Exploring Exponential Growth and Decay Models Sections 8.5 and 8.6

What am I going to learn? Concept of an exponential function Models for exponential growth Models for exponential decay Meaning of an asymptote 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. 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! 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 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.

Exponential Growth 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.

Exponential Growth Models When you deposit money into a bank savings account, the bank pays you interest for using your money. The interest the bank pays you is added into your account, and you earn interest on the interest. This is called compound interest. Compound interest is an Exponential Growth Function. Exponential Growth Function y = a(1 + r)t a = Initial Amount r = Growth Rate 1 + r = Growth Factor t = Time (usually in years)

Example: In 1980 about 2,180,000 U.S. workers worked at home. During the next ten years, the number of workers working at home increased 5% per year. a.Write a model giving the number w (in millions) of workers working at home t years after b.Find the number of workers working at home in a.y = a(1 + r)t a = 2.18 r = 0.05 w = 2.18( )t w = 2.18(1.05)t b.y = a(1 + r)t w = 2.18(1.05)t t = 10 w = 2.18(1.05)10 w = million workers

The Exponential Growth Function works fine if all we need to find the new amount only once during the growth period. But, interest earned in bank accounts is typically computed monthly. The interest earned this month is added to your account and will earn interest next month, and so on. This is called compound interest. We need a new formula to compute compound interest. A = new Amount P = Principal (initial amount) r = interest rate (as a decimal) t = time (in years) n = number of compounding periods/year

Example: You deposit $1,500 in an account that pays 6% annual interest. Find the balance after 5 years if the interest is compounded: a.Quarterly. b.Monthly. P = 1500 r = 0.06 t = 5 a.n = 4 = $ b.n = 12 = $ 

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 top 64 teams in Division I are invited to play each spring. When a team loses, it is out of the tournament. 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. Then be prepared to interpret what is happening in the graph. After round x Number of teams in tournament (y)

Exponential Decay 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.

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!

End Behavior Notice the end behavior of the first graph-exponential growth. Go back and look at your graph. as you move to the right, the graph goes up without bound. as you move to the left, the graph levels off-getting close to but not touching the x-axis (y = 0).

End Behavior Notice the end behavior of the second graph- exponential decay. Go back and look at your graph. as you move to the right, the graph levels off-getting close to but not touching the x-axis (y = 0). as you move to the left, the graph goes up without bound.

Asymptotes One side of each of the graphs appears to flatten out into a horizontal line. An asymptote is a line that a graph approaches but never touches or intersects.

Asymptotes Notice that the left side of the graph gets really close to y = 0 as. We call the line y = 0 an asymptote of the graph. Think about why the curve will never take on a value of zero and will never be negative.

Asymptotes Notice the right side of the graph gets really close to y = 0 as. We call the line y = 0 an asymptote of the graph. Think about why the graph will never take on a value of zero and will never be negative.

Let’s take a second look at the base of an exponential function. (It can be helpful to think about the base as the object that is being multiplied by itself repeatedly.) Why can’t the base be negative? Why can’t the base be zero? Why can’t the base be one?

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

Example 4 Writing an Exponential Function Write an exponential function for a graph that includes (0, 4) and (2, 1). (We’ll write out each step.)

Example 5 Writing an Exponential Function Write an exponential function for a graph that includes (2, 2) and (3, 4). (Do each step on your own. We’ll show the solution step by step.) Use the general form. Substitute using (2, 2). Solve for a. Substitute using (3, 4). Substitute in for a.

Example 5 Writing an Exponential Function Write an exponential function for a graph that includes (2, 2) and (3, 4). Simplify. Backsubstitute to get a. Plug in a and b into the general formula to get equation.

What’s coming up tomorrow? Applications of growth and decay functions using percent increase and decrease Translations of y = ab x The number e Continuously Compounded Interest

Homework Problems Worksheet