families/Images/8_3-exponential-decay.png)
Rules of Exponents
Students begin to develop understanding for the rules of exponents by examining patterns in a powers table for the first 10 whole numbers.
Example
By examining the multiplicative structure of the bases:
82 = (2 x 2 x 2)2 = (23)2 = 26; the general pattern is (bm)n = bmn
9 x 27 = 243 or 32 x 33 = 35; in general, (bm)(bn) = bm+n
4 x 25 = 22 x 52 = (2 x 5)2 = 102 = 100; in general, (ambm) = (ab)m
Similar explorations lead to the rule am/an = am-n.