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LESSON 13.1 Name Using Graphs and Properties to Solve Equations with Exponents A1.9.D …graph exponential functions that model growth...in...real-world problems. Also A1.9.B, A1.9.C In previous lessons, variables have been raised to rational exponents and you have seen how to simplify and solve equations containing these expressions. How do you solve an equation with a rational number raised to a variable? In x certain cases, this is not a difficult task. If 2 x = 4 it is easy to see that x = 2 since 2 2 = 4. In other cases, like 3(2) = 96, where would you begin? Let’s find out. Graph exponential functions that model growth and decay and identify key features, including y-intercept and asymptote, in mathematical and real-world problems. Also A1.9.B, A1.9.C Mathematical Processes Solve for 3(2) = 96 for x. x Let ƒ(x) = 3(2) . Complete the table for ƒ(x). x A1.1.C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. x f(x) 1 6 2 3 4 Language Objective 5 6 1.B, 3.E, 3.F, 3.H When the bases are equal, use the Equality of Bases Property. When the bases are not equal, graph each side of the equation as its own function and find the intersection. Solving Exponential Equations Graphically Explore 1 A1.9.D 7 © Houghton Mifflin Harcourt Publishing Company Essential Question: How can you solve equations involving variable exponents? Resource Locker Essential Question: How can you solve equations involving variable exponents? The student is expected to: ENGAGE Date 13.1 Using Graphs and Properties to Solve Equations with Exponents Texas Math Standards Explain to a partner how to use a graph to find the solution to an equation with a variable exponent. Class g(x) 1 96 2 6 96 96 96 96 96 7 96 4 5 12 24 48 96 192 384 x The graphs intersect at point(s): Module 13 (5, 96) 96 72 (4, 48) f(x) 48 (3, 24) 24 (1, 6) 0 -2 Let g(x) = 96. Complete the table for g(x). 3 Using the table of values, graph ƒ(x) on the axes provided. y (2, 12) x 1 2 3 4 5 6 7 8 Using the table, graph g(x) on the same axes as ƒ(x). y 96 72 g(x) (5 , 96) f(x) 48 24 x 0 -2 1 2 3 4 5 6 7 8 (5, 96) This means that ƒ(x) = g(x) when x = 5 . ges EDIT--Chan DO NOT Key=TX-A Correction must be Lesson 1 605 gh “File info” made throu Date Class rties d Prope Graphs an th 13.1 Usinglve Equations wi to So Exponents Name PREVIEW: LESSON PERFORMANCE TASK ion: How Quest Essential A1_MTXESE353879_U5M13L1 605 can you solve h exponential A1.9.D …grap A1.9.C Also A1.9.B, functions that ms. al-world proble ...in...re model growth Graphic Equations 96 for for 3(2) = x x. Solve lete the table (2) . Comp ƒ(x) = 3 Let for ƒ(x). 72 12 24 48 96 192 384 2 3 4 5 6 7 y g Compan Harcour t Publishin 96 n Mifflin 2 © Houghto 3 4 5 6 96 96 96 96 96 7 s interse The graph L1 605 9_U5M13 g(x) 48 x 24 -2 0 96 ) s): (5, 96 ct at point( (5 , 96) f(x) 72 96 1 SE35387 g(x table, graph Using the as ƒ(x). same axes y g(x) x (3, 24) x 24 (2, 12) (1, 6) 8 5 6 7 0 1 2 3 4 ) on the lete the table 96. Comp Let g(x) = for g(x). (4, 48) f(x) 48 -2 A1_MTXE Lesson 13.1 Using the provided. on the axes y (5, 96) 96 6 1 Turn to these pages to find this lesson in the hardcover student edition. table of f(x) x Module 13 605 ally fy and solve In l how to simpli to a variable? x Exponentia ents and you have seennumbe r raised = like 3(2) 1 Solving rational rational expon other cases, Explore on with a raised to 22 = 4. In an equati have been x = 2 since do you solve s, variables sions. Howx 4 it is easy to see that us lesson = In previo these expres task. If 2 containing a difficult equations this is not find out. certain cases, begin? Let’s would you ƒ(x) 96, where values, graph x View the Engage section online. Discuss why a town government might need to know the rate at which the town’s population is growing. Then preview the Lesson Performance Task. HARDCOVER PAGES 449456 Resource Locker ents? le expon ing variab involv equations This means that ƒ(x) 7 8 4 5 6 1 2 3 when x = = g(x) 5 . Lesson 1 605 20/02/14 1:10 AM 20/02/14 1:10 AM Reflect EXPLORE 1 Discussion Consider the function h(x)= -96. Where do ƒ(x) and h(x) intersect? The graphs would not intersect as f(x) is always greater than 0. Raising any positive 1. Solving Exponential Equations Graphically number to a positive exponent yields a positive number. Divide both sides of the equation 3( 2 = 96 by 3 (an algebraic step) and utilize the same method as )x 2. in Explore 1 to graph each side of the equation as a function. The point of intersection would be: (5, 32) . Is this the same point of intersection? Is this the same answer? Can this be done? Elaborate as to why or why not. INTEGRATE TECHNOLOGY It is not the same point of intersection. The y-values of the points are different. They Students can use graphing calculators to solve an exponential equation by the method shown in the Explore activity. Students should enter the appropriate exponential function and constant function, graph both functions, and use the calculator’s intersect feature to find their point of intersection. do represent the same solution because the equations are equivalent by the Division Property of Equality. Solving Exponential Equations Algebraically Explore 2 Recall the example 2 = 4, with the solution x = 2. What about a slightly more complicated equation? Can an x equation like 5 (2) = 160 be solved using algebra? x A Solve 5 (2) = 160 for x. The first step in isolating the term containing the variable x on one side of the equation is to divide each side of the equation by 5 . QUESTIONING STRATEGIES 5( 2 ) x _ _ = 160 5 B When solving an exponential equation of the form ab x = c graphically, what two functions do you graph? the exponential function f(x) = ab x and the horizontal line g(x) = c 5 Simplify. C (2) = 32 x Rewrite the right hand side as a power of 2. 5 x (2) = (2) D Solve. x= 5 3. Discussion The last step of the solution process seems to imply that if b x = b y then x = y. Is this true for all values of b? Justify your answer. No, it is not true. For example, 0 5 = 0 8 but 5 ≠ 8, or 1 7 = 1 958 but 7 ≠ 958. 4. In Reflect 2, we started to solve 3(2) = 96 algebraically. Finish solving for x. x 3(2) = 96 x 2 x = 32 2x = 25 How does graphing these two functions on the same grid help you determine the value of the exponent? The value of the exponent is the x-value of the point of intersection. © Houghton Mifflin Harcourt Publishing Company Reflect EXPLORE 2 Solving Exponential Equations Algebraically x=5 QUESTIONING STRATEGIES Module 13 606 Lesson 1 PROFESSIONAL DEVELOPMENT A1_MTXESE353879_U5M13L1.indd 606 13/11/14 4:09 PM Learning Progressions In this lesson, students continue to build on their understanding of geometric sequences and exponential functions. They learn the Equality of Bases Property, which states that If b > 0 and b ≠ 1, then b x = b y if and only if x = y. They learn to solve equations involving variable exponents either by using the Equality of Bases Property or by graphing. They also begin to model real-world situations using exponential equations, which can then be solved by either method. Work with exponential functions will continue as students learn about exponential growth and decay models and exponential regression. In the equation 4 x = 64, how can you evaluate x? Write 64 as a power of 4. 3 64 = 4 , so x = 3. Assuming that x is an integer in the equation b x = c, what must be true for this method to work? The value of c must be a power of b. Using Graphs and Properties to Solve Equations with Exponents 606 Solving Equations by Equating Exponents Explain 1 EXPLAIN 1 Solving the previous exponential equation for x used the idea that if 2 x = 2 5, then x = 5. This will be a powerful tool for solving exponential equations if it can be generalized to if b x = b y then x = y. However, there are values for which this is clearly not true. For example, 0 7 = 0 3 but 7 ≠ 3. If the values of b are restricted, we get the following property. Solving Equations by Equating Exponents Equality of Bases Property Two powers with the same positive base other than 1 are equal if and only if the exponents are equal. Algebraically, if b > 0 and, b ≠ 1, then b x = b y if and only if x = y. QUESTIONING STRATEGIES x 5· Multiply both sides by _ 2 Simplify. Rewrite the right side as a power of 5. x = 4 Equality of Bases Property. ( ) 5 x=_ 250 2 _ 27 3 ( ) 5 x 250 _ 2 _ 3 27 _=_ 2 Divide both sides by 2 . 2 (_35 ) © Houghton Mifflin Harcourt Publishing Company Discuss with students the limitations on the Equality of Bases Property. Have students give examples to show why the property does not apply when the base is 0, 1, or –1. For example: 05 = 0 8 = 0 but 5 ≠ 8; 4 6 1 20 = 1 99 = 1 but 20 ≠ 99; and (–1) = (–1) = 1 but 4 ≠ 6. 2 ( 5 ) x = 250 _ 5 5·_ 5 2 (5) x = 250 · _ _ 2 5 2 5 x = 625 5x = 54 How does this step compare to isolating a variable on one side of a linear equation? It is done for the same reason. By isolating the power, you have isolated the variable as well. Then you can compare the exponents in the final equivalent expression. INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning Solve by equating exponents and using the Equality of Bases Property. Example 1 In an equation such as 36(2) = 576, what property of equality can you use to isolate 2 x? Explain how. Division Property of Equality; divide both sides by 36. (_35 ) x x 125 =_ 27 = (_53) Simplify. 3 x= 3 Rewrite the right side as a power of 5 _ . 3 Equality of Bases Property. Reflect 5. Suppose while solving an equation algebraically you are confronted with: 5 x = 15 5x = 5 Can you find x using the method in the examples above? No, you cannot. It is not possible because 15 is not a whole number power of 5. AVOID COMMON ERRORS Some students may misread the base b in an expression b x as a coefficient of x and try to divide both sides of the equation by b to isolate the variable. Remind them that when a number is raised to a power, it cannot be treated as a single factor. They must use the properties of equality to isolate b x, then use the Equality of Bases Property to solve for the variable. 607 Lesson 13.1 Module 13 607 Lesson 1 COLLABORATIVE LEARNING A1_MTXESE353879_U5M13L1 607 Peer-to-Peer Activity Have students work in pairs. Have each student write an equation involving a variable exponent in the form b x = c. After students exchange equations, each partner should first decide whether c can be expressed as a whole number power of b. If so, the student should rewrite c as a power of b and solve for x. If not, the student should use a graphing calculator to graph each side of the equation as a separate function and use the intersect feature to find the x-coordinate of the intersection point, which is the solution to the original equation. Have students check each other’s work. 20/02/14 1:10 AM Your Turn EXPLAIN 2 Solve by equating exponents and using the Equality of Bases Property. 6. 2 (3) x = 18 _ 3 7. _3 (_4 ) = _8 _2 (3) = 18 x 3 2 3 3 x x 2 3 Solving a Real-World Exponential Equation by Graphing x _2 ⋅ _3 (_4 ) _3 ⋅ _2 (3) = 18 ∙ _3 2 3 3 x = 27 () x 8 3 _ 4 =_ _ 2 3 3 2 3 = 16 (_43 ) = _ 9 4 4 _ _ (3) = (3) _2 ∙ _8 3 3 QUESTIONING STRATEGIES x 3x = 33 x x=3 If a population grows by 5% each year, by what factor is the population multiplied each year? Explain. 1.05; if the population is p one year, it will be p + 0.05p = 1.05p the next year. 2 x= 2 Explain 2 Solving a Real-World Exponential Equation by Graphing Why is it appropriate to round a prediction involving time to the nearest year? A prediction is usually just an estimate, so rounding is appropriate. Some equations cannot be solved using the method in the previous example because it isn’t possible to write both sides of the equation as a whole number power of the same base. Instead, you can consider the expressions on either side of the equation as the rules for two different functions. You can then solve the original equation in one variable by graphing the two functions. The solution is the input value for the point where the two graphs intersect. Solve by graphing two functions. An animal reserve has 20,000 elk. The population is increasing at a rate of 8% per year. There is concern that food will be scarce when the population has doubled. How long will it take for the population to reach 40,000? Analyze Information Identify the important information. • The starting population is • The ending population is • The growth rate is 20,000. 40,000 . 8% or 0.08 . Formulate a Plan With the given situation and data there is enough information to write and solve an exponential model of the population as a function of time. Write the exponential equation and then solve it using a graphing calculator. Set ƒ(x) = the target population and g(x) = the exponential model . Input Y 1 = ƒ(x) and Y 2 = g(x) into a graphing calculator, graph the functions, and find their intersection . Module 13 608 © Houghton Mifflin Harcourt Publishing Company · Image Credits: © James Prout/Alamy Example 2 Lesson 1 DIFFERENTIATE INSTRUCTION A1_MTXESE353879_U5M13L1.indd 608 13/11/14 10:43 PM Graphic Organizers Have students complete a graphic organizer that shows when to solve an equation involving a variable exponent algebraically and when to solve it graphically. Solving b x = c (where b > 0, b ≠ 1, and c > 0) Value of c Solution Method d c = b for some Algebraic: whole number d. c is not a power of b. b x = b d, so x = d. Graph: intersection of ƒ(x) = b x and g(x) = c Using Graphs and Properties to Solve Equations with Exponents 608 Solve INTEGRATE TECHNOLOGY Write a function P(t) = ab t, where P(t) is the population and t is the number of years since the population was initially measured. When solving exponential equations graphically, have a student demonstrate how to identify the two functions to be graphed, enter them into a graphing calculator, and find the solution by finding the point of intersection. Discuss how to adjust the viewing window so that the graph and the point of intersection are clearly visible. a represents the initial population of elk a = 20,000 b represents the yearly growth rate of the elk population b = 1.08 ( 1.08 ) . t The function is P(t) = 20,000 () To find the time when the population is 40,000, set the function or P t equal to 40,000 and solve for t . ( 1.08 ) . t 40,000 = 20,000 Write functions for the expressions on either side of the equation. 40,000 ƒ(x) = ) ( g(x) = 20,000 1.08 x Using a graphing calculator, set Y 1 = ƒ(x) and Y 2 = g(x). View the graph. Use the intersect feature on the CALC menu to find the intersection of the two graphs. The approximate x-value where the graphs intersect is 9.006468 . Therefore, the population will double in just a little over 9 years. © Houghton Mifflin Harcourt Publishing Company Justify and Evaluate Check the solution by evaluating the function at t = 9 . P ( 9 ) = 20,000 ⋅ (1.08) = 20,000 ⋅ ( 1.9990 ) 9 = 39,980 Since 39,980 ≈ 40,000, it is in 9 years. accurate to say the population will double This prediction is reasonable because 1.08 Module 13 9 609 ≈ 2 . Lesson 1 LANGUAGE SUPPORT A1_MTXESE353879_U5M13L1.indd 609 Connect Context Support students in interpreting the language used in problem statements. Explain that the word suppose at the beginning of a problem signals that what follows is a hypothetical example, meaning that readers should use their imaginations to consider a possible scenario. Often, a problem will be followed by the question, Why or why not? Explain that the question is phrased this way so as not to give away the answer. Students should understand that they need to explain either why a result is true or why it is not true, depending on the situation. 609 Lesson 13.1 13/11/14 4:24 PM Your Turn ELABORATE Solve using a graphing calculator. 8. 9. There are 225 wolves in a state park. The population is increasing at the rate of 15% per year. You want to make a prediction for how long it will take the population to reach 500. Y 1 = 500 Graph The intersection point is (5.713341, 500). The wolf x Y 2 = 225(1.15) population will reach 500 in approximately 5.7 years. There are 175 deer in a state park. The population is increasing at the rate of 12% per year. You want to make a prediction for how long it will take the population to reach 300. Y 1 = 300 The intersection point is (4.756046, 300). The deer Graph x Y 2 = 175(1.12) population will reach 300 in approximately 4.8 years. QUESTIONING STRATEGIES What is the shape of the graph of an exponential function of the form f(x) = b x when b > 1? It is a curve that rises in greater and greater amounts as x increases. What is the shape of the graph of a function ƒ(x)= b x when 0 < b < 1? It is a curve that falls more and more gradually as x increases. Elaborate 10. Explain how you would solve 0.25 = 0.5 x. Which method can always be used to solve an exponential equation? Possible answer: Algebraically. 0.25 = 0.5 x (0.5)2 = (0.5) → x = 2 SUMMARIZE THE LESSON Exponential equations can always be solved graphically. How can you solve an equation where the variable is an exponent? First, use the properties of equality to isolate the number raised to a variable power. Then check whether the constant on the other side of the equation can be written as a whole-number power of the same base. If it can, use the Equality of Bases Property to solve. If not, graph each side of the equation as its own function and find the x-value of the point of intersection. x x 11. What would you do first to solve the equation __14 (6) = 54? Multiply each side of the equation by 4 to isolate the power. 12. How does isolating the power in an exponential equation like __14 (6) = 54 compare to isolating the variable in a linear equation? Both are done for the same reason. By isolating the power, you have isolated the variable x as well. Then you can compare the exponents in the final equivalent expressions. situation, 2 = 0.99 x, has no solution for x > 0. The graphing calculator will show a horizontal line at 2 and an exponential function with a y-intercept of 1 decreasing towards the positive x-axis. 14. Solve 0.5 = 1.01 x graphically. Suppose this equation models the point where a population increasing at a rate of 1% per year is halved. When will the population be halved? Since x = -69.66072, you would have to go back in time, which is not possible. Seventy years or so ago the population was half of what it is now. © Houghton Mifflin Harcourt Publishing Company 13. Given a population decreasing by 1% per year, when will the population double? What will this type of situation look like when graphed on a calculator? It will never double as the population is decreasing. The equation representing this 15. Essential Question Check-In How can you solve equations involving variable exponents? When the bases are equal, use the Equality of Bases Property. When there are not equal bases on both sides of the equation, graph each side of the equation as its own function and find the intersection. Module 13 A1_MTXESE353879_U5M13L1 610 610 Lesson 1 20/02/14 1:10 AM Using Graphs and Properties to Solve Equations with Exponents 610 Evaluate: Homework and Practice EVALUATE 1. • Online Homework • Hints and Help • Extra Practice Would it have been easier to find the solution to the equation in Explore 1, x 3(2) = 96, algebraically? Justify your answer. In general, if you can solve an exponential equation graphing by hand, why can you solve it algebraically? Yes, 3(2) = 96 becomes (2) = 32 after dividing both sides of the equation by 3 and 32 is an integer power of 2. x x In general, the input-output tables for f(x) and g(x) have integers in the domain and the values in the range are easy to calculate. ASSIGNMENT GUIDE Concepts and Skills Practice Explore 1 Solving Exponential Equations Graphically Exercises 2–3 Explore 2 Solving Exponential Equations Algebraically Exercises 1, 3 Example 1 Solving Equations by Equating Exponents Exercises 4–16, 24 –25 Example 2 Solving a Real-World Exponential Equation by Graphing Exercises 17–23 2. The equation 2 = (1.01) models a population that has doubled. What is the rate of increase? What does x represent? x The rate of increase is 1% per unit time. x is number of units of time. 3. Can we solve equations using both algebraic and graphical methods? Yes. We can simplify the equation algebraically and then use graphing. Solve the given equation. x 4. 4(2) = 64 5. 4(2) 64 ____ = __ 4 INTEGRATE MATHEMATICAL PROCESSES Focus on Communication Circulate as students solve the practice exercises. Invite students to explain their reasoning as they begin a new problem. © Houghton Mifflin Harcourt Publishing Company 7. 7(3) = 63 x 7 3x = 9 2x = 24 3x = 32 x=4 x=2 75 (_14 )(_56 ) = _ 432 x 8. 5 75 1 _ = 4 ⋅ ___ 4⋅ _ 4 6 432 x x x 2 x=2 () Exercise 611 Lesson 13.1 6 x = 216 6x = 63 x=3 x 49 7 =_ 2_ 2 2 9. 2( 2 ) 2 ____ = __ _7 2 _7 2 _7 2 x 2 49 __ = 4 x 7 = _ 2 3(11) = 3993 x 3(11) 3993 ______ = ____ x 49 __ 3 3 () () () 11 x = 1331 x 11 x = 11 3 2 x=3 x=2 Module 13 A1_MTXESE353879_U5M13L1 611 x 7 2 x = 16 6 x = 54 _ 4 6 = 4 · 54 4 · __ 4 x 4 ( )( ) 25 (_56) = __ 36 5 (_6) = (_56) 6. 7(3) 63 ____ = __ x Lesson 1 611 Depth of Knowledge (D.O.K.) Mathematical Processes 1 2 Skills/Concepts 1.D Multiple representations 2 2 Skills/Concepts 1.A Everyday life 3 2 Skills/Concepts 1.D Multiple representations 4–12 1 Recall 1.F Analyze relationships 13–16 2 Skills/Concepts 1.F Analyze relationships 17–22 2 Skills/Concepts 1.A Everyday life 20/02/14 1:10 AM () x 1 2 _ 9 9x = 92 9 81 9 9 (_21 )(_32 ) = (_41 )(_1627 ) ( )( ) ( )( ) 8 (_23) = __ 27 (_23) = (_23) 16 1 _ 2 1 __ 2 _ =2 _ 4 27 2 3 x x x 3 x=3 x=2 () x ( ) 15. 16 ___ 8 8 13 13 When solving an equation involving a variable exponent, suggest that students try to structure the solution so that they are solving an equation of the form b x = c y. If c = b, then x = y; if c ≠ b, then they should solve by graphing. 8 (_25 )(_25 ) = _ 125 x x 27 x x 3 (_2) = (_2) 3 169 x x (_2) = __8 3 13 8 (_52)(_25)(_25) = (_52)___ 125 4 (_25) = __ 25 (_25) = (_25) 8(3) 4( 27 ) ____ = ____ x 2 x=2 16 2 = (4) _ 14. (8) _ 27 3 _2 2 x 2 (__4 ) = (__4 ) x 2 (_1) = (_1) x x x 16 (__4 ) = ___ x (_1) = __1 x=2 16. 2 INTEGRATE MATHEMATICAL PROCESSES Focus on Patterns 32 ___ ( ) ___ ____ = 169 4 2 __ 13 x 2 x 32 4 =_ 12. 2 _ 13 169 2 __ ( ) __ ____ = 81 9 x = 81 13. ( ) x 1 =_ 2 11. 2 _ 81 9 10. 2(9) = 162 3 2 x=2 x=3 8 _ 8 (_25 ) (_52 ) = (_ 125 )( 125 ) x 2x 2x 2x ( ) 2 = ((_ 5) ) 2 = (_ 5) 8 = ___ 125 2 3 2 6 © Houghton Mifflin Harcourt Publishing Company · Image Credits: ©prudkov/ Shutterstock (_25) (_25) (_25) x 2x = 6 6 2x __ =_ 2 2 x=3 17. There is a draught and the oak tree population is decreasing at the rate of 7% per year. If the population continues to decrease at the same rate, how long will it take for the population to be half of what it is? The model for the oak tree population is P(t) = P i(0.93) , t where t is the time in years, P i is the initial population, and P(t) is the population in year t. To find when the i population is half of its initial value, solve P(t) = __ for t. 2 P Pi t __ = P i(0.93) 2 P __ P (0.93) __2 = ______ i t i Pi Pi _1 = (0.93) t 2 t ≈ 9.55 Using the calculator solution method from Explain 2, the population will reach half of its original value in approximately 9.6 years. Module 13 Exercise A1_MTXESE353879_U5M13L1.indd 612 Lesson 1 612 Depth of Knowledge (D.O.K.) Mathematical Processes 23 3 Strategic Thinking 1.A Everyday life 24–25 3 Strategic Thinking 1.G Explain and justify arguments 13/11/14 4:29 PM Using Graphs and Properties to Solve Equations with Exponents 612 18. An animal reserve has 40,000 elk. The population is increasing at a rate of 11% per year. How long will it take for the population to reach 80,000? AVOID COMMON ERRORS The model for population is P(t) = 40,000(1.11) , where t is the time in t Students may be confused by complicated equations that involve variable exponents as well as additional factors. Remind them to first apply the properties of equality to isolate the number with the variable exponent, then use the Equality of Bases Property to solve. years and P(t) is the population in year t. To find when the population is 80,000, solve P(t) = 80,000 for t. 80,000 = 40,000(1.11) t Using the calculator solution method from Explain 2, t ≈ 6.64. The population will reach 80,000 in approximately 6.6 years. 19. A lake has a small population of a rare endangered fish. The lake currently has a population of 10 fish. The number of fish is increasing at a rate of 4% per year. When will the population double? How long will it take the population to be 80 fish? t The model for population is P(t) = 10(1.04) , where t is the time in years and P(t) is the population in year t. Solve P(t) = 20 for t. 20 = 10(1.04) t Using the calculator solution method from Explain 2, t ≈ 17.67. The population of the fish will double in 18 years. To find when the population will be 80, you can solve P(t) = 80 for t. © Houghton Mifflin Harcourt Publishing Company Alternatively, note that 80 = 10 · 8 = 10 · 2 3. This corresponds to the population doubling three times, from 10 to 20, from 20 to 40, and from 40 to 80. The population will be 80 in 54 years (3 · 18). 20. Tim has a savings account with the bank. The bank pays him 1% per year. He has $5000 and wonders when it will reach $5200. When will his savings reach $5200? The model is S(t) = 5000(1.01) . Solve S(t) = 5200 for t. t 5200 = 5000(1.01) t Using the calculator solution method from Explain 2, t ≈ 3.94. Graphing f(x) and g(x), we get the point of intersection (3.941648, 1.04). Rounding up and considering interest is calculated yearly, it will take Tim 4 years. Module 13 A1_MTXESE353879_U5M13L1 613 613 Lesson 13.1 613 Lesson 1 2/14/15 12:17 PM 21. Tim is considering a different savings account that pays 1%, but this time it is compounded monthly. AVOID COMMON ERRORS Some students may be unsure how to raise a fraction to a power. Remind them that both the numerator and the denominator must be raised to the same power. (When interest is compounded monthly, the bank pays interest every month instead nt of every year. The function representing compounded interest is S(t) = P(1 + __nr ) , where P is the principal, or initial deposit in the account, r is the interest rate, n is the number of times the interest is compounded per year, t is the year, and S(t) is the savings after t years.) How many years will it take Tim to earn $200 at this bank? Should he switch? ( 0.01 The model is S(t) = 5000 1 + ___ 12 5200 = 5000(1.00083) 12t ) 12t 12t or S(t) = 5000(1.00083) . Solve S(t) = 5200 for t. CURRICULUM INTEGRATION Encourage students to research applications of exponential functions. They should consider applications in science and business as well as uses in other math courses. Using the calculator solution method from Explain 2, t ≈ 3.92. t is approximately 4 years. Both accounts will reach $5200 in about 4 years. Switching won’t make much difference. INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning 22. Lisa has a credit card that charges 3% interest on a monthly balance. She buys a $200 bike and plans to pay for it by making monthly payments of $100. How many months will it take her to pay it off? Assume the first payment she makes is charged no interest because she paid it before the first bill. As students solve real-world problems involving time, have them make predictions before calculating their results. Write the predictions on the board, then compare them to the solutions found algebraically or graphically. Encourage students to improve their predictions by analyzing whether their predictions tend to be too high or too low and by considering how they can change their estimation methods. Her first payment is $100. At that time she owes $100 plus interest or $103. The second month she pays $100 and the third month she pays the rest. It takes her three months to pay it off. You do not have to solve an exponential because 3% is not a very high interest. 23. Analyze Relationships A city has 175,000 residents. The population is increasing at the rate of 10% per year. © Houghton Mifflin Harcourt Publishing Company a. You want to make a prediction for how long it will take for the population to reach 300,000. Round your answer to the nearest tenth of a year. b. Suppose there are 350,000 residents of another city. The population of this city is decreasing at a rate of 3% per year. Which city’s population will reach 300,000 sooner? Explain. a. 300,000 = 175,000(1.1) x Using the calculator solution method from Explain 2, x ≈ 5.7. The population will reach 300,000 in approximately 5.7 years. b. 300,000 = 350,000(0.97) x Using the calculator solution method from Explain 2, x ≈ 5.1. 5.1 < 5.7 The second city’s population will reach 300,000 sooner. Module 13 A1_MTXESE353879_U5M13L1 614 614 Lesson 1 2/14/15 12:17 PM Using Graphs and Properties to Solve Equations with Exponents 614 VISUAL CUES H.O.T. Focus on Higher Order Thinking 24. Explain the Error Jean and Marco each solved the equation 9(3) = 729. Whose solution is incorrect? Explain your reasoning. How could the person who is incorrect fix the work? x Have students create posters as visual reminders of how to solve equations involving exponents. Remind students to include examples as well as step-by-step procedures. Jean Marco 9(3) = 729 9(3) = 729 x x (_91 ) ⋅ 9(3) = (_19 ) ⋅ 729 3 2 ⋅ (3) = 729 x x 3 x = 81 = 3 4 JOURNAL 3 2 + x = 729 = 3 6 x=4 In their journals, have students explain how to use the Equality of Bases Property to solve an equation with a variable exponent. x=6 Jean is completely correct and Marco could correct his work as follows: Marco 9(3) = 729 x 3 2 ⋅ (3) = 729 x 2+x 3 = 729 = 3 6 x+2=6 x=4 He substituted for x = 6 instead of x + 2 = 6, which yields x = 4. 25. Critical Thinking Without solving, state the column containing the equation with the greater solution for each pair of equations. Explain your reasoning. 1 (3) x = 243 1 (9) x = 243 _ _ 3 3 () () 1( ) The equation _ 3 = 243 has a greater solution. Since the values of the 3 x powers of 3 increase less quickly than the values of the powers of 9, the © Houghton Mifflin Harcourt Publishing Company 1( ) 1( ) value of x in _ 3 = 243 will be greater than the value of x in _ 9 = 243. 3 3 x Module 13 A1_MTXESE353879_U5M13L1 615 615 Lesson 13.1 x 615 Lesson 1 27/02/14 7:48 AM Lesson Performance Task INTEGRATE MATHEMATICAL PROCESSES Focus on Modeling A town has a population of 78,918 residents. The town council is offering a prize for the best prediction of how long it will take the population to reach 100,000. The population rate is increasing 6% per year. Find the best prediction in order to win the prize. Write an exponential equation in the form y = abx and explain what a and b represent. Before students write an equation for the situation in the Lesson Performance Task, discuss how they know that the base to be raised to a power in the exponential equation is 1.06 and not 0.06. Have them consider “What factor multiplied by the population makes the number 6% greater?” Then ask what the base would be if the population were decreasing by 6% per year. Students should recognize that it would be 1 – 0.06 = 0.94. Discuss what a graph showing each growth rate would look like. Write an exponential equation. Let y represent the population and x represent time in years. a represents the initial population, 78,918. b represents the rate of increase in the population per year. y = 78,918(1 + 0.06) x Substitute the target population for y: 100,000 = 78,918(1 + 0.06) x Write the expressions from the two sides of the equation as functions. f(x) = 100,000 g(x) = 78,918(1 + 0.06) x Use the intersect feature on a graphing calculator to find the point of intersection. INTEGRATE MATHEMATICAL PROCESSES Focus on Technology The point of intersection is (4.063245, 100,000). The population will reach 100,000 in just over 4 years. © Houghton Mifflin Harcourt Publishing Company · Image Credits ©Jim West/ Alamy Images Module 13 616 As students use their graphing calculators to graph the two functions and find their intersection, remind them to adjust the viewing window so that the intersection is shown clearly. INTEGRATE MATHEMATICAL PROCESSES Focus on Communication Have students share their reasons for why the point where the graphs of the right- and left-hand sides of the equation intersect is the solution. Lesson 1 EXTENSION ACTIVITY A1_MTXESE353879_U5M13L1.indd 616 Have students research the current population of their community or state and the rate at which it is growing or decreasing. Then have students write an exponential equation in which y represents the population and x represents time in years. Finally, have students choose a future population size and predict when the population will reach that size. 13/11/14 4:46 PM Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Using Graphs and Properties to Solve Equations with Exponents 616