Slide 1

Exponential Growth and Exponential Decay Section 8.1 and 8.2

Slide 2

WHAT YOU WILL LEARN: 1.How to graph exponential growth functions. 2.How to graph exponential decay functions.

Slide 3

Exponential Growth This is demonstrated by the classic riddle in which a child is offered two choices for an increasing weekly allowance: the first option begins at 1 cent and doubles each week, while the second option begins at $1 and increases by $1 each week.

Slide 4

Exponential Growth W01234567891011121314151617 1.01.02.04.08.16.32.641.282.565.1210.2 4 20.4 8 40. 96 81. 92 163.84 327.68 655.36 131 0.7 2 2$1$2$3$4$5$6$7$8$9$10$11$12$13$14$15$16$17$18 This is demonstrated by the classic riddle in which a child is offered two choices for an increasing weekly allowance: the first option begins at 1 cent and doubles each week, while the second option begins at $1 and increases by $1 each week. Although the second option, growing at a constant rate of $1/week, pays more in the short run, the first option eventually grows much larger:

Slide 5

Why! Exponential Growth! The equation for option 1 is: y = 2 n where n is the number of weeks. The equation for option 2 is y = 1 + n where n is the number of weeks.

Slide 6

Oh Boy! Vocabulary An exponential function involves the expression b x where the base b is a positive number other than 1. The variable is going to be in the position of the exponent.

Slide 7

Let s Graph an Example Question: Will the graph ever pass below y of 0?

Slide 8

Let s Graph an Example Question: Will the graph ever pass below y of 0? We say that there is an asymptote at y = 0.

Slide 9

Let s Graph an Example Question: Will the graph ever pass below y of 0? We say that there is an asymptote at y = 0. An asymptote is a line that a graph approaches as you move away from the origin.

Slide 10

Try the following on your graphing calculator Group 1:Group 2: How does a in the function affect the graph?

Slide 11

A Definition y = ab x is an exponential growth function. When a is greater than 0 and b is greater than 1.

Slide 12

Graphing Examples Graph

Slide 13

Another Example Graph

Slide 14

Graphing by Translation The generic form of an exponential function is: y = ab x-h + k Where h is movement along the x axis and k is movement along the y axis.

Slide 15

An Example of Graphing by Translation Graph

Slide 16

You Try Graph

Slide 17

Exponential Growth Model We will use the formula: y = a(1 + r) t a is the initial amount, r is the percent increase expressed as a decimal and t is the number of years. The term 1 + r is called the growth factor.

Slide 18

An Example Problem In January 1993, there were about 1,313,000 Internet hosts. During the next five years, the number of hosts increased by about 100% per year. Write a model. How many hosts were there in 1996? Graph the model. When will there be 30 million hosts?

Slide 19

Section 8.2 Exponential Decay These functions will have the form y = ab x where a is greater than zero and b is between 0 and 1. 19

Slide 20

Example 1 20 State whether the function is an exponential growth or exponential decay function.

Slide 21

You Try State whether the function is an exponential decay or growth function.

Slide 22

A Basic Graph A graph of

Slide 23

Graphing Exponential Functions again Graph:

Slide 24

Another Example Graph:

Slide 25

Graphing by Translation The generic form of an exponential function is: y = ab x-h + k Where h is movement along the x axis and k is movement along the y axis.

Slide 26

Graphing by Translation Graph:

Slide 27

An Exponential Decay Word Problem We will use the formula: y = a(1 - r) t (1-r) is called the decay factor.

Slide 28

The Word Problem You buy a new car for $24,000. The value y of the car decreases by 16% each year. 1. Write an exponential decay model for the value of the car. 2. Use the model to estimate the value after 2 years. 3. Graph the model. 4. When will the car have a value of $12,000.

Slide 29

Homework : Page 469, 14-18 even, 19-24 all, 34, 36, 38, 43-45 all Page 477, 12, 16, 18, 19-24 all, 36, 40, 42, 47-49 all