Table of Contents Click on the topic to go to that section Exponential Growth Introduction Exponential Growth vs. Linear Growth Exponential Relationships in Equations, Tables and Graphs Growth Rates and Growth Factors Exponential Decay Rules of Exponents
Exponential Growth Introduction Return to Table of Contents
Making Confetti The members of the Drama Club need to make confetti to throw after the final act in the play. They start by cutting a sheet of colored paper in half. Then, they stack the two pieces and cut them in half. They stack the resulting four pieces and cut them in half. They repeat this process, creating smaller and smaller pieces of paper. Cut Three Cut One Cut Two How many pieces of confetti are made after 1 cut? How many pieces of confetti are made after 2 cuts? How many pieces of confetti are made after 3 cuts?
After each cut, the members of the Drama Club count the pieces of confetti and record the results in a table. How many pieces of confetti after 4 cuts? How many pieces of confetti after 5 cuts?
The members of the Drama Club want to predict the number of pieces of confetti after any number of cuts. Look at the pattern in the way the number of pieces of confetti changes with each cut. Use your observations to extend your table to show the number of pieces of confetti for up to 10 cuts.
Suppose the members of the Drama Club make 20 cuts. How many pieces of confetti would they have? How many pieces of confetti would they have if they made 30 cuts? 20 cuts = 1,048,576 30 cuts = 1,073,741,824 answer
As opening night quickly approaches, the members of the Drama Club need to speed up the process of making confetti. They decide to cut the sheet of colored paper into thirds instead of cutting it in half. Then, they stack the three pieces and cut them in thirds. They repeat this process of cutting into thirds, creating smaller and smaller pieces of paper.
First Cut Second Cut Third Cut How many pieces of confetti are made after 1 cut? How many pieces of confetti are made after 2 cuts? How many pieces of confetti are made after 3 cuts?
After each cut, the members of the Drama Club count the pieces of confetti and record the results in a table. 3 9 3 cuts = 27 pieces 4 cuts = 81 pieces 5 cuts = 243 pieces 10 cuts = 59,049 pieces How many pieces of confetti are made after: 3 cuts? 4 cuts? 5 cuts? 10 cuts? answers
How is the process the same when the members cut the original sheet into halves and when they cut the first sheet into thirds? How is the process different? Is there a way to predict how many pieces of confetti they will have after any number of cuts? These problems are an example of exponential growth.
Exponential Form 3 is the Exponent 53 5 is the Base 53 = 5 x 5 x 5 = 125
Vocabulary In the expression 53: 5 is the Base: The number being repeatedly multiplied 3 is the Exponent: How many times to multiply the base 53 = 5 x 5 x 5 = 125 53 is in Exponent Form 5 x 5 x 5 is in Expanded Form 125 is in Standard Form
Common Powers An exponent is also called a Power An exponent of 2 is called a "square" 72 is read "7 squared" A power of 3 is called a "cube" 53 is read "5 cubed"
Example Write each expression in exponential form. a. 2 x 2 x 2 a. b. answers b. 6 x 6 x 6 x 6 x 6 c. c. 9 x 9 x 9 x 9 x 9 x 9 x 9 x 9
Example Write each expression in standard form. a. 27 a. 2 x 2 x 2 x 2 x 2 x 2 x 2 b. 3 x 3 x 3 x 3 x 3 answers c. 1.5 x 1.5 x 1.5 b. 35 c. 1.54
Explain how the meanings of 52, 25 and 5(2) differ. : base = 5, 2 = exponent => 5 x 5 : base = 2, exponent = 5 => 2 x 2 x 2 x 2 x 2 : no base/exponent => 5 x 2 answer 5(2)
1 Evaluate A 53 B 15 C 125 C answer D 35
2 Evaluate A 35 B 243 answer B C 15 D 53
3 Evaluate 4096 answer
Evaluate eight squared 4 64 answer
Evaluate three cubed 5 27 answer
Evaluate four raised to the seventh power 6 16,384 answer
Exponential Growth vs Linear Growth Return to Table of Contents
Let's Imagine... What would your dream job be? Any profession Working for any person/company Write down your dream job at the top of your page You will be working this job for 30 days straight so choose wisely!!!
Payroll Options You must decide what payment option you would like before beginning your dream job: Option 1: You receive $35,000 a day for the next thirty days Option 2: You make $0.01 on the first day and then your salary will double every day for the next thirty days (You receive $0.01 on the first day of work, $0.02 on the second day of work, $0.04 on the third day of work, etc.) Write down the number of the payroll option you prefer next to your dream job at the top of the page
7 What payment option did you choose? A Option 1: $35,000 a day B Option 2: $0.01 day 1 and doubles each day after
Getting Paid Why did you choose Option 1? Why did you choose Option 2? Let's see who would get paid more by the end of the 30 days.
Option 1 30 days x $35,000 a day = $1,050,000 CONGRATULATIONS, you are a MILLIONAIRE!!!
Option 2 This means for 7 days worth of work you earned $1.27. If you worked 40 hours in week 1, a typical number of hours for a work week, how much money have you made per hour? Do you want to keep this payroll option? answer
Week 2 Although this is more money than the previous week, this is still a small amount of money for working 7 days. Anyone want to change to Option 1?!?
Let's Keep Going What patterns/trends do you notice with this payment plan? What do you predict will happen by day 30?
8 Now, what payment Option would you prefer? A Option 1: $35,000 a day B Option 2: $0.01 day 1 and doubled each day after
After 30 days, those that chose payment Option 1 will only have $1,050,000. After 4 weeks we are up to $2,684,354.55 for payment Option 2 and we still have two more days to get paid!
Two More Days of Pay!!! Although those workers that chose Option 2 got paid $0.01 on day 1 of work, they ended up making significantly more money than the workers that chose Option 1. Why do you think this occurred?
Linear Growth Payment Option 1 is an example of Linear Growth. Linear Growth: Constant rate of change during a given interval Rate of Change = $35,000 Given Interval = Every day We can display linear growth in three forms: Table Equation Graph
Different Representations: Linear Growth Table Equation y = 35000x Graph
Linear Growth Payment Option 1 pays the same amount of money each day. How does each form of the payment option represent the linear growth? Table: Day increases by 1 and Pay for the Day remains the same ($35,000) Graph: Days increase and Money Earned is a straight Line Equation: Variable is being multiplied by a constant y = 35000x answer
Exponential Growth Payment Option 2 is an example of Exponential Growth. Exponential Growth: Rate of change increases at a constantly growing rate Constantly Growing Rate = Doubling previous days pay We can also display the exponential growth in three forms: Table Equation Graph
Different Representations: Exponential Growth Table Equation Graph
Exponential Growth Payment Option 2 increases the Pay for the Day EACH day How does each form of the payment option represent the exponential growth? Table: Pay for the Day is double the previous amount Graph: Increasing at a growing rate (curve) Equation: Variable is part of the exponent answer
Linear vs Exponential y = 35000x
9 What type of growth has an increasing rate of change? A Exponential Growth B Linear Growth answer A
10 Which equation(s) below represents linear growth? (Choose all that apply) A B y = 25x + 3 C B, C and D answer D y = 2x
11 Which graph(s) below depicts linear growth? (Choose all that apply) B A B and C answer C D
12 Choose the table(s) below that represents exponential growth. (Choose all that apply) B A A and B answer D C
Exponential Relationships in Equations, Tables and Graphs Return to Table of Contents
Related Materials Fish Exponential Growth.pdf Exponential Table "Graph" Equation.pdf There are handouts that can be used along with this section. They are located under the heading labs on the Exponential page of PMI Algebra. Click for link to materials.
By now you have explored exponential growth and seen how it compares to linear growth. We will take a closer look at tables, graphs and patterns found in exponential relationships. Our exploration will lead us to recognize how the starting values and growth factors for exponential relationships are reflected in tables, graphs, and equations.
Equations Exponential Equations can be written in the form: y = a(bx) y-intercept Location where the graph of the equation will intersect the y-axis y = a(bx) Growth Factor The quantity increasing at a growing rate