Check Skills You ll Need (For help, go to Lesson 1-2.) Evaluate each expression for the given value of x.

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1 A_3eSE_00X 0/6/005 :3 AM Page - Eploring Eponential Models Lesson Preview What You ll Learn To model eponential growth To model eponential deca... And Wh To model a car s depreciation, as in Eample 6 Check Skills You ll Need (For help, go to Lesson -.) Evaluate each epression for the given value of.. for = 3. + for = for = for = 9 5. Q R for = 0 6. for =- New Vocabular eponential function growth factor deca factor asmptote Eponential Growth Interactive lesson includes instant self-check, tutorials, and activities. Need Help? If the value of a function depends on the value of, then is the independent variable. For some data, the best model is a function that uses the independent variable as an eponent. An eponential function is a function with the general form = ab, where is a real number, a 0, b. 0, and b. You can use an eponential function to model growth when b.. When b., b is the growth factor. Eponential Growth ab (0, a) Growth factor b. a. 6 Graphing Eponential Growth Graph =. b. O O Step Make a table of values. Step Graph the coordinates. Connect the points with a smooth curve O Check Understanding Graph each function. a b. See left. a. = () b. = 3 Chapter Eponential and Logarithmic Functions

2 A_3eSE_00X //03 :0 PM Page 3 You can use an eponential function to model population growth. If ou know the rate of increase r, ou can find the growth factor b using the equation b = + r. Real-World Connection Population Refer to the graph. In 000, the annual rate of increase in the U.S. population was about.%. a. Find the growth factor for the U.S. population. b. Suppose the rate of increase continues to be.%. Write a function to model U.S. population growth. 300 U.S. Population Real-World Connection The United States counts its population ever 0 ears. Population (millions) Year SOURCE: U.S. Census Bureau. Go to for a data update. Web Code: agg-0 a. Find the growth factor. b = + r = Substitute.%, or 0.0, for r. =.0 Simplif. b. Write a function. Relate The population increases eponentiall, so use the general form of an eponential function, = ab. Define Let = number of ears after 000. Write Let = the population in millions. = a(.0) = a(.0) 0 To find a, substitute the 000 values:, 0. = a? An number to the zero power equals. = a Simplif. = (.0) Substitute a and b into ab. The function = (.0) models U.S. population growth. Check Understanding a. Predict U.S. population in 05 to the nearest million. about 33 million b. Critical Thinking Eplain wh the model and our prediction ma not be valid for 05. The growth factor ma change. c. Suppose the rate of population increase changes to.%. Write a function to model population growth and use it to predict the 05 population to the nearest million. (.0) ; about 36 million Lesson - Eploring Eponential Models 3

3 A_3eSE_00X //03 :0 PM Page You can write an eponential function from two points on the function s graph. 3 Writing an Eponential Function Check Understanding 3 Eponential Deca Write an eponential function = ab for a graph that includes (, ) and (3, ). = ab Use the general form. = a? b Substitute for and using (, ). = a b Solve for a. = ab Use the general form. = Substitute for and using (3, ) and for a using. b b3 b = b 3 - Division Propert of Eponents = b Simplif. b = Solve for b. Use our equation for a. b Substitute for b. Simplif. =? Substitute for a and for b in ab. The eponential function for a graph that includes (, ) and (3, ) is =?. Write an eponential function = ab for a graph that includes (, ) and (3, 6). 0.5() Real-World Connection On a da in March, the NCAA announces the selection of 6 teams for its annual tournament.. No; the change from one point to the net is not constant. Investigation: Tournament Pla The National Collegiate Athletic Association (NCAA) holds an annual basketball tournament. The top 6 teams in Division I are invited to pla each spring. When a team loses, it is out of the tournament.. How man teams are left in the tournament after the first round of basketball games? 3 teams. a. Cop, complete, and etend the table until onl one team is left. a b. See margin. b. Graph the points from our table on graph paper. 3. How man rounds are plaed in the tournament? 6 rounds After Round 0. Does the graph represent a linear function? Eplain. Number of Teams Left in Tournament () 5. How does the number of teams left in each round compare to the number of teams in the previous round? There are half as man teams as in the previous round Chapter Eponential and Logarithmic Functions

4 A_3eSE_00X //03 :0 PM Page 5 An eponential function can be used to model deca when 0, b,. When b,, b is a deca factor. Eponential Deca ab (0, a) Deca factor b, Analzing a Function Without graphing, determine whether the function = (0.95) represents eponential growth or eponential deca. In = (0.95), b = Since b,, the function represents eponential deca. Check Understanding Without graphing, determine whether each function represents eponential growth or eponential deca. a. = 00(0.) b. = 0.(5) c. = 6Q R eponential deca eponential growth eponential deca Reading Math In the word asmptote, asm means not together. 5 An asmptote value. is a line that a graph approaches as or increases in absolute Graphing Eponential Deca Graph = Q R. Identif the horizontal asmptote. Step Make a table of values. 5a Step Graph the coordinates. Connect the points with a smooth curve. b. O, ,000 O As increases, approaches 0. O The asmptote is the -ais, = 0. Check Understanding 5 Graph each deca function. Identif the horizontal asmptote. For graphs, see left. a. = Q 3 R 0 b. = 00(0.) 0 Lesson - Eploring Eponential Models 5

5 A_3eSE_00X 3/5/03 :3 PM Page 6 Reading Math Depreciate comes from a prefi meaning lower and a root meaning price. 6 Depreciation is the decline in an item s value resulting from age or wear. When an item loses about the same percent of its value each ear, ou can use an eponential function to model the depreciation. Real-World Connection Depreciation The eponential deca graph shows the epected depreciation for a car over four ears. Estimate the value of the car after si ears. The deca factor b equals + r, where r is the annual rate of decrease. The initial value of the car is $0,000. After one ear the value of the car is about $7,000. r = final value initial value initial value Write an equation for r. 7,000 0,000 = 0,000 Substitute. =-0.5 Simplif. b = + r Use r to find b. = + (-0.5) = 0.5 Simplif. Write a function, and then evaluate it for = 6. Relate The value of the car decreases eponentiall; b = 0.5. Define Let = number of ears. Let = value of the car. Write = ab 0,000 = a(0.5) 0 Substitute using (0, 0,000). 0,000 = a Solve for a. = 0,000(0.5) Substitute a and b into ab. = 0,000(0.5) 6 Evaluate for 6. < Simplif. The car s value after si ears will be about $750. Value ($) 0,000 5,000 0,000 5, Years since purchase Check Understanding 6 Eercises Estimate the value of the car after 0 ears. about $3900 EXERCISES Practice and Problem Solving For more practice, see Etra Practice. A Practice b Eample Eample (page ) Eample (page 3) Graph each function.. See margin.. = 6. = 3(0) 3. = 000(). = 9(3) 5. f() = (3) 6. s(t) =.5 t 7. = (5). = 9. Population The world population in 000 was approimatel 6.0 billion. The annual rate of increase was about.6%. a. Find the growth factor for the world population..06 b. Suppose the rate of increase continues to be.6%. Write a function to model world population growth. 6.0(.06), where 0 corresponds to Chapter Eponential and Logarithmic Functions