Name ______________ Date ______Section ______ CW

Aim: How do we graph exponential functions?

Do Now: If you could fold a paper 100 times, how many rectangles would you form?

Hint: If you can’t figure it out, follow the instructions below and fill in the chart:1. Fold a sheet of paper in half. Notice that the fold line divides the paper into 2 rectangles. 2. Fold the paper in half again. Now how many rectangles are there?3. Fill in the following table that corresponds to the number of rectangles. 4. Determine how to write this with exponents

Number of Folds Number of Rectangles

Written as exponents

0 1

1 2

2

3

4

100

x

Challenge Question: If you could fold a paper in half 100 times, how tall would the paper be? (note: 500 pieces of paper (1 ream) is about 2 inches tall)

1

Classwork: Fill in the table of values that correspond to the function. Sketch the function in the axis provided. Note: The graph doesn’t need to be perfect, as the numbers will get really big.

Function Table of values sketchy = 3x

Type of function:

_______

x y-2-101234

  y

x

y = x3

Type of function:

_______

x y-2-101234

  y

x

y = 3x

Type of function:

_______

x y-2-101234

 y

x

Questions1. For which function do the y values grow the fastest? Why do you think that is?

2. What does it mean when we say that something is growing “exponentially” ie, the deficit in our economy is growing at exponential rates?

3. Which of the functions that you graphed do you think is an exponential function?

2

Class Notes: Exponential FunctionDefinition/equation Graph of an exponential function

y

x

Domain Range

Practice: Graph each function on the axes provided.

1. g(x) = 2x and f(x) = 2x 2. g(x) = 2x and f(x) = 1.5x

y

x

y

x

3. g(x) = 2x and f(x) = x2

y

x

4. g(x) = 3 2x and f(x) = 2x

y

x

3

Write down any observations you made from the above graphs.

Application Example #1: Suppose you put $100 in a savings account at bank #1. Your friend puts $100 in a savings account at bank #2. Write a function for each equation. Note: In both cases the interest rate is 5%

Bank #1Year Amount ($) 0 1001 1052 1103 1154 1205 125

Bank #2Year Amount ($)0 1001 1052 110.253 115.764 121.555 127.63

1. Graph each function on a graphing calculator (one at a time). (put years in L1 and the amount of investment in L2). Don’t forget to turn stat plots on and press zoom 9.

2. Take the appropriate regression of the function (To graph the regression, select vars, y-vars, function, f1) . What is the equation of the function for bank #1? _____________Check your table and confirm that you are correct. What is the equation of the function for bank #2? _____________Check your table and confirm that you are correct.

3. a. Find the correlation coefficient (make sure your diagnostics is turned on): ______ b. Why do you think this is the correlation coefficient? What does this correlation coefficient tell you about about future predictions based on this function?

4. How much money will each person have if they leave the money in the account for 10 years? 30 years?

5. In both cases, the interest rate is 8%. How is each function different?

Simple Interest Compound Interest

4

Application Ex #2:

Part I: Suppose you invest $1000 when you turn 18 in the stockmarket. The average annual percent return (interest) is approximately 9%.

1. Fill in the following table to determine how much money you will have if you leave the money for 10 years.

year amount of investment0 10001 1000 + 1000(.09) = 1000(1.09) = $1090 2 1090(1.09) = 1188.10345678910

2. Graph:

5

3. Figuring out the equation of the function: Method #1: Try to find a pattern from the table. Hint: The original investment is multiplied by _______________Your guess for the equation: _____________. Test your guess by entering the equation in the y= in the calculator. Check the table to see if you are correct. Method #2: Use the same method we used for example #1 Part II: Questions based on the function:

1. How much will the investment be if you leave it for 20 years?

2. Suppose you leave the investment until you retire at the age of 65. Assuming you are 18, how much money will you have?

3. Challenge: How many years would you have to leave your money at this rate in order to make a million dollars?

4. Super Challenge: Suppose you kept investing another $1000 each year at the same rate. How long would you have to keep investing in order to have a million dollars?

Part III: Answer the following questions:

1. Were you surprised by how quickly $1000 will grow with compound interest? Why or why not?

2. Does this change your views on savings and investing? ________________ Explain:

3. Why is it helpful to know how to write an exponential function to model the various interest examples?

Part IV: Answer the following question: Albert Einstein said “ The most powerful force in the universe is compound interest”. What do you think he meant?

Part V: Compound interest Compounded Period per

yearRates per period

Function given an initial investment of $1000

Amount of money you’d have after 10 years

Annually 1 8%Semi-Annually

2 8%/2 = 4%

Quarterly 4Monthly 12Every 2 years 1/2Every 4 years 1/4

How much money would you need to invest now if you wanted a million dollars by the time you are 65 (8% interest compounded monthly)?

Compound Interest Equation:

Part VI: Practice Problems1. Suppose you have $1000 now and you want to keep it as savings. Instead of putting it in the bank or investing, you decide to put it in an envelope in the back of your sock drawer. If the inflation rate is approximately 3% each year, how much will the $1000 be worth in 10 years? Determine a function to model the above situation.

2. a. Michelle invested $5000.00 in a CD with the interest rate of 4.8%. (The 4.8% is added on once at the end of the CD’s term. If the CD’s term is 2 years, how much interest would she earn after 2 years?b. Suppose she invested it in a money market that had a return rate of 4.8% compounded monthly. How much would she have at the end of 2 years?

3. Jeff has one savings account with the interest rate of 3.3%, and one money market account with the interest rate of 5.1% in a bank. If he deposits $1200.00 to the savings account, and $1800.00 to the money market account, how much money will he have after 6 years?

Part VI. Read the following article and answer the questions at the end:

The Power of Long-term CompoundingIn 1626, Dutchman Peter Minuit purchased the entire island of Manhattan for $24 from the Wappinger Indians. In other words, for what it would cost to order a bagel and café latte at a midtown hotel today, Monsieur Minuit owned the entire Big Apple.

While there are many outside of Gotham that would look at neither as a bargain, our point is to demonstrate the power of compound interest over time. Compound interest has been called the "eighth wonder of the world" and, with the help of the "ninth wonder of the world," the HP 12C, we can calculate whether Peter Minuit got a good deal or not.

Obviously, the key variable to determine the answer is interest rate that we apply to the $24. Or what we could have earned in an alternative investment.

The difference between a 5% return and a 10% return isn’t a simple doubling but a compounding that becomes staggering over time. If the $24 was invested at 5% interest over the past 377 years it would have grown to $2.3 billion today, implying a good price given Rockefeller Center sold for $1.9 billion in 2000.

At a 10% return however, the $24 doesn’t just double the 5% return, or to $4.2 billion, but magnifies it to $97 quadrillion!

At the foundation of our investment philosophy is that over time, share prices are nearly 100% correlated with earnings. Hence, our objective is to identify companies that can grow their earnings at a high and sustainable rate and hold on for the ride.

In the world of investing, few stocks have accomplished the returns of Peter Minuit, yet, consider that Microsoft went from a $500 million market cap company at the time of its IPO to nearly $400 billion today by growing its earnings at approximately a 40% compound annual growth rate over the past 17 years.

The trick, of course, is that it is almost impossible to grow at a rate that high, for that long of a period, as the laws of compounding cause growth to diminish with size. Bearing this out is the fact that there are fewer than 30 companies that managed to grow their earnings in excess of 20% annually during the past 10 years – out of a universe of more than 12,000 companies!

Questions on the article:

1) The $24 grew to billions of dollars in 377 years due to compound interest. Suppose we were dealing with simple interest instead. What would the price of Manhattan be assuming a 5, 7.5, and 10% interest rate?

2) The article says “laws of compounding cause growth to diminish with size”. Why do you think that’s true?

3) Challenge: From the article, is it possible to determine the number of periods per year in which the principle was compounded?