15.4 Comparing Linear and Exponential Models

Name Class Date 15.4 Comparing Linear and Exponential Models Essential Question: How can you recognize when to use a linear model or an exponential model? Resource Locker Explore 1 Comparing Constant Change and Constant Percent Change Suppose that you are offered a job that pays you $1000 the first month with a raise every month after that. You can choose a $100 raise or a 10% raise. Which option would you choose? What if the raise were 8%, 6%, or 4%? A Find the monthly salaries for the first three months. Record the results in the table, rounded to the nearest dollar. For the $100 raise, enter 1000 into your graphing calculator, press ENTER, enter +100, press ENTER, and then press ENTER repeatedly. For the 10% raise, enter 1000, press ENTER, enter 1.10, press ENTER, and then press ENTER repeatedly. For the other raises, multiply by 1.08, 1.06, or 1.04. Monthly Salary after Indicated Monthly Raise Month $100 10% 8% 6% 4% 0 $1000 $1000 $1000 $1000 $1000 1 $1100 $1100 $1080 $1060 $1040 2 3 B For each option, find how much the salary changes each month, both in dollars and as a percent of the previous month s salary. Round the each percent to the nearest whole number. Record the values in the table. Change in Salary per Month for Indicated Monthly Raise Interval $100 10% 8% 6% 4% $ % $ % $ % $ % $ % 0 1 100 10 100 10 80 8 60 6 40 4 1 2 100 10 8 6 4 2 3 100 10 8 6 4 Module 15 749 Lesson 4

C Continue the calculations you did in Part A until you find the number of months it takes for each salary with a percent raise to exceed the salary with the $100 raise. Record the number of months in the table below. Number of Months until Salary with Percent Raise Exceeds Salary with $100 Raise 10% 8% 6% 4% 2 Reflect 1. Discussion Compare and contrast the salary changes per month for the raise options. Explain the source of any differences. 2. Discussion Would you choose a constant change per month or a percent increase per month? What would you consider when deciding? Explain your reasoning. Explore 2 Exploring How Linear and Exponential Functions Grow Linear functions change by equal differences, while exponential functions change by equal factors. Now you will explore the proofs of these statements. x 2 - x 1 and x 4 - x 3 represent two intervals in the x-values of a function. A Complete the proof that linear functions grow by equal differences over equal intervals. Given: x 2 - x 1 = x 4 - x 3 ƒ is linear function of the form ƒ (x) = mx + b. Prove: ƒ ( x 2 ) - ƒ ( x 1 ) = ƒ ( x 4 ) - ƒ ( x 3 ) Proof: 1. x 2 - x 1 = x 4 - x 3 Given 2. m ( x 2 - x 1 ) = ( x 4 - x 3 ) Multiplication Property of Equality 3. mx 2 - = mx 4 - Distributive Property 4. mx 2 + b - mx 1 - b = Addition & Subtraction Properties of Equality mx 4 + - mx 3-5. mx 2 + b - ( mx 1 + b) = Distributive property mx 4 + b - ( ) 6. f ( x 2 ) - f ( x 1 ) = ( ) Definition of ƒ (x) Module 15 750 Lesson 4

B Complete the proof that exponential functions grow by equal factors over equal intervals. Given: x 2 - x 1 = x 4 - x 3 Prove: g is an exponential function of the form g (x) = ab x. g ( x _ 2 ) g ( x 1 ) = g (x 4 _) g (x 3 ) Proof: 1. x 2 - x 1 = x 4 - x 3 Given 2. b ( x 2 - x 1 ) = b ( x 4 - x 3 ) If x = y, then b x = b y. 3. _ b x 2 = _ b x 4 x 1 b Quotient of Powers Property 4. 5. _ ab x 2 ab = ab x 4 _ x 1 g ( x _ 2 ) g ( x 1 ) = g (x 4 _) Multiplication Property of Equality Definition of g (x) 3. In the previous proofs, what do x 2 - x 1 and x 4 - x 3 represent? Explain 1 Comparing Linear and Exponential Functions When comparing raises, a fixed dollar increase can be modeled by a linear function and a fixed percent increase can be modeled by an exponential function. Example 1 Compare the two salary plans listed by using a graphing calculator. Will Job B ever have a higher monthly salary than Job A? If so, after how many months will this occur? A Job A: $1000 for the first month with a $100 raise every month thereafter Job B: $1000 for the first month with a 1% raise every month thereafter Write the functions that represent the monthly salaries. Let t represent the number of elapsed months. Job A: S A (t) = 1000 + 100t Job B: S B (t) = 1000 (1.01) t Graph the functions on a calculator using Y 1 for Job A and Y 2 for Job B. Estimate the number of months it takes for the salaries to become equal using the intersect feature of the calculator. At x 364 months, the salaries are equal. Module 15 751 Lesson 4

Go to the estimated intersection point in the table feature. Find the first x-value at which Y 2 exceeds Y 1. Job B will have a higher monthly salary than Job A after 364 months. B Job A: $1000 for the first month with a $200 raise every month thereafter Job B: $1000 for the first month with a 4% raise every month thereafter Write the functions that represent the monthly salaries. Let t represent the number of elapsed months. t Job A: S A (t) = + t Job B: S B (t) = ( ) Graph the functions on a calculator and use this graph to estimate the number of months it takes for the salaries to become equal. At x months, the salaries are equal. Job B will have a higher salary than Job A after months. Reflect 4. In Example 1A, which job offers a monthly salary that reflects a constant change, and which offers a monthly salary that reflects a constant percent change? 5. Describe an exponential increase in terms of multiplication. Your Turn 6. Job A: $2000 for the first month with a $300 raise every month thereafter Job B: $1500 for the first month with a 5% raise every month thereafter Module 15 752 Lesson 4

Explain 2 Choosing between Linear and Exponential Models Both linear equations and exponential equations and their graphs can model real-world situations. Determine whether the dependent variable appears to change by a common difference or a common ratio to select the correct model. A model may not fit real-world data exactly, so differences or factors between successive intervals may not be constant, but may be nearly so. Example 2 Determine whether each situation is better described by an increasing or decreasing function, and whether a linear or exponential regression should be used. Then find a regression equation for each situation by using a graphing calculator. Evaluate the fit. A The size of an elk population is studied each year during a period in which there is an increase in its predator population. Population over Time Year Population, P 0 9739 Change per Interval Difference P (t n ) - P (t n - 1 ) Factor P (t n ) _ P (t n - 1 ) 1 4637 5102 0.48 2 2007 2630 0.43 3 997 1010 0.50 4 458 539 0.46 5 226 232 0.49 The dependent variable is population, and it is decreasing while the number of years is increasing. This means that the function is decreasing. Note that because the factor changes are relatively close to equal while the difference changes are not, an exponential regression model should be used. Perform the exponential regression analysis and evaluate the fit. The r-value suggests a good fit. To draw a residual plot, you can calculate the residuals, enter them in column L3, and make a scatter plot using L1 as the XList and L3 as the YList. The analysis of residuals suggests a good fit. Module 15 753 Lesson 4

B The size of a raccoon population is studied each year during a period in which there is an increase in its predator population. Population over Time Year Population, P 0 190 Change per Interval Difference P (t n ) - P (t n - 1 ) Factor P (t n ) _ P (t n - 1 ) 2 256 66 1.35 4 338 82 1.32 6 451 113 1.33 8 611 160 1.35 10 801 190 1.31 Is the function increasing or decreasing? Explain. Which changes are closer to being equal, the differences or the factors? Which type of regressions should be used? Perform the regression analysis and evaluate the fit. Reflect Note that the r-value suggests a fit. The analysis of residuals suggests a fit. 7. What would the residual plot look like if an exponential regression was not a good fit for a function? Module 15 754 Lesson 4

Your Turn Determine whether this situation is better described by an increasing or decreasing function, and whether a linear or exponential regression should be used. Then find a regression equation. Evaluate the fit. 8. The price of a barrel of oil is recorded each month. Price over Time Year Price, P (dollars) 0 42.00 Change per Interval Difference P ( t n ) - P ( t n - 1 ) Factor _ P ( t n ) P ( t n - 1 ) 1 50.40 8.40 =1.20 2 60.41 10.01 1.20 3 72.58 12.17 1.20 4 87.16 14.58 1.20 5 105.25 18.09 1.21 Elaborate 9. In the long term, which type of raise will guarantee a larger paycheck: a fixed raise or a percentage raise? Image Credits: Greg Lawler/Alamy 10. What type of function is typically represented by a linear function? 11. Essential Question Check-In An exponential growth model is appropriate when consecutive function values appear to be changing by a constant. Module 15 755 Lesson 4

Evaluate: Homework and Practice State whether each situation is best represented by an exponential or linear function. Then write an exponential or linear function for the model and state whether the model is increasing or decreasing. Online Homework Hints and Help Extra Practice 1. Enrollment at a school is initially 454 students and grows by 3% per year. 2. A salesperson initially earns $50,434 dollars per year and receives a yearly raise of $675. 3. A customer borrows $450 at 5% interest compounded annually. 4. A wildlife park has 35 zebras and sends 1 zebra to another wildlife park each year. 5. The value of a house is $546,768 and decreases by 3% each year. 6. The population of a town is 66,666 people and decreases by 160 people each year. 7. A business has a total income of $236,000 and revenues go up by 6.4% per year. Use a graphing calculator to answer each question. 8. Statistics Companies A and B each have 100 employees. If Company A increases its workforce by 31 employees each month and Company B increases its workforce by an average of 10% each month, when will Company B have more employees than Company A? 9. Finance Employees A and B each initially earn $18.00 per hour. If Employee A receives a $1.50 per hour raise each year and Employee B receives a 4% raise each year, when will Employee B make more per hour than Employee A? 10. Finance Account A and B each start out with $400. If Account A earns $45 each year and Account B earns 5% of its value each year, when will Account B have more money than Account A? 11. Finance Stock A starts out with $900 and gains $50 each month. Stock B starts out with $800 and gains 11% each month. When will Stock B be worth more money than Stock A? Module 15 756 Lesson 4

12. Finance Accounts A and B both start out with $800. If Account A earns $110 per year and Account B earns 3% of its value each year, when will Account B have more money than Account A? 13. Finance Two factory workers, A and B, each earn $24.00 per hour. If Employee A receives a $0.75 per hour raise each year and Employee B receives 1.9% raise each year, when will Employee B make more per hour than Employee A? 14. Statistics Two car manufacturers, A and B, each have 500 employees. If Manufacturer A increases its workforce by 15 employees each month and Manufacturer B increases its workforce by 1% each month, when will Manufacturer B have more employees? Image Credits: Vasily Smirnov/Shutterstock 15. Finance Stock A is initially worth $1300 and loses $80 each month. Stock B is initially worth $400 and gains 9.5% each month. When will Stock B be worth more than Stock A? Module 15 757 Lesson 4

Biology Each table shows an animal population s change over time. Determine whether each situation is best described by an increasing or decreasing function and whether a linear or exponential regression should be used. Then find a regression equation for each situation. Evaluate the fit. 16. x y 1 49 2 58 3 70 4 83 5 101 difference y 2 - y 1 factor y 2 y 1 17. x y 1 31 2 32 3 34 4 35 difference y 2 - y 1 factor y 2 y 1 5 37 Module 15 758 Lesson 4

18. x y 1 46 2 61 3 83 4 107 5 143 difference y 2 - y 1 factor y 2 y 1 19. x y 1 22 2 35 3 60 4 104 5 189 difference y 2 - y 1 factor y 2 y 1 Module 15 759 Lesson 4

20. Using the given exponential functions, state a and b. a. y = 3 (4) x b. y = -5 (8) x c. y = 4 (0.6) x d. y = -5 (0.9) x e. y = 2 x 21. Suppose that you are offered a job that pays you $2000 the first month with a raise every month after that. You can choose a $400 raise or a 15% raise. Which option would you choose? What if the raise were 10%, 8%, or 5%? Monthly Salary after Indicated Monthly Raise Month $400 15% 10% 8% 5% 0 $2000 $2000 $2000 $2000 $2000 1 $2400 $2300 $2200 $2160 $2100 2 3 Change in Salary per Month for Indicated Monthly Raise Interval $400 15% 10% 8% 5% $ % $ % $ % $ % $ % 0 1 400 20 300 15 200 10 160 8 100 5 1 2 400 15 10 8 5 2 3 400 15 10 8 5 Number of Months until Salary with Percent Raise Exceeds Salary with $400 Raise 15% 10% 8% 5% 5 Module 15 760 Lesson 4

H.O.T. Focus on Higher Order Thinking 22. Draw Conclusions Liam would like to put $6000 in savings for a 5-year period. Should he choose a simple interest account that pays an interest rate of 5% of the principal (initial amount) each year or a compounded interest account that pays an interest rate of 1.5% of the total account value each month? 23. Critical Thinking Why will an exponential growth function always eventually exceed a linear growth function? 24. Explain the Error JoAnn analyzed the following data showing the number of cells in a bacteria culture over time. Time (min) 0 6.9 10.8 13.5 15.7 17.4 Cells 8 16 24 32 40 48 She concluded that since the number of cells showed a constant change and the time did not, neither a linear function nor an exponential function modeled the number of cells over time well. Was she correct? Module 15 761 Lesson 4

Lesson Performance Task Two major cities each have a population of 25,000 people. The population of City A increases by about 150 people per year. The population of City B increases by about 0.5% per year. a. Find the population increase for each city for the first 5 years. Round to the nearest whole number, if necessary. Then compare the changes in the populations of each city per year. b. Will City B ever have a larger population than City A? If so, what year will this occur? Yearly Population Yearly Population Increase Image Credits: Greg Henry/Shutterstock Module 15 762 Lesson 4