6.6 Eponential Deca eponential deca? What are the characteristics of 1 ACTIVITY: Comparing Tpes of Deca Work with a partner. Describe the pattern of deca for each sequence and graph. Which of the patterns represent eponential deca? Eplain our reasoning. a. 0.0,., 19., 1.7,., b. 0, 7,, 1, 1, 15,, 9, 6,, 0 7.5,.,.7, 1., 0., 0.0 0 0 6 6 0 0 0 0 1 5 6 7 9 0 0 1 5 6 7 9 c. 0.0,.0, 19., 15.,., d. 0.0, 9.7,., 7., 5.,.5, 9., 7.9, 6., 5.0,.0,. 19., 15.,., 5.7, 0.0 0 0 COMMON CORE Eponential Functions In this lesson, ou will identif eponential growth and deca. write, interpret, and graph eponential deca functions. Learning Standards A.SSE.1a A.SSE.1b F.IF.7e 6 0 0 0 1 5 6 7 9 6 0 0 0 1 5 6 7 9 00 Chapter 6 Eponential Equations and Functions
Math Practice Simplif a Situation How can ou organize the given information to simplif this problem? How is the answer affected? ACTIVITY: Describing a Deca Pattern Work with a partner. Newton s Law of Cooling states that when an object at one temperature is eposed to air of another temperature, the difference in the two temperatures drops b the same percent each hour. A forensic pathologist was called to estimate the time of death of a person. At midnight, the bod temperature was 0.5 F and the room temperature was 60 F. One hour later, the bod temperature was 7.5 F. a. B what percent did the difference between the bod temperature and the room temperature drop during the hour? b. Assume that the original bod temperature was 9.6 F. Use the percent decrease found in part (a) to make a table showing the decreases in bod temperature. Use the table to estimate the time of death. Time Temperature ( F) 0 9.6 1 5 6 7 9. IN YOUR OWN WORDS What are the characteristics of eponential deca? How can ou distinguish eponential deca from other deca patterns?. Sketch a graph of the data from the table in Activit. Do the data represent eponential deca? Eplain our reasoning. 5. Suppose the pathologist arrived at 5:0 a.m. What was the bod temperature at 6 a.m.? Use what ou learned about eponential deca to complete Eercises and on page 0. Section 6.6 Eponential Deca 01
6.6 Lesson Lesson Tutorials Ke Vocabular eponential deca, p. 0 eponential deca function, p. 0 Stud Tip Notice that an eponential deca function is of the form = ab, where b is replaced b 1 r and is replaced b t. Eponential deca occurs when a quantit decreases b the same factor over equal intervals of time. Eponential Deca Functions A function of the form = a(1 r) t, where a > 0 and 0 < r < 1, is an eponential deca function. Final amount Initial amount = a(1 r) t Rate of deca (in decimal form) Time Deca factor For eponential growth, the value inside the parentheses is greater than 1 because r is added to 1. For eponential deca, the value inside the parentheses is less than 1 because r is subtracted from 1. EXAMPLE 1 Identifing Eponential Growth and Deca Determine whether each table represents an eponential growth function, an eponential deca function, or neither. a. b. + 1 + 1 + 1 + 1 + 1 + 1 0 70 1 90 0 1 1 As increases b 1, 1 is multiplied b 1. So, the table represents an eponential deca function. 0 1 5 0 0 As increases b 1, is multiplied b. So, the table represents an eponential growth function. Eercises 1 Determine whether the table represents an eponential growth function, an eponential deca function, or neither. 1.. 0 1 1 5 7 6 1 11 1 5 0 Chapter 6 Eponential Equations and Functions
EXAMPLE Interpreting an Eponential Deca Function The function P = 70(0.9) t represents the population P of a town after t ears. B what percent does the population decrease each ear? Use the deca factor 1 r to find the rate of deca. 1 r = 0.9 Write an equation. r = 0.06 Solve for r. So, the population of the town decreases b 6% each ear. Eercises 15 17. The function A = 75 ( ) 9 t represents the area A (in square miles) of a coral reef after t ears. B what percent does the area of the coral reef decrease each ear? EXAMPLE Real-Life Application The value of a car is $1,500. It loses % of its value ever ear. a. Write a function that represents the value (in dollars) of the car after t ears. = a(1 r) t Write eponential deca function. = 1,500(1 0.) t Substitute 1,500 for a and 0. for r. = 1,500(0.) t Simplif. Check = 1,500(0.) t = 1,500(0.) 6 995 b. Graph the function from part (a). Use the graph to estimate the value of the car after 6 ears. From the graph, ou can see that the -value is about,000 when t = 6. So, the value of the car is about $,000 after 6 ears. Value (dollars) Value of a Car,500 0,000 1,500(0.) t 17,500 15,000,500,000 7500 5000 500 0 0 1 5 6 7 9 Year t Eercise. WHAT IF? The car loses 9% of its value ever ear. a. Write a function that represents the value (in dollars) of the car after t ears. b. Graph the function from part (a). Estimate the value of the car after ears. Round our answer to the nearest thousand. Section 6.6 Eponential Deca 0
6.6 Eercises Help with Homework 1. WRITING When does the function = ab represent eponential growth? eponential deca?. VOCABULARY What is the deca factor in the function = a(1 r) t? 9+(-6)= +(-)= +(-9)= 9+(-1)= Describe the pattern of deca for the sequence.., 6,,, 0, 1,, 1,,,. 56, 19, 1,, 1, 60., 5.6,., 5.6, 19., 1. Determine whether the graph represents an eponential growth function, an eponential deca function, or neither. 5. 6. 7. 1 1 1 Determine whether the table represents an eponential growth function, an eponential deca function, or neither. 1. 0 1 9. 1 17 51 15 59 0. 1 11. 6 65 5 5 5 56 6. 6 1. 5 7 9 5 9 6 7776 79,96 1. CAMPER The table shows the value of a camper t ears after it is purchased. a. Determine whether the table represents an eponential growth function, an eponential deca function, or neither. b. What is the value of the camper after 5 ears? t Value 1 $,000 $19,00 $15,60 $, 0 Chapter 6 Eponential Equations and Functions
Write the rate of deca of the function as a percent. 15. = (0.) t. f (t) = 0(0.95) t 17. g(t) = ( Match the eponential function with its graph. 1. = (1.) t 19. h(t) = 6 ( 7 ) t 0. = (0.6) t A. B. 6 C. ) t 6 6 6 6 6 6 6 1. CHOOSE TOOLS When would ou graph an eponential deca function b hand? When would ou use a graphing calculator? Eplain our reasoning.. POPULATION A cit has a population of 50,000. The population is epected to decrease b 1.5% annuall for the net decade. Write a function that represents this situation. Then predict the population in ears.. TIRE PRESSURE At noon on Monda, the air pressure of a tire is pounds per square inch (psi). The tire loses % of its air ever da. The tire pressure monitoring sstem (TPMS) will alert the driver when the tire pressure is less than or equal to psi. On what da of the week will the TPMS alert the driver? Use the trace feature of a graphing calculator to help find the answer.. The graph of an eponential function passes through (, ) ( and, ). a. Do the -values increase or decrease as increases? How do ou know? b. Find the -intercept of the graph. c. Write an eponential function that represents the graph. Write an equation for the nth term of the arithmetic sequence. Then find a 15. (Section 5.6) 5. 9,, 15, 1,... 6., 1, 1,,... 7. 7, 11, 15, 19,.... MULTIPLE CHOICE What is the solution of the linear sstem? (Section.) A (, ) B (, ) 5 = 11 C (, ) D (, ) 5 = 1 Section 6.6 Eponential Deca 05