Slide 1 / 175 Slide 2 / 175 Algebra I Algebraic Exponents & Exponential Functions 2015-11-02 www.njctl.org Slide 3 / 175 Slide 4 / 175 Table of Contents Click on the topic to go to that section Exponential Growth Exponential Growth Introduction · Exponential Growth vs. Linear Growth · Introduction Exponential Relationships in Equations, Tables and Graphs · Growth Rates and Growth Factors · Exponential Decay · Rules of Exponents · Return to Table of Contents Slide 5 / 175 Slide 6 / 175 Making Confetti Making Confetti After each cut, the members of the Drama Club count the pieces of The members of the Drama Club need to make confetti confetti and record the results in a table. 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 One Cut Two Cut Three How many pieces of confetti are made after 1 cut? How many pieces of confetti after 4 cuts? How many pieces of confetti are made after 2 cuts? How many pieces of confetti are made after 3 cuts? How may pieces of confetti after 5 cuts?
Slide 7 / 175 Slide 8 / 175 Making Confetti Making Confetti The members of the Drama Club want to predict the number of pieces of confetti after any number of cuts. Suppose the members of the Look at the pattern in the way the number of pieces of confetti Drama Club make 20 changes with each cut. Use your observations to extend your table to cuts. How many pieces of show the number of pieces of confetti for up to 10 cuts. confetti would they have? How many pieces of confetti would they have if they made 30 cuts? Slide 9 / 175 Slide 10 / 175 Making Confetti Making Confetti 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 Cut One Cut Two the three pieces and cut them in thirds. Cut Three They repeat this process of cutting into thirds, creating smaller and smaller pieces of paper. 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? Slide 11 / 175 Slide 12 / 175 Making Confetti Making Confetti How is the process the same when the members cut After each cut, the members of the Drama Club count the the original sheet into halves and when they cut the pieces of confetti and record the results in a table. first sheet into thirds? How is the process different? 3 9 VS How many pieces of confetti are made after: Is there a way to predict how many pieces of confetti 3 cuts? they will have after any number of cuts? 4 cuts? 5 cuts? These problems are an example of exponential growth. 10 cuts?
Slide 13 / 175 Slide 14 / 175 Vocabulary Exponential Form 5 3 In the expression 5 3 : 5 is the Base: The number being repeatedly multiplied 3 is the Exponent 5 3 3 is the Exponent: How many times to multiply the base 5 3 = 5 x 5 x 5 = 125 5 is the Base 5 3 is in Exponent Form 5 x 5 x 5 is in Expanded Form 5 3 = 5 x 5 x 5 = 125 125 is in Standard Form Slide 15 / 175 Slide 16 / 175 Example Common Powers Write each expression in exponential form. An exponent is also called a Power a. 2 x 2 x 2 An exponent of 2 is called a "square" 7 2 is read "7 squared" b. 6 x 6 x 6 x 6 x 6 A power of 3 is called a "cube" c. 9 x 9 x 9 x 9 x 9 x 9 x 9 x 9 5 3 is read "5 cubed" Slide 17 / 175 Slide 18 / 175 Example Check Your Understanding Explain how the meanings of 5 2 , 2 5 and 5(2) differ. Write each expression in standard form. a. 2 7 b. 3 5 c. 1.5 4
Slide 19 / 175 Slide 20 / 175 1 Evaluate 5 3 2 Evaluate 3 5 A 53 A 35 B 15 B 243 C 125 C 15 D 35 D 53 Slide 21 / 175 Slide 22 / 175 3 Evaluate 4 6 Evaluate eight squared 4 Slide 23 / 175 Slide 24 / 175 Evaluate three cubed Evaluate four raised to the seventh power 5 6
Slide 25 / 175 Slide 26 / 175 Exponential Growth The Five Million Dollar Mission vs Linear Growth Adapted from Presentation Created By: Mr. Kanauss Algebra Teacher at Monongahela Middle School Return to Table of Contents Slide 27 / 175 Slide 28 / 175 Let's Imagine... Payroll Options You must decide what payment option you would like before What would your dream job be? beginning your dream job: Any profession Working for any person/company Option 1: You receive $35,000 a day for the next thirty days Write down your dream job at the top of your page 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 You will be working this job for 30 days straight so choose on the third day of work, etc.) wisely!!! Write down the number of the payroll option you prefer next to your dream job at the top of the page Slide 29 / 175 Slide 30 / 175 7 Now, what payment Option would you prefer? Getting Paid A Option 1: $35,000 a day Why did you choose Option 1? B Option 2: $0.01 day 1 and doubles each day after Why did you choose Option 2? Let's see who would get paid more by the end of the 30 days.
Slide 31 / 175 Slide 32 / 175 Option 2 Option 1 30 days x $35,000 a day = $1,050,000 CONGRATULATIONS, you are a MILLIONAIRE!!! 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? Slide 33 / 175 Slide 34 / 175 Week 2 Let's Keep Going Although this is more money than the previous week, this is still a small amount of money for working 7 days. What patterns/trends do you notice with this payment plan? Anyone want to change to Option 1?!? What do you predict will happen by day 30? Slide 35 / 175 Slide 36 / 175 Week 4 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!
Slide 37 / 175 Slide 38 / 175 Two More Days of Pay!!! 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 Although those workers that chose Option 2 got paid $0.01 on Given Interval = Every day day 1 of work, they ended up making significantly more money than the workers that chose Option 1. We can display linear growth in three forms: -Table Why do you think this occurred? -Equation -Graph Slide 39 / 175 Slide 40 / 175 Different Representations: Linear Growth Linear Growth Payment Option 1 pays the same amount of money each day. How does each display-form of the payment option represent the linear growth? Graph Table Day Pay for the Day Answer Day Pay for the Day 1 $35,000 1 $35,000 2 $35,000 2 $35,000 3 $35,000 3 $35,000 4 $35,000 4 $35,000 5 $35,000 5 $35,000 6 $35,000 6 $35,000 7 $35,000 7 $35,000 Equation y = 35000 x y = 35000 x Slide 41 / 175 Slide 42 / 175 Different Representations: Exponential Growth Exponential Growth Payment Option 2 is an example of Exponential Growth. Exponential Growth: Rate of change increases at a Table Graph constantly growing rate Day Pay for the Day 1 $0.01 2 $0.02 Constantly Growing Rate = Doubling previous days pay 3 $0.04 4 $0.08 5 $0.16 6 $0.32 7 $0.64 We can also display the exponential growth in three forms: -Table Equation -Equation -Graph
Slide 43 / 175 Slide 44 / 175 Exponential Growth Answer Payment Option 2 increases the Pay for the Day EACH day How does each form of the payment option represent the exponential growth? Day Pay for the Day 1 $0.01 2 $0.02 3 $0.04 4 $0.08 5 $0.16 6 $0.32 7 $0.64 Slide 45 / 175 Slide 46 / 175 9 What type of growth has an increasing rate of change? 10 Which equation(s) below represents linear growth? (Choose all that apply) A Exponential Growth A y = x 2 B Linear Growth B y = 25x + 3 C y = 25x + 3 2 D y = 2x Slide 47 / 175 Slide 48 / 175 11 Which graph(s) below depicts linear growth? 12 Choose the table(s) below that represents exponential growth. (Choose all that apply) (Choose all that apply) A C A B x y x y Answer 0 0 0 0.00 1 3 1 0.50 2 11 2 1.00 3 27 3 1.50 4 59 4 2.00 5 123 5 2.50 C D B x y D x y 0 1.00 0 25 1 1.50 1 125 2 2.25 2 225 3 3.38 3 325 4 425 4 5.06 5 7.59 5 525
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