Graphing Exponential Functions

Topic

Exponential Graphs and Exponential Growth

Objectives:

Level One: Students will be able to identify an exponential function from an equation and graph and be able to apply the formula of exponential growth and where it applies in every day life.

Level Two: Students will be able to identify an exponential function from a graph and understand how it will change depending on the coefficient. Students will also be able to understand where the formula of exponential growth comes from and where it applies in every day life.

Level Three: Students will be able to identify and understand the variations of an exponential function. Students will also be able to create the formula of exponential growth, be able to apply it to problems, and see where it applies in every day life.

Materials

Coordinate Plane Transparency

Homework Answers Transparency

Graph paper

Bill Gates Offer Transparency

“Million or Double?” Worksheet

Double Your Money Worksheet

Sources

Motivation

Do Now:

Use a piece of graph paper and graph the following equations. Find the y-intercepts and the change of increase of each equation. Create a table to plot the equations.

1.

2.

3.

Procedure

  1. Begin class by reviewing homework from the previous lesson. The answers will be put up on an overhead. Students will check their answers with the transparencies and compile a list of problems the students had trouble with.
  2. After going over the homework, have the students go over the Do Now. Have a coordinate plane on a transparency so that it is easy to display to the class. Briefly discuss graphing since some students had trouble graphing on the last quiz. Have students discuss how the rate of change differs in #1 and #2 from #3. Explain to the students that #1 and #2 of the Do Now are called Linear Functions and #3 is an example of an Exponential Function.

Does #3 seem change at a constant rate?

If students do not realize that the rate changes by a constant percentage, guide them by asking them to refer back to their table of values. Have the students find the ratio of one value to the next. Have students find the percentage of one value to the next.

How does calculating the rate of change in #3 differ from #1 and #2?

A function is growing exponentiallywhen it increases by the same percent in each unit of time

  1. Present the Bill Gates Offer transparency.
  2. Have the students fill out the ballot,“Million or Double?” Explain both offers completely.

-Option 1: one million dollars.

-Option 2: a cent on the first day, two on the second, with the amount doubling and accumulating each day for one month.

  1. Have students classify themselves into groups by raising their hands, Flat Fee or Pennies. Have students debate and defend which group is right. Then have the students compute the problem as a whole class by filling out Double Your Money Worksheet.

What kind of function is presented for the second option?

  1. Have the students observe the graph that they plotted. Discuss the shape of the graph, the concept of doubling, and the idea of exponential growth. Engage students in a discussion about how the amount of money became enormous so fast, and why, assessing the level of understanding from student responses

Why do you think the amount of money became enormous so fast?

  1. Have students discuss different real life situations that involve exponents.

Where do exponents play a role in every day life?

Be sure students name some of the following:

Population

Growth Rate of Bacteria

Interest rates

  1. Pose a problem:

You deposit $1000 in an account that pays you 8% annual interest compounded yearly. What is the account balance after 6 years?

  1. Have students attempt to solve this problem without giving them a formula. Guide the students by asking them how they can calculate what they will be paid in interest after the first year, the second year, and then the 6th year.

Y1= (1000) + (1000)(0.08) = 1000(1+0.08) = 1000(1.08) = 1,080

Y2= [(1000)(1.08)](1.08) = 1000(1.08)2 = 1116.40

Y3= [(1000)(1.08)(1.08)](1.08) = 1000(1.08)3 =1259.71

Do you notice a pattern? Can we right a general formula for this model?

How can we create a formula to model exponential growth?

Yt = C(1+r)t

C = initial amount

r = growth rate

t = time period

Be sure to explain to students that the time period is affected by the rate.

  1. Have students use the formula to solve the problem.

Y6 = 1000(1+ .08)6 =

  1. Have students solve the next problem on their own.

You deposit $500 in an account that pays you 6% annual interest compounded yearly. What is the account balance after 8 years?

How would the formula change if we were to compound the interest monthly?

  1. Pose another question:

A newly hatched channel catfish typically weighs about 0.3 gram. During the first six weeks of its life, its growth rate is approximately exponential increasing by about 10% each day. Round to the nearest tenth.

6 weeks = 42 days

W42=0.3(1.10)42= 16.4

Therefore the weight is about 16.4 grams.

Students will most likely make mistakes with the term. Because the rate affects the term, we will be computing the term in days, not weeks.

  1. There will most likely be little time available. If students need more examples, tell them to refer to 8.5 in their textbooks. If there is time left, pose a few more examples:

A population of 20 rabbits is released into a wild-life region. The population triples each year for 5 years.

What is the percent of increase each year?

What is the population after 5 years?

Students will most likely make mistakes with the rate. The population triples, therefore common sense says 20(3)t. The population increases by 200% each year, therefore

P5 = 20(1 + 2.00)5= 4860

Closure

Have students respond to the following question:

How does a linear function differ from an exponential function?

Where do you feel that this knowledge will apply to you in the future?

Homework

Pg 480 #1-5, #7-15 odd

You're sitting in math class, minding your own business, when in walks a Bill Gates kind of guy - the real success story of your school. He's made it big, and now he has a job offer for you.

He doesn't give too many details, mumbles something about the possibility of danger. He's going to need you for 30 days, and you'll have to miss school. (Won't that just be too awful?) And you've got to make sure your passport is current. But do you ever sit up at the next thing he says.

You'll have your choice of two payment options:

  1. One cent on the first day, two cents on the second day, and double your salary every day thereafter for the thirty days

or

  1. Exactly $1,000,000
    That's one million dollars!

I jump up out of my seat at that. You've got your man, Bill, right here.

I'll take…