# 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

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 = 2^{1} |
2 |

2 | 1 x 2 x 2 = 2^{2} |
4 |

3 | 1 x 2 x 2 x 2 = 2^{3} |
8 |

: | : | : |

6 | 1 x 2 x 2 x 2 x 2 x 2 x 2 = 2^{6} |
64 |

: | : | : |

n |
1 x 2 x 2 x ... x 2 = 2^{n} |
2^{n} |

On the nth day, the reward R will be R = 1 x 2* ^{n}*. 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

## 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:

8^{2} = (2 x 2 x 2)^{2} = (2^{3})^{2} = 2^{6}; the general pattern is (b* ^{m}*)

*= b*

^{n}

^{mn}9 x 27 = 243 or 3^{2} x 3^{3} = 3^{5}; in general, (b* ^{m}*)(b

*) = b*

^{n}

^{m+n}4 x 25 = 2^{2} x 5^{2} = (2 x 5)^{2} = 10^{2} = 100; in general, (*a ^{m}b^{m}*) = (

*ab*)

^{m}Similar explorations lead to the rule a* ^{m}/*a

*= a*

^{n}

^{m-n}.