6.5 Eponential Growth growth? What are the characteristics of eponential ACTIVITY: Comparing Tpes of Growth Work with a partner. Describe the pattern of growth for each sequence and graph. How man of the patterns represent eponential growth? Eplain our reasoning. a.,, 7, 0, 3,, 9, b..0,., 2.0, 2.7, 3., 5., 7.5, 0.5, 22, 25,, 3.,.7,.9 0 0 2 2 0 0 2 3 5 6 7 9 0 0 0 2 3 5 6 7 9 0 c..0,.3, 2.3,.0, 6.3, 9.3, 3.0, d..0,.6, 2., 3.,.7, 6.,.7 7.3, 22.3,.0, 3.3.5, 5.3,.2, 26.6 COMMON CORE Eponential Functions In this lesson, ou will write, interpret, and graph eponential growth functions. Learning Standards A.SSE.a A.SSE.b F.IF.7e 0 2 0 2 0 0 2 3 5 6 7 9 0 0 0 2 3 5 6 7 9 0 29 Chapter 6 Eponential Equations and Functions
Math Practice Consider Similar Problems How can ou use the results from the previous activit to help ou solve this problem? 2 ACTIVITY: Predicting a Future Event Work with a partner. It is estimated that in 72 there were about 00,000 nesting pairs of bald eagles in the United States. B the 960s, this number had dropped to about 500 nesting pairs. This decline was attributed to loss of habitat, loss of pre, hunting, and the use of the pesticide DDT. The 90 Bald Eagle Protection Act prohibited the trapping and killing of the birds. In 967, the bald eagle was declared an endangered species in the United States. With protection, the nesting pair population began to increase, as shown in the graph. Finall, in 07, the bald eagle was removed from the list of endangered and threatened species. Describe the growth pattern shown in the graph. Is it eponential growth? Assume the pattern continues. When will the population return to the levels of the late 700s? Eplain our reasoning. Bald Eagle Nesting Pairs in Lower States Number of nesting pairs,000 0,000 979 9000 000 7000 66 6000 509 5000 000 3399 3000 00 75 000 0 97 92 96 990 99 99 02 06 Year 3. IN YOUR OWN WORDS What are the characteristics of eponential growth? How can ou distinguish eponential growth from other growth patterns?. Which of the following are eamples of eponential growth? Eplain. a. Growth of the balance of a savings account b. Speed of the moon in orbit around Earth c. Height of a ball that is dropped from a height of 00 feet Use what ou learned about eponential growth to complete Eercises 3 and on page 29. Section 6.5 Eponential Growth 295
6.5 Lesson Lesson Tutorials Eponential growth occurs when a quantit increases b the same factor over equal intervals of time. Ke Vocabular eponential growth, p. 296 eponential growth function, p. 296 compound interest, p. 297 Eponential Growth Functions A function of the form = a( + r)t, where a > 0 and r > 0, is an eponential growth function. Stud Tip Initial amount Notice that an eponential growth function is of the form = ab, where b is replaced b + r and is replaced b t. Final amount Rate of growth (in decimal form) = a( + r)t Time Growth factor EXAMPLE Using an Eponential Growth Function The function = 50,000(.)t represents the attendance at a music festival t ears after 0. a. B what percent does the festival attendance increase each ear? Use the growth factor + r to find the rate of growth. + r =. Write an equation. r = 0. Subtract from each side. So, the festival attendance increases b 0% each ear. b. How man people will attend the festival in? Round our answer to the nearest ten thousand. The value t = represents. = 50,000(.)t Write eponential growth function. = 50,000(.) Substitute for t. = 29,65 Use a calculator. About 2,000 people will attend the festival in. Eercises 5 0. The function = 500,000(.5)t represents the number of members of a website t ears after 0. a. B what percent does the website membership increase each ear? b. How man members will there be in? Round our answer to the nearest hundred thousand. 296 Chapter 6 Eponential Equations and Functions
Compound Interest Compound interest is interest earned on the principal and on previousl earned interest. The balance of an account earning compound interest is = P ( + r n ) nt. P = principal (initial amount) r = annual interest rate (in decimal form) t = time (in ears) n = number of times interest is compounded per ear EXAMPLE 2 Writing a Function Stud Tip For interest compounded earl, ou can substitute for n in the formula to get = P( + r) t. You deposit $00 in a savings account that earns 5% annual interest compounded earl. Write a function for the balance after t ears. = P ( + n) r nt Write compound interest formula. = 00 ( + 0.05 ) ()(t) = 00(.05) t Substitute 00 for P, 0.05 for r, and for n. Simplif. EXAMPLE 3 Real-Life Application Balance Saving Mone 0 00(.) t 75 50 5 00 75 00(.05) t 50 25 0 0 2 3 5 6 7 Year t The table shows the balance of a mone market account over time. a. Write a function for the balance after t ears. From the table, ou know the balance increases 0% each ear. = a( + r) t Write eponential growth function. = 00( + 0.) t Substitute 00 for a and 0. for r. = 00(.) t Simplif. Year, t Balance 0 $00 $0 2 $ 3 $33.0 $. 5 $.05 b. Graph the functions from part (a) and Eample 2 in the same coordinate plane. Compare the account balances. The mone market account earns 0% interest each ear and the savings account earns 5% interest each ear. So, the balance of the mone market account increases faster. Eercises 2. You deposit $500 in a savings account that earns % annual interest compounded earl. Write and graph a function that represents the balance (in dollars) after t ears. Section 6.5 Eponential Growth 297
6.5 Eercises Help with Homework. VOCABULARY When does the eponential function = a( + r) t represent an eponential growth function? 2. VOCABULARY The population of a cit grows b 3% each ear. What is the growth factor? 9+(-6)=3 3+(-3)= +(-9)= 9+(-)= Describe the pattern of growth for the sequence. 3..0,.2,.,.7, 2., 2.5, 3.0, 3.6,., 7, 3, 9, 25, 3, 37, 3,.3, 5.2, 6.2 9, 55, 6 Identif the initial amount a and the rate of growth r (as a percent) of the eponential function. Evaluate the function when t = 5. Round our answer to the nearest tenth. 5. = 25(.2) t 6. f (t) = (.05) t 7. d(t) = 500(.07) t. = 75(.0) t 9. g(t) = 6.7(2) t 0. h(t) =. t 2 3 Write and graph a function that represents the situation.. You deposit $00 in an account that earns 7% annual interest compounded earl.. Your $35,000 annual salar increases b % each ear. 3. A population of 20,000 increases b.5% each ear.. Sales of $0,000 increase b 70% each ear. 5. ERROR ANALYSIS The growth rate of a bacteria culture is 50% each hour. Initiall, there are 0 bacteria. Describe and correct the error in finding the number of bacteria in the culture after hours. b(t) = 0(.5) t b() = 0(.5) 256.3 After hours, there are about 256 bacteria in the culture.. INVESTMENT The function = 7500(.0) t represents the value of an investment after t ears. a. What is the initial investment? b. What is the value of the investment after 6 ears? 7. POPULATION The population of a cit has been increasing b 2% annuall. In 00, the population was 35,000. Predict the population of the cit in. Round our answer to the nearest thousand. 29 Chapter 6 Eponential Equations and Functions
Write a function that represents the situation. Find the balance in the account after the given time period.. $00 deposit that earns 5% annual interest compounded quarterl; 5 ears 9. $60 deposit that earns.% annual interest compounded monthl; months. NUMBER SENSE During a flu epidemic, the number of sick people triples ever week. What is the growth rate as a percent? Eplain our reasoning. 2. SAVINGS You deposit $9000 in a savings account that earns 3.6% annual interest compounded monthl. You also save $0 per month in a safe at home. Write a function C(t) = b(t) + h(t), where b(t) represents the balance of our savings account and h(t) represents the amount in our safe after t ears. What does C(t) represent? 22. REASONING The number of concert tickets sold doubles ever hour. After hours, all of the tickets are sold. After how man hours are about one-fourth of the tickets sold? Eplain our reasoning. 23. YOU BE THE TEACHER The balance of a savings account can be modeled b the function b(t) = 5000(.02) t, where t is the time in ears. To model the monthl balance, a student writes ( b(t) = 5000(.02) t = 5000(.02) ) t t = 5000 (.02 ) 5000(.002) t. Is the student correct? Eplain our reasoning. 2. Gordon Moore stated that the number of transistors that can be placed on an integrated circuit will double ever 2 ears. This trend is known as Moore s Law. In 97, the Intel 06 held 29,000 transistors on an integrated circuit. a. Write a function that represents Moore s Law, where t is the number of ears since 97. b. How man transistors could be placed on an integrated circuit in 5? Simplif the epression. (Section 6.2) 25. ( 2 3) 2 26. ( ) 3 27. ( 3 5). MULTIPLE CHOICE The domain of the function = 3 is,, 7, 0, and 3. Which number is not in the range of the function? (Section 5.) A B 0 C 3 D 25 Section 6.5 Eponential Growth 299