BLOCK 2 ~ EXPONENTIAL FUNCTIONS

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BLOCK 2 ~ EXPONENTIAL FUNCTIONS TIC-TAC-TOE Looking Backwards

Write a story where the character(s) encounter exponential

finding similarities and differences. Which Bank?

Create a display to show why a bank should be chosen.

interest formula to answer realworld questions.

1 BLOCK 2 ~ EXPONENTIAL FUNCTIONS TIC-TAC-TOE Looking Backwards Recursion Mix-Up Story Time Use exponential functions to look into the past to answer questions. Write arithmetic and geometric recursive routines. Write a story where the character(s) encounter exponential situations. See page 73 for details. See page 51 for details. See page 72 for details. How-To Brochure Create a brochure on how to write and graph exponential functions. Exponential vs. Linear Functions Compare exponential and linear functions by finding similarities and differences. Which Bank? Examine interest rates from different banks. Create a display to show why a bank should be chosen. See page 61 for details. See page 61 for details. See page 67 for details. Compound Interest Real-World Data Asymptotes Use the compound interest formula to answer realworld questions. Create a scatter plot of realworld data that forms an exponential curve. Find an equation of best fit. Research functions that have asymptotes and create a display of these functions. See page 68 for details. See page 62 for details. See page 62 for details. Block 2 ~ Exponential Functions ~ Tic-Tac-Toe 45

2 Tic-Tac-Toe ~ R ecu rsion M i x-up Recursive routines have a start value and a repeated operation. Recursive routines can be arithmetic or geometric. Arithmetic recursive routines use either addition or subtraction as their repeated operation. Geometric recursive routines use a constant multiplier as their repeated operation. Determine the type of recursive routine that is represented in each problem (arithmetic or geometric). Give the start value and repeated operation for each situation. Find the value for the situation after 10 days. 1. Jan has $50 in an envelope. Each day she puts an additional $20 into the envelope. 2. Kurt ran 3 blocks today. Each day after, Kurt runs twice as many blocks as the previous day. 3. A virus has spread to 400 people. Each day, the virus spreads to an additional 8 people. 4. A swimming pool loses 4.5 liters of water each day. It holds 5,000 liters of water when full. 5. There are 10 spiders living in a barn. Each day, there are 1.9 times the number of spiders in the barn compared to the previous day. Write an application situation that could be modeled by each of the following recursive routines. 6. Start Value = Start Value = 0.5 Repeated Operation = + 6 Repeated Operation = 4 8. Start Value = 20, Start Value = 48 Repeated Operation = 1.2 Repeated Operation = 1 _ 2 Lesson 2.1 ~ Recursive Routines 51

Tic-Tac-Toe ~ How-To Brochure Create a brochure that would help teach a fellow student how to write and graph exponential functions. Include step by step procedures as well as examples.

Tic-Tac-Toe ~ E x pone n ti a l v s. L i near F unctions Linear functions can be written using slope-intercept form, y = mx + b. Exponential functions can be written in the form y = bm x.

3 Tic-Tac-Toe ~ How-To Brochure Create a brochure that would help teach a fellow student how to write and graph exponential functions. Include step by step procedures as well as examples. Include at least one real-world example in your brochure. The brochure should be typed or clearly printed. Use color in your brochure to make important facts stand out. Tic-Tac-Toe ~ E x pone n ti a l v s. L i near F unctions Linear functions can be written using slope-intercept form, y = mx + b. Exponential functions can be written in the form y = bm x. Create a poster showing the similarities and differences of linear and exponential functions. Compare their equations, graphs and real-world situations. Include graphs or other visual displays when helpful. Include an example of each that uses the same b and m values. Lesson 2.3 ~ Graphing Exponential Functions 61

4 Tic-Tac-Toe ~ R e a l-wor l d Data Scatter plots are used to show if there is a correlation between two items. In the past you may have worked with data that shows a linear trend. There are times when real-world situations create data points that can be represented by exponential growth or decay. 1. Use the data in the table below. Determine which item will be represented by the variable x and which item will be represented by the variable y. Explain how you made your decision. Bug Population Days Display your data in a scatter plot drawn on graph paper. 3. Draw a curve that best fits the data. Explain how you chose where to draw the curve. 4. Find an exponential function that approximately fits the curve drawn in #3. Graph this function on the same coordinate plane in a different color. 5. Use your function to predict the size of the bug population after 20 days. Tic-Tac-Toe ~ A sy m ptote s Look up the term asymptote. What types of functions have asymptotes? Create a display/poster that shows at least four different functions with asymptotes. On your poster you should include the following: Definition of asymptote. A minimum of four types of functions that have asymptotes. Sample graphs and equations for each of your functions. 62 Lesson 2.3 ~ Graphing Exponential Functions

Imagine that you have $1,000 to deposit in a savings account. Which bank would you choose?

5 Tic-Tac-Toe ~ Wh ich Ba n k? Visit three different banks and ask for information about their savings accounts. Ask for information regarding minimum balances, interest rates, fees and benefits. Create a display showing the information you collected. Imagine that you have $1,000 to deposit in a savings account. Which bank would you choose? Support your choice with reasoning and calculations. NOTE: Most banks give compound interest. See the Tic-Tac-Toe activity on the next page to learn how to calculate compound interest. Lesson 2.4 ~ Growth and Decay 67

6 Tic-Tac-Toe ~ Com pound I n te r e st Compound interest is when you earn interest on your deposit as well as any interest you have earned in the past. Interest can be compounded after different time periods (yearly, monthly, daily, etc). Compound interest is the most common type of payment you will earn when depositing your money in a bank account. The formula below can be used to calculate the future value of your account. P = C (1 + n r )nt where P = future value C = initial deposit r = annual interest rate (expressed as a decimal, not a percent) n = number of times per year interest is compounded t = number of years invested Use the compound interest formula to solve these real-world situations. Assume each rate given is the annual interest rate. 1. Susie deposited $1,000 in an account that is compounded annually (once per year). She earns 2% interest each year. How much will she have in the account after 5 years? 2. Colt earns 3.5% in an account that is compounded monthly. He deposited $600 four years ago. How much is in the account now? 3. Lamar has $8,000 to deposit in an account for two years. He has three options. Which option should he choose? Support your answer with calculations. Option 1: 4% interest compounded annually Option 2: 3.8% interest compounded every 6 months Option 3: 3.6% interest compounded monthly 4. Jennifer earns 3% interest compounded daily in her savings account. If she deposited $2,000, how much will she have after 10 years (assume 365 days per year)? 5. Rosita deposited $500 in an account that earns 4.8% interest compounded monthly. How much interest will she earn after 6 years? 6. Uma deposits $1,000 into two different accounts. One earns 2% interest compounded daily. The other earns 2.4% compounded monthly. a. Which account will earn more after 4 years? b. How much more will it earn? 68 Lesson 2.4 ~ Growth and Decay

The character(s) in your book should encounter situations that require them to write exponential functions and

7 Tic-Tac-Toe ~ Story Time Exponential functions occur in many real-world situations. Create a children s book that incorporates exponential functions in everyday life. The character(s) in your book should encounter situations that require them to write exponential functions and solve them for a given value in a variety of real-world situations. Your book should have a cover, illustrations and a story line that is appropriate for children. 72 Block 2 ~ Review

8 Tic-Tac-Toe ~ L ook i ng Back wa r ds If a real-world quantity has experienced consistent growth or decay over time, you can use an exponential function to look back on a past value of the quantity. You do this by setting the current value as the start value (b) and using a negative number in the exponent to look back in time. For example, Rita has a car that is currently worth $6,400. It has experienced 7% depreciation over the 5 years she has owned it. How much did she buy it for 5 years ago (to the nearest dollar)? Function: y = 6400(0.93) x Looking Back Five Years: y = 6400(0.93) ⁵ Purchase Price: 6400(0.93) ⁵ $9,200 Solve each problem using an exponential function. Round answers to the nearest dollar. 1. Sarah bought a new car 5 years ago. Each year it has decreased in value by 12%. The car is now worth about $9235. How much did she pay for the car? 2. John knows that his laptop loses 17% of its value each year. If it is worth about $457 now, how much was it worth three years ago? 3. Housing prices in an area have been decreasing by about 8% each year for the last two years. If a home is now worth $380,000, how much was it worth two years ago? 4. In another area, housing prices had been increasing by about 12% each year for the previous four years. If a home was worth $280,000 in 2007, how much was it worth four years earlier? 5. Marla borrowed some money from her parents. Each month Marla s parents make her pay back 5% of what she still owes them. She has been repaying the loan for 7 months and still owes $349. About how much did Marla originally borrow from her parents? 6. When Jared was born, his parents bought him a savings bond that increases in value by 6% each year. Jared is now 9 years old and his savings bond is worth $ a. How much did Jared s parents pay for his savings bond? b. How much will the savings bond be worth when Jared is 18 years old? 7. Jonah s car loses 9% of its value each year. He has owned the car for 4 years and it is currently worth $15,086. a. How much did Jonah s car cost when he bought it? b. How much will the car be worth in 3 years? Block 2 ~ Review 73

Exponential Growth and Decay

Exponential Growth and Decay Identifying Exponential Growth vs Decay A. Exponential Equation: f(x) = Ca x 1. C: COEFFICIENT 2. a: BASE 3. X: EXPONENT B. Exponential Growth 1. When the base is greater than