Graph each function See back of book. 1. y = 3x 2. y = 4x 3. y =-2x Simplify each expression. New Vocabulary exponential function

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1 -7. Plan bjectives To evaluate eponential To graph eponential Eamples Evaluating an Eponential Function Real-World Problem Solving Graphs of Eponential Functions Real-World Problem Solving -7 What You ll Learn To evaluate eponential To graph eponential... And Wh To use an eponential model for a population of rabbits, as in Eample Eponential Functions Check Skills You ll Need G for Help Lessons - and - Graph each function.. See back of book.. =. =. =- Simplif each epression ? 7.? -.? -. 0? 0 New Vocabular eponential function Math Background Eponential can model man naturall occurring phenomena, such as the growth of a colon of bacteria and the deca of radioactive polonium. More Math Background: p. D Part Evaluating Eponential Functions The rules ou wrote in Lesson - to describe geometric sequences, such as A(n) =? n -, are eamples of eponential. Ke Concepts Definition Eponential Function An eponential function is a function in the form = a? b, where a is a nonzero constant, b is greater than 0 and not equal to, and is a real number. Eamples = 0.5? f() =-? 0.5 Lesson Planning and See p. E for a list of the resources that support this lesson. Bell Ringer Practice Check Skills You ll Need For intervention, direct students to: Slope-Intercept Form Lesson -: Eample Etra Skills and Word Problem Practice, Ch. Zero and Negative Eponents Lesson -: Eample Etra Skills and Word Problem Practice, Ch. For: Eponential Functions Activit Use: Interactive Tetbook, -7 You can evaluate an eponential function for given values of the domain to find the corresponding values of the range. EXAMPLE Evaluating an Eponential Function Evaluate each eponential function. a. = 5 for =,, b. t(n) =? n for the domain {-, } Evaluate each eponential function for the domain {-, 0, }. a. = b. f() = 0? 5 c. g() =-?,, 5, 0, 50,, 5 n n t(n) 7 Chapter Eponents and Eponential Functions Special Needs L Have students search for eamples of eponential in, for eample, newspapers and magazines and science tets. Ask volunteers to present and describe their findings, including graphs and. learning stle: verbal Below Level L Ask students what f() and represent in the function f() =.5 in Eample. Help them to see what effect the value of the base,.5, has on the graph of the function. learning stle: visual

2 Real-World Connection Rabbits were brought to Australia in 0. Their numbers increased eponentiall. You can evaluate eponential to solve real-world problems. EXAMPLE Real-World Problem Solving Gridded Response Suppose 0 rabbits are taken to an island. The rabbit population then triples ever half ear. The function f() = 0?, where is the number of half-ear periods, models this situation. How man rabbits would there be after ears? f() = 0? = 0? In ears, there are half ears. Evaluate the function for. = 0? Simplif powers. = 0 Simplif. After two ears, there would be 0 rabbits. 0 / / Suppose 0 animals are taken to an island, and then the population of these animals quadruples ever ear. Use the function f() = 0?. How man animals would there be after ears? 0,0 animals. Teach Guided Instruction Additional Eamples Evaluate each eponential function. a. = for =,,, 7, b. p(q) =? for the domain {-, }, Suppose two mice live in a barn. If the number of mice quadruples ever months, how man mice will be in the barn after ears?,07 Graphing Eponential Functions Here are two graphs that show what eponential generall look like. ( ) ( ) ( ) ( ) To graph an eponential function, make a table of values. Plot the points. Then join the points to form a smooth curve. Additional Eamples Graph =?. See back of book. The function ƒ() =.5 models the increase in size of an image being copied over and over at 5% on a photocopier. Graph the function. See back of book. Dail Notetaking Guide -7 L Dail Notetaking Guide -7 Adapted Instruction L A B C D E A B C D E A B C D E A B C D E 5 A B C D E B C D E Test-Taking Tip rganize our work. Use a column to show our work when evaluating a function. EXAMPLE Graph =?. Graphs of Eponential Functions (, ) a, b a, b 0 0 (0, ) (, ) (, ) Graph each eponential function. a. = 0.5? b. =-0.5? a b. See margin p. 70. Lesson -7 Eponential Functions 0 Closure Ask students to eplain wh = is not an eponential function. The eponent, not the base, must be a variable. Have students eplain how the graphs of = and = differ. The graph of is a u-shape in which the -values increase on both sides of the -ais as ou move awa from the -ais. The graph of is a smooth curve in which the -values increase quickl as ou move to the right side of the -ais and decrease slowl as ou move to the left of the -ais. Advanced Learners L Lead students in a discussion of the graph of = a, when a is greater than or equal to and when a is between 0 and. learning stle: visual English Language Learners ELL Be sure students understand the term eponential function. Ask them to write a function that describes as a function of, and then a function for as an eponential function of. Discuss how the differ. learning stle: verbal

3 . Practice Assignment Guide A B -, 5-, - A B -, 5, C Challenge -50 Test Prep 5-55 Mied Review 5-5 Homework To check students understanding of ke skills and concepts, go over Eercises,,,,. Careers Eercises A financial planner helps people plan how to manage and invest their mone to pa for present and future needs. Investments ma be used to earn mone for a business venture, to bu propert, to save for retirement, or to pa for an large purchase. GPS Enrichment Guided Problem Solving Reteaching Adapted Practice Practice Name Class Date Practice -7 Eponential Functions Complete the table for each eercise.. Investment increases b. The number of animals. The amount of matter.5 times ever 5 r. doubles ever mo. halves ever ear. Value of Number of Amount Investment Animals of Matter Initial $00 5 r $00 0 r $00 5 r $700 0 r 7 5 r 7 Evaluate each function for the domain {, 0,,, }.. = 5. =.. = =?. = 0?. = 5? 5 0. =. = 00? a a. =? b 0 b Graph each function. Initial mo mo 7 mo 7 mo 7 Initial. =. = 5. =.5. = 7 7. = 0? 5. =? 0.5. =? 0. =?. =? a 5 b Evaluate each function rule for the given values.. = 5.5 for =,, and. =?.5 for =,, and 5. =? for =,, and 5 5. = for =,, and. = 0.7 for =,, and 7. =. for =,, and. = 0? 0.5 for = 0, -, and. =. for = -, -, and 0 0. = 00? 0. for = -, -, and. = 5 for = -, -, and 00 g r 00 g r 00 g r 7 L L L L L Pearson Education, Inc. All rights reserved. EXERCISES Practice and Problem Solving A G Practice b Eample for Help Eample (page ) Eample (page ). $0,000; $0, $000; $000 Eample (page ) 70 Chapter Eponents and Eponential Functions You can graph eponential to model real-world situations. EXAMPLE Real-World Problem Solving Photocoping Man photocopiers allow ou to choose how large ou want an image to be. The function f() =.5 models the new size of an image being copied over and over at 50%, where is the number of enlargements. Graph the function..5.5 a. You can also make images that are smaller than the original on a photocopier. The function ƒ() = 0. models the new size of an image being copied over and over at 0%. Graph the function. See margin. b. Critical Thinking Eplain wh the function in Eample models discrete data. It is not possible to have a fractional number of enlargements. Evaluate each function rule for the given value.. f() = for =. g(t) =? t for t =-. = 0? (0.5) for =.5. h(w) = 0.5? w for w = 5. = 50? (0.) for =.5. f() =.? for = = 00? Q R for =- 00. =? Q 5 R for = Finance Suppose an investment of $0,000 doubles in value ever ears. How much is the investment worth after 5 ears? After 5 ears? See left. 0. Finance Suppose an investment of $500 doubles in value ever 5 ears. How much is the investment worth after 0 ears? After 5 ears? See left.. Finance Suppose an investment of $000 doubles in value ever ears. How much is the investment worth after ears? After ears? $,000, $,000 Match each table with the function that models the data.. = A. = C. = B A. B. C..5 (, f()) (,.5).5.5. (,.) (,.) (, 5.) (5, 7.) 7 f() For more eercises, see Etra Skill and Word Problem Practice. 7 Solve each equation.. 5 = 5.? =. = 5.? 5 = 5 70 page a. b. a.

4 . B Eample (page 70) Appl Your Skills , 0.,, 5, 5, 5; increase. 0., 0.,,.5,.5, 5.5; increase 7. 00, 0,, 0., 0.0, 0.00; decrease. 0.5,.5, 5, 0, 0, 0; increase.,,, 0.5, 0.5, 0.5; decrease 0.,,,,, 7 ; decrease. 0.0, 0.,, 0, 00, 000; increase..,., 00, 0,,.7; decrease Match each function rule with the graph of the function. 5. = B. =- A B D 7. = Q R C. = Q R A A. B. C. D. Graph each function.. See margin.. = 0? 0. = 0.?. =?. =. Photocoping Suppose ou are photocoping an image, reducing it to 5% its original size. The function = 0.5 models the size of an image after number of times it is reduced. Graph the function. See margin.. Science A population of 00 insects triples in size ever month. The function = 00? models the population after months. Graph the function. See left above. Evaluate each function for the domain {,, 0,,, }. As the values of the domain increase, do the values of the range increase or decrease? 5. See left. 5. f() = 5. =.5 7. h() = 0.. f() = 5?. = = Q R. g() =? 0. = 00? 0.. Multiple Choice The population of Teas in 000 was about 0.5 million people. The function p(n) = 0.5(.007) n estimates the population where n = 0 corresponds to the ear 000. Which is a reasonable estimate in millions of the population of Teas in 00? B Biolog A certain species of bacteria in a laborator culture begins with GPS 75 cells and doubles in number ever 0 min. a. Cop, complete, and etend the table to find when there will be more than 5,000 bacteria cells. See back of book. (min) Initial Number of 0-min Periods 0 Pattern 75 b. Write a function rule to model the situation. 75?, where is the number of 0-min time periods Number of Bacteria Cells Lesson -7 Eponential Functions 7 Eercises 5 Suggest students review graphs from the lesson and make generalizations about their shapes before doing these eercises. Technolog Tip Eercise Suggest that students write an equation for the problem. Let equal the number of 0-min time periods. Let equal the number of bacteria cells. Students can check their answer to the eercise b using the TABLE function on a graphing calculator. Error Prevention! Eercise Some students ma begin b multipling 00 b 0 and then squaring the product. Remind them that the order of operations is: parentheses, eponents, multiplication and division, addition and subtraction. pages Eercises. 7

5 . Assess & Reteach Lesson Quiz. Evaluate each function rule for the given value. a. = 0.5 for = 0.5 b. ƒ() =? for =-. Suppose an investment of $5000 doubles ever ears. a. How much is the investment worth after ears? $0,000 b. After ears? $0,000. Graph = 0.5?.. Graph =-0.5?. Alternative Assessment Write = on a transparenc and project it with an overhead projector. Give students three seconds to look at the equation and write on their own paper whether the equation is eponential or not eponential. Repeat with various. Cover each function as ou proceed. Include eponential, linear, quadratic, and absolute value. At the end of the activit, uncover the whole list of. Have students compare their answers with those of classmates and determine which are correct. Test Prep G 5c. No; there is no value of for which 0. d. If the base is S, the graph gets steeper as the base increases. If the base is R, the graph gets steeper as the base decreases. 7a. nline Homework Help Visit: PHSchool.com Web Code: ate b. Ever other value is negative. The absolute value of one term is double the previous term. C Challenge Standardized Test Prep Test Prep Multiple Choice 7 Chapter Eponents and Eponential Functions 5. a. Graph =, =, and = (0.5). See margin. b. What point is on each graph? (0, ) c. Does the graph of an eponential function intersect the -ais? Eplain. d. Critical Thinking How does the graph of an eponential function change as the base increases or decreases? c d. See left.. Ecolog In 50 das, a water hacinth can generate 000 offspring (the number of plants is multiplied b 000). a.,000,000,000 plants a. How man hacinth plants could there be after 50 das? b. How man hacinth plants could there be after 00 das?,000,000,000,000 plants 7. a. Make a table of values for the domain {,,,, 5} of the function = (-). See left. b. What pattern do ou see in the outputs? See left. c. Critical Thinking Is = (-) an eponential function? Justif our answer. No; in a? b, b S 0. R 0, so it is not eponential. Which function is greater at the given value?. 5. f(t) 00? t. = 5 or = 5 at =. f(t) = 0? t or f(t) = 00? t at t = 7 0. = or = at =. f() = or f() = 00 at = 0 f() 00. Writing Analze the range of the function f() = 500? using the domain {,,,, 5}. Eplain wh the definition of eponential function includes the restriction that b. {500}; b produces a linear graph.. a. Graphing Calculator Graph the = and =. b. What happens to the graphs between = and =? a b. See margin. c. Critical Thinking How do ou think the graph of = would compare to the graphs of = and =? The graph of is steeper than and. Solve each equation.. = 5. = 7. = 7.? =.? =. 5? - 5 = Suppose (0, ) and (, ) are on the graph of an eponential function. a. Use (0, ) in the general form of an eponential function = a? b to find the value of the constant a. b. Use our answer from part (a) along with (, ) to find the value of the constant b. c. Write a rule for the function.? d. Evaluate the function for =- and =., 5. For the function =-, what is the value of when =-? B A. - B. C. D. 5. Which function contains the points (, ) and (,.75)? G F. = G. =?.5 H. =.5? J. = Which function has the same -intercept as =? A A. = + B. = C. = D. = ( + ) For additional practice with a variet of test item formats: Standardized Test Prep, p. Test-Taking Strategies, p. Test-Taking Strategies with Transparencies 7 pages 70 7 Eercises 5a. (0.5) a. Xmin=- Xma= Ymin=- Yma= b. Between and, the graph of rises faster than the graph of. The graphs intersect at.

6 G Short Response Mied Review for Help Lesson - Lesson ; 50, 50,, ; 57, 70, ;.,.,. 5. ;,, 7 5. A population of 000 doubles in size ever 0 ears. Which equation relates the size of the population to the number of 0-ear periods? H F. = 000? 0 G. = 0? H. = 000? J. =? Between what two integer values of do the graphs of = 0(0.5) and = 0.5? intersect? Show our work. See margin. Find each common ratio. Then find the net three terms in each sequence , 0, 50, 50, c 57. 7, -,, -, c See left , -0., -0., -., c 5. 7, -,, -, c 0. 50, 5,.5, 0.5, c. 7, 7,,, c 0.; 0.05, 0.005, ;, 7,.75 Write an equation for the line that passes through the given point and is parallel to the given line.. = 5 + ; (0, 0) 5. = - ; (0, ) ±. =- + 5; (, 0) ± 5. = ; (, -) 0.. Use this Checkpoint Quiz to check students understanding of the skills and concepts of Lessons -5 through -7. Grab & Go Checkpoint Quiz Checkpoint Quiz Lessons -5 through -7 Real-World Connection About 0 million vehicles cross the George Washington Bridge between New York and New Jerse in a ear. Simplif each epression..... Q t Q Q 5 Q 0m n r 0 t R 5n R R 5 R 0 5m t 0 Determine whether each sequence is arithmetic or geometric. 5.,, 5.5,.75, c. 5, 0, 0, 0, 0, c 7. 5, 0, 5, 0, 5, c geometric geometric arithmetic. Use the sequence -00, 0, -, c. a. What is the first term? 00 b. What is the common ratio? 5 or 0. c. Write a rule for the sequence. A(n) 00? ( 0.) n d. Use our rule to find the fifth and seventh terms in the sequence. 0.; Phsics n the first swing, a pendulum swings through an arc of length 0 cm. n each successive swing, the length of the arc is 5% of the length of the previous swing. a. Write a rule to model this situation. A(n) 0? (0.5) n b. Find the length of the arc on the fifth swing. Round our answer to the nearest millimeter. 0 mm 0. Commuting Refer to the information at the left. a. Write the number of vehicles that crossed the George Washington Bridge in scientific notation..0 0 b. The Port Authorit collected about $ million in tolls from this bridge. Write this number in scientific notation.. 0 c. What was the average toll per vehicle? about $. lesson quiz, PHSchool.com, Web Code: ata-007 Lesson -7 Eponential Functions [] The graphs intersect between and (R equivalent eplanation). [] answer with no work shown 7