8.8 Exponential Growth

What am I going to learn? Concept of an exponential function Concept of an exponential function Models for exponential growth Models for exponential growth Models for exponential decay Models for exponential decay Meaning of an asymptote Meaning of an asymptote Finding the equation of an exponential function Finding the equation of an exponential function

Recall Independent variable is another name for domain or input, which is typically but not always represented using the variable, x. Independent variable is another name for domain or input, which is typically but not always represented using the variable, x. Dependent variable is another name for range or output, which is typically but not always represented using the variable, y. Dependent variable is another name for range or output, which is typically but not always represented using the variable, y.

What is an exponential function? Obviously, it must have something to do with an exponent! Obviously, it must have something to do with an exponent! An exponential function is a function whose independent variable is an exponent. An exponential function is a function whose independent variable is an exponent.

What does an exponential function look like? Base Exponent and Independent Variable Just some number that’s not 0 Why not 0? Dependent Variable

The Basis of Bases The base of an exponential function carries much of the meaning of the function. The base of an exponential function carries much of the meaning of the function. The base determines exponential growth or decay. The base determines exponential growth or decay. The base is a positive number; however, it cannot be 1. We will return later to the reason behind this part of the definition. The base is a positive number; however, it cannot be 1. We will return later to the reason behind this part of the definition.

Exponential Growth An exponential function models growth whenever its base > 1. (Why?) An exponential function models growth whenever its base > 1. (Why?) If the base b is larger than 1, then b is referred to as the growth factor. If the base b is larger than 1, then b is referred to as the growth factor.

What does Exponential Growth look like? x 2x2x2x2xy ¼ 2 -1 ½ Consider y = 2 x Table of Values: Graph: Cool Fact: All exponential growth functions look like this!

Investigation: Tournament Play The NCAA holds an annual basketball tournament every March. The NCAA holds an annual basketball tournament every March. The top 64 teams in Division I are invited to play each spring. The top 64 teams in Division I are invited to play each spring. When a team loses, it is out of the tournament. When a team loses, it is out of the tournament. Work with a partner close by to you and answer the following questions. Work with a partner close by to you and answer the following questions.

Investigation: Tournament Play Fill in the following chart and then graph the results on a piece of graph paper. Fill in the following chart and then graph the results on a piece of graph paper. Then be prepared to interpret what is happening in the graph. Then be prepared to interpret what is happening in the graph. After round x Number of teams in tournament (y)

8.8 – Exponential Growth Compound interest Compound interest is the money that a bank pays you for putting money in their bank. This interest is paid over an interest period.

8.8 – Exponential Growth To figure Compound Interest, use the general formula. y = a * b y is the balance a is the initial deposit b is the interest rate (Always add 1 to the rate) x is the number of interest periods. x

8.8 – Exponential Growth y = 2000 * means what? 12

8.8 – Exponential Growth Interest can be calculated for periods other than a year. Annually means 1 time a year Semi-annually means 2 times a year Quarterly means 4 times a year Monthly means 12 times a year

8.8 – Exponential Growth You need to calculate your interest and interest periods differently for these values. The interest rate is divided by the periods. Example: 6% quarterly would be

8.8 – Exponential Growth The interest period = number of years x the number of times a year Example 5 years paid monthly would be 5 * 12 or 60

Exponential Decay

An exponential function models decay whenever its 0 < base < 1. (Why?) An exponential function models decay whenever its 0 < base < 1. (Why?) If the base b is between 0 and 1, then b is referred to as the decay factor. If the base b is between 0 and 1, then b is referred to as the decay factor.

What does Exponential Decay look like? Consider y = (½) x Table of Values: x (½) x y -2 ½ -2 4 ½ ½0½0½0½01 1 ½1½1½1½1½ 2 ½2½2½2½2¼ 3 ½3½3½3½31/8 Graph: Cool Fact: All exponential decay functions look like this!

8.8 – Exponential Growth The equation for exponential decay is y = a * b a is the starting value b is the DECAY factor (between 0 and 1) x is the exponent. x

Decay Factor The constant factor that each value in an exponential decay pattern is multiplied by to get the next value. Decay factor = the base in an exponential decay equation, y = a(b x ). Example: y = 15(.25 x ).25 is the decay factor. The decay factor is always less than 1.

To find it in a table, take any y-value and divide it by the previous y-value. Example: xy divided by 80 =.5 20 divided by 40 =.5 10 divided by 20 =.5 The decay factor is.5

Decay Rate Factor to Decay rate - subtract the decay factor from 1. Example: Decay factor is.25 so the decay rate is =.75 or 75%. ALWAYS Decay Factors are ALWAYS less than one (1) They are NOT negative.

Practice Find the Decay Factor and Rate from this table xy )Divide a Y value by the previous value. 2)Repeat with different values. Are they the same? 3)That is your Decay Factor. 4)Convert to a Decay Rate (%) 1)Subtract from 1. 2)Convert to percent.

Find the Equation xy y= 80(.75) x Decay rate is =.25 = 25%

Find the Equation and Decay Rate xy y = 192(.5) x Decay rate is =.5 = 50%

Solve How much is a car worth in 10 years if the value decays at 9% per year? The initial value is $10,000. Equation v = 10,000(.91) n Insert 10 for the variable n v = 10,000(.91) 10 v = 10,000 ( ) v = $

Or Make a Table xy 010, v = 10,000(.91) n Why is the Decay Factor.91 and not.09?

8.8 – Exponential Growth Since 1980, the number of gallons of whole milk each person in the U.S. drinks each year has decreased 4.1% each year. In 1980, each person drank an average of 16.5 gallons of whole milk per year. Write an equation to model the graph How much milk will the average person drink in 2011?

8.8 – Exponential Growth Start with the Exponential Decay formula: y = a * b

Examples Determine if the function represents exponential growth or decay Exponential Growth Exponential Decay Exponential Decay