Exponential Growth and Exponential Decay

Exponential Growth and Exponential Decay • Section 8.1 and 8.2

What You Will Learn: How to graph exponential growth functions. How to graph exponential decay functions.

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.

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: Exponential Growth

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

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

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

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.

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.

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

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

Graph Graphing Examples

Graph Another Example

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

Graph An Example of Graphing by Translation

Graph You Try

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.

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?

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

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

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

A graph of A Basic Graph

Graph: Graphing Exponential Functions…again

Graph: Another Example

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

Graph: Graphing by Translation

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

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.

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

Exponential Growth and Exponential Decay