Essential Question: How do you graph an exponential function of the form f (x) = ab x? Explore Exploring Graphs of Exponential Functions. 1.

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1 Locker LESSON 4.4 Graphing Eponential Functions Common Core Math Standards The student is epected to: F-IF.7e Graph eponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Also F-IF.8b Mathematical Practices MP.4 Modeling Language Objective Eplain the domain, range, and end behavior of the graphs of eponential functions of the form ƒ () = ab with a < 0 and 0 < b <. Name Class Date 4.4 Graphing Eponential Functions Essential Question: How do ou graph an eponential function of the form f () = ab? Eplore Eploring Graphs of Eponential Functions Eponential functions follow the general shape = ab. Graph the eponential functions on a graphing calculator, and match the graph to the correct function rule.. = 3 (). = 0.5 () 3. = 3 (0.5) 4. = () a c b d a. c. b. d. Resource Locker ENGAGE Essential Question: How do ou graph an eponential function of the form f () = ab? Use the value of a to find the -intercept. Choose several values of other than 0 to plot a few more ordered pairs and connect them with a smooth curve. Use the values of a and b to determine the end behavior of the function. Houghton Mifflin Harcourt Publishing Compan In all the functions 4 above, the base b > 0. Use the graphs to make a conjecture: State the domain and range of = ab if a > 0. In all of the functions 4 above, the domain values are all real numbers, or - < <. If a > 0 and b > 0, then the range values are all positive, or > 0. In all the functions 4 above, the base b > 0. Use the graphs to make a conjecture: State the domain and range of = ab if a < 0. In all of the functions 4 above, the domain values are all real numbers, or - < <. If a < 0 and b > 0, then the range values are all negative, or < 0. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and the fact that the weight of a pumpkin might grow eponentiall. Then preview the Lesson Performance Task. What is the -intercept of ƒ () = 0.5 ()? 0.5 Module Lesson 4 Name Class Date 4.4 Graphing Eponential Functions Essential Question: How do ou graph an eponential function of the form f () = ab? F-IF.7e Graph eponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Also F-IF.8b Eplore Eploring Graphs of Eponential Functions Eponential functions follow the general shape = ab. Houghton Mifflin Harcourt Publishing Compan Graph the eponential functions on a graphing calculator, and match the graph to the correct function rule.. = 3 () a. b.. = 0.5 () 3. = 3 (0.5) 4. = () In all the functions 4 above, the base b > 0. c. Use the graphs to make a conjecture: State the domain and range of = ab if a > 0. a c b d In all the functions 4 above, the base b > 0. Use the graphs to make a conjecture: State the domain and range of = ab if a < 0. What is the -intercept of ƒ () = 0.5 ()? d. Resource In all of the functions 4 above, the domain values are all real numbers, or - < <. If a > 0 and b > 0, then the range values are all positive, or > 0. In all of the functions 4 above, the domain values are all real numbers, or - < <. If a < 0 and b > 0, then the range values are all negative, or < Module Lesson 4 HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition. 677 Lesson 4.4

2 Note the similarities between the -intercept and a. What is their relationship? The value of a in = ab is the -intercept. Reflect. Discussion What is the domain for an eponential function = ab? all real numbers, or - < <.. Discussion Describe the values of b for all functions = ab. The base b is positive, or b > 0, in functions of the form = ab Eplain Graphing Increasing Positive Eponential Functions The smbol represents infinit. We can describe the end behavior of a function b describing what happens to the function values as approaches positive infinit ( ) and as approaches negative infinit ( - ). Eample ƒ () = Graph each eponential function. After graphing, identif a and b, the -intercept, and the end behavior of the graph. Choose several values of and generate ordered pairs. EXPLORE Eploring Graphs of Eponential Functions INTEGRATE TECHNOLOGY Have students complete the Eplore activit in either the book or online lesson. QUESTIONING STRATEGIES What is the relationship between the value of a and the -intercept of the graph of ƒ () = ab? The -intercept is equal to a. f () = a = b = -intercept: (0, ) End Behavior: As -values approach positive infinit ( ), -values approach positive infinit ( ). As -values approach negative infinit ( - ), -values approach zero ( 0). Using smbols onl, we sa: As,, and as -, 0. 4 (-, 0.5) (, 4) (, ) (0, ) f() = 0 4 Houghton Mifflin Harcourt Publishing Compan EXPLAIN Graphing Increasing Positive Eponential Functions AVOID COMMON ERRORS When generating ordered pairs for eponential functions of the form ƒ () = ab, students ma multipl a b b and then raise that product to. Remind students to use the correct order of operations. Module Lesson 4 PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices This lesson provides an opportunit to address Mathematical Practice MP.4, which calls for students to use modeling. Students use tables and graphs to represent eponential functions. The can use the graphs to determine the end behavior of the functions, and make generalizations about the effect of parameters a and b on the end behavior of an eponential function of the form f() = ab. Graphing Eponential Functions 678

3 QUESTIONING STRATEGIES If an eponential function is of the form ƒ () = ab, both a and b are positive real numbers, and the domain is the set of all real numbers, what is the range of the function? The range is set of all positive real numbers. B ƒ () = 3 (4) Choose several values of and generate ordered pairs. f () = 3 (4) a = 3 b = 4 -intercept: ( 0, 3 ) End Behavior: As, and as -, 0. Reflect (, ) (-, 0.75) (0, 3) (, 48) f() = 3(4) 3. If a > 0 and b >, what is the end behavior of the graph? If a > 0 and b >, as approaches infinit approaches infinit, and as approaches negative infinit approaches Describe the -intercept of the eponential function ƒ () = a b in terms of a and b. A graph of the eponential f () = a b will have a -intercept at (0, a). Your Turn 5. Graph the eponential function ƒ () = () After graphing, identif a and b, the -intercept, and the end behavior of the graph. Houghton Mifflin Harcourt Publishing Compan a = b = f () = () intercept: (0, ) End Behavior: As, and as -, 0. (-, ) -4-9 (, 8) 6 f() = () (, 4) 3 (0, ) Module Lesson 4 COLLABORATIVE LEARNING Peer-to-Peer Activit In pairs, have students develop and record their own rules for determining whether a graph represents one of the four main tpes of eponential functions: increasing positive, decreasing negative, decreasing positive, and increasing negative. For eample, the rules might be a > 0 and b >, a < 0 and 0 < b <, a > 0 and 0 < b <, and a < 0 and b >. Then, give each pair one of each tpe of function and its graph. Have the students use their rules to determine which tpe of eponential function each graph represents. As a class, discuss the rules that worked. 679 Lesson 4.4

4 Eplain Graphing Decreasing Negative Eponential Functions You can use end behavior to discuss the behavior of a graph. Eample Graph each eponential function. After graphing, identif a and b, the -intercept, and the end behavior of the graph. Use end behavior to discuss the behavior of the graph. EXPLAIN Graphing Decreasing Negative Eponential Functions ƒ () = -(3) Choose several values of and generate ordered pairs. f () = -(3) a = - b = 3 -intercept: (0, -) End Behavior: As, - and as -, 0. ƒ () = (4) Choose several values of and generate ordered pairs. f () = (4) a = b = 4 -intercept: ( 0, ) End Behavior: As, - and as -, (0, -) -6 (, -6) - -8 f() = -(3) (, -8) 40 0 f() = (4) (0, ) (, -) (, -48) -60 Houghton Mifflin Harcourt Publishing Compan QUESTIONING STRATEGIES How can the value of a tell ou in which quadrants the graph of an eponential function in the form ƒ () = ab is located? If a is greater than 0, the graph is located in quadrants I and II. If a is less than 0, the graph is located in quadrants III and IV. AVOID COMMON ERRORS Students ma forget that b the order of operations rules, ƒ () = - means that ƒ () = - ( ), and therefore the negative sign is not raised to the power. Tell students that the base of an eponential function cannot be negative. Module Lesson 4 DIFFERENTIATE INSTRUCTION Cognitive Strategies Guide students to see that for eponential functions of the form f () = ab, b > represents growth (the graph is increasing), and b < represents decline (the graph is decreasing). Remind students that if b =, the function is linear and not eponential; the function represents no change: no growth or decline. Graphing Eponential Functions 680

5 EXPLAIN 3 Graphing Decreasing Positive Eponential Functions QUESTIONING STRATEGIES If 0 < b < and a is a positive constant, how can ou alter b to make the graph of = ab decrease more graduall? Increase the value of b. Reflect 6. If a < 0 and b >, what is the end behavior of the graph? If a < 0 and b >, as approaches infinit approaches negative infinit, and as approaches negative infinit approaches 0. Your Turn 7. Graph the eponential function. ƒ () = (3) After graphing, identif a and b, the -intercept, and the end behavior of the graph. f () = (3) a = b = 3 -intercept: (0, ) End Behavior: As, - and as -, (0, ) (, -9) -0 0 f() = (3) (, -7) Eplain 3 Graphing Decreasing Positive Eponential Functions Eample 3 Graph each eponential function. After graphing, identif a and b, the -intercept, and the end behavior of the graph. Houghton Mifflin Harcourt Publishing Compan ƒ () = (0.5) Choose several values of and generate ordered pairs. f () = (0.5) a = b = 0.5 -intercept: (0, ) End Behavior: As, 0 and as -,. f() = (-, ) (, 0.5) (0, ) (, 0.5) Module 4 68 Lesson 4 LANGUAGE SUPPORT Connect Vocabular Make sure that students understand what is meant b the end behavior of a function. The end behavior of a function describes what happens to the function values as gets larger and larger (for eample, as becomes 00, 000, and,000,000) and what happens to the function values as the values of get smaller and smaller (for eample, as becomes 00, 000, and,000,000). So, end behavior describes what happens to f() when is farther and farther to the right and what happens to f() when is farther and farther to the left. 68 Lesson 4.4

6 B ƒ () = (0.4) Choose several values of and generate ordered pairs. - 0 f () = (0.4) COGNITIVE STRATEGIES Show students that the graph of = a ( b ) is the reflection of = ab over the -ais b graphing = 3 ( ) and = 3 (). a = b =.4 -intercept: ( 0, ) End Behavior: As, 0 and as -,. Reflect 6 (-, 5) 4 f() = (0.4) (0, ) (, 0.8) If a > 0 and 0 < b <, what is the end behavior of the graph? If a > 0 and 0 < b <, as approaches infinit approaches zero, and as approaches negative infinit approaches infinit. Your Turn 9. Graph the eponential function. After graphing, identif a and b, the -intercept, and the end behavior of the graph. ƒ () = 3 (0.5) f () = 3 (0.5) a = 3 b = 0.5 -intercept: (0, 3) End Behavior: As, 0 and as -,. (-, 6) 6 f() = 3(0.5) 4 (0, 3) (,.5) (, 0.75) Houghton Mifflin Harcourt Publishing Compan Module 4 68 Lesson 4 Graphing Eponential Functions 68

7 EXPLAIN 4 Graphing Increasing Negative Eponential Functions Eplain 4 Eample 4 ƒ () = -0.5 Graphing Increasing Negative Eponential Functions Graph each eponential function. After graphing, identif a and b, the -intercept, and the end behavior of the graph. Choose several values of and generate ordered pairs. QUESTIONING STRATEGIES Wh does it make sense for the graph of f () = ab to stretch when a >, and for the graph to shrink when 0 < a <, compared to the graph of f () = b? Multipling b a factor a greater than increases the outputs, compared with the function f () = b, and multipling b a factor a between 0 and decreases the outputs, compared with the function f () = b. f () = a = - b = 0.5 -intercept: (0, -) End Behavior: As, 0 and as -, -. ƒ () = (0.4) Choose several values of and generate ordered pairs. (, -0.5) (, -0.5) - f() = -0.5 (0, -) (-, -) f () = (0.4) Houghton Mifflin Harcourt Publishing Compan a = b = 0.4 -intercept: ( 0, ) End Behavior: As, 0 and as -, (, -.) f() = (0.4) (0, ) (-, -7.5) -6-9 Module Lesson Lesson 4.4

8 Reflect 0. If a < 0 and 0 < b <, what is the end behavior of the graph? If a < 0 and 0 < b <, as approaches infinit approaches zero, and as approaches negative infinit approaches negative infinit. Your Turn. Graph the eponential function. After graphing, identif a and b, the -intercept, and the end behavior of the graph. ƒ () = -(0.5) f () = -(0.5) a = - b = 0.5 -intercept: (0, -) End Behavior: As, 0 and as -, -. Elaborate (, -) - f() = -(0.5) (0, -) (-, -4). Wh is ƒ () = 3 (-0.5) not an eponential function? f () = 3 (-0.5) is not an eponential function because its values alternate between negative and positive, creating a line that jumps between increasingl smaller positive and negative values as approaches infinit. An eponential function has a smooth curve that takes on increasingl smaller positive or negative values as approaches infinit. 3. Essential Question Check-In When an eponential function of the form ƒ () = a b is graphed, what does a represent? a represents the -intercept of the function Houghton Mifflin Harcourt Publishing Compan INTEGRATE TECHNOLOGY Using graphing calculators can allow students to compare the graphs of eponential functions quickl, but the must enter the functions correctl. Show students that = ( is not the same as = - ( 3 Eponential Functions: = ab a > 0, b > = 3() 3) both functions on a calculator. ELABORATE a < 0, b > = -(3) 3) b graphing QUESTIONING STRATEGIES Which of the following functions has the greatest -intercept and which has the least -intercept: =.5, = (.5), = (.5), or = 4 (.5)? Eplain. = 4 (.5) has the greatest -intercept (4), and = (.5) has the least -intercept ( ). SUMMARIZE THE LESSON Cop and complete the graphic organizer with our students. In each bo, give an eample of an appropriate eponential function and sketch its graph. 0 3 Module Lesson a > 0, 0 < b < = a < 0, 0 < b < = Graphing Eponential Functions 684

9 EVALUATE Evaluate: Homework and Practice State a, b, and the -intercept then graph the function on a graphing calculator.. ƒ () = (3). ƒ () = -6() Online Homework Hints and Help Etra Practice ASSIGNMENT GUIDE Concepts and Skills Eplore Eploring Graphs of Eponential Functions Practice Eercises, 9 a = b = 3 -intercept: (0, ) a = -6 b = -intercept: (0, -6) Eample Graphing Increasing Positive Eponential Functions Eercises 5, 9, 3, 5, 8, 0 3. ƒ () = -5(0.5) 4. ƒ () = 3 (0.8) Eample Graphing Decreasing Negative Eponential Functions Eercises 8,, 6, 3 Eample 3 Graphing Decreasing Positive Eponential Functions Eample 4 Graphing Increasing Negative Eponential Functions INTEGRATE TECHNOLOGY Eercises 4, 7, 0, 7, 5 Eercises 3, 6,, 4, 4 On a graphing calculator, have students graph = 3 (), = (), = 3 ( ), and = ( ) Have them keep track of the changes b drawing sketches of each graph on the same coordinate grid.. Houghton Mifflin Harcourt Publishing Compan a = -5 b = 0.5 -intercept: (0, -5) 5. ƒ () = 6 (3) a = 6 b = 3 -intercept: (0, 6) a = 3 b = 0.8 -intercept: (0, 3) 6. ƒ () = -4(0.) a = -4 b = 0. -intercept: (0, -4) Module Lesson 4 Eercise Depth of Knowledge (D.O.K.) Mathematical Practices 8 Recall of Information MP. Reasoning 9 8 Skills/Concepts MP. Reasoning 9 Recall of Information MP.5 Using Tools 0 Skills/Concepts MP.4 Modeling 5 3 Strategic Thinking MP.3 Logic 685 Lesson 4.4

10 7. ƒ () = 7 (0.9) 8. ƒ () = () QUESTIONING STRATEGIES How do ou know the eponent in ab applies to b and not ab? B the order of operations, bases are raised to eponents before multipling. a = 7 b = 0.9 -intercept: (0, 7) a = b = -intercept: (0,) State a, b, and the -intercept then graph the function and describe the end behavior of the graphs. 9. ƒ () = 3 (3) Ordered pairs: f() = 3(3) 30 (, 7) 0 0 (, 9) (-, ) (0, 3) f () = 3 (3) Graph the ordered pairs and connect them with a smooth curve. a = 3 b = 3 -intercept: (0, 3) End Behavior: As, and as -, ƒ () = 5 (0.6) Ordered pairs: (-, 8.3) (0, 5) f () = 5 (0.6) Graph the ordered pairs and connect them with a smooth curve. a = 5 b = 0.6 -intercept: (0, 5) 3 (, 3) (,.8) 0-6 f() = 5(0.6) End Behavior: As, 0 and as -,. Houghton Mifflin Harcourt Publishing Compan Module Lesson 4 Graphing Eponential Functions 686

11 AVOID COMMON ERRORS Students ma have trouble raising a number to a negative power. Remind them that a number raised to a negative power is the reciprocal of the number raised to the opposite power. For eample, 4 - = ( 4) = 6.. ƒ () = -6(0.7) 3. ƒ () = 5 () f () = -6 (0.7) (, -.9) f() = -6(0.7) (, -4.) -6 (0, -6) (-, -8.6) -9 a = -6; b = 0.7 -intercept: (0, -6) End Behavior: As, 0 and as -, -.. ƒ () = -4(3) 4. ƒ () = -(0.8) f () = -4 (3) (-, -.3) -4 (0, -4) f() = -4(3) -8 - a = -4; b = 3 -intercept: (0, -4) (, -) End Behavior: As, - and as -, 0. Houghton Mifflin Harcourt Publishing Compan f () = 5 () (, 0) 4 f() = 5() (, 0) 7 (-,.5) (0, 5) f () = -(0.8) (, -.3) f() = -(0.8) (, -.6) - (-, -.5) (0, -) a = 5; b = -intercept: (0, 5) End Behavior: As, and as -, 0. a = -; b = 0.8 -intercept: (0, -) End Behavior: As, 0 and as -, -. Module Lesson Lesson 4.4

12 5. ƒ () = 9 (3) a = 9; b = 3 7. ƒ () = 7 (0.4) 90 (, 8) 60 f() = 9(3) 30 (, 7) (0, 9) (-, 3) intercept: (0, 9) f () = 9 (3) 7 8 End Behavior: As, and as -, ƒ () = -5() a = -5; b = 8. ƒ () = 6 () (-, -.5) -7 (0, -5) f() = -5() (, -0) -4 -intercept: (0, -5) - f () = -5 () (, -0) End Behavior: As, - and as -, 0. COGNITIVE STRATEGIES Show students that the graph of = -ab is the reflection of = ab over the -ais. INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP. Introduce students to the term horizontal asmptote. A horizontal asmptote is a line that the graph gets closer and closer to, but never reaches. The line = 0 is the horizontal asmptote of the graphs in this lesson. (-, 7.5) 8 f() = 7(0.4) f () = 7 (0.4) (0, 7) (,.8) 0 (,.) f () = 6 () f() = 6() 0 (, 4) 0 (, ) (-, 3) (0, 6) Houghton Mifflin Harcourt Publishing Compan a = 7; b = 0.4 -intercept: (0, 7) End Behavior: As, 0 and as -,. a = 6; b = -intercept: (0, 6) End Behavior: As, and as -, 0. Module Lesson 4 IN_MNLESE38976_U6M4L /9/4 :9 AM Graphing Eponential Functions 688

13 JOURNAL Have students describe the four basic shapes of the graphs of eponential functions, including the quadrants each graph occupies and the end behavior of each graph. Also have them write an eponential function of each tpe. 9. Identif the domain and range of each function. Make sure to provide these answers using inequalities. a. ƒ () = 3 () b. ƒ () = 7 (0.4) c. ƒ () = -(0.6) d. ƒ () = (4) e. ƒ () = () Domain: - < <, Range: > 0 Domain: - < <, Range: > 0 Domain: - < <, Range: < 0 Domain: - < <, Range: < 0 Domain: - < <, Range: > 0 0. Statistics In 000, the population of Massachusetts was 6.3 million people and was growing at a rate of about 0.3% per ear. At this growth rate, the function ƒ () = 6.3 (.003) gives the population, in millions ears after 000. Using this model, find the ear when the population reaches 7 million people. Using a graphing calculator and the graph of f () = 6.3 (.003), ou can find that when the graph reaches a -value of 7, is approimatel 33, and the ear is = 033. The population will reach approimatel 7 million people during the ear 033. Houghton Mifflin Harcourt Publishing Compan Image Credits: Spirit of America/Shutterstock. Phsics A ball is rolling down a slope and continuousl picks up speed. Suppose the function ƒ () =. (.) describes the speed of the ball in inches per minute. How fast will the ball be rolling in 0 minutes? Round the answer to the nearest whole number. f () =. (.) f () =. (.) 0 f (). (8.06) f () 9.67 The ball will be rolling at a rate of about 0 inches per minute after 0 minutes. H.O.T. Focus on Higher Order Thinking. Draw Conclusions Assume that the domain of the function ƒ () = 3 () is the set of all real numbers. What is the range of the function? The range of the function is all positive real numbers since f () is alwas positive. 3. What If? If b = in an eponential function, what will the graph of the function look like? The graph of the eponential function will be a horizontal line at = a. 4. Critical Thinking Using the graph of an eponential function, how can b be found? Identif two points (, ) and (, ) on the graph of the eponential function with <. If - =, then b =, otherwise b = -. Module Lesson Lesson 4.4

14 5. Critical Thinking Use the table to write the equation for the eponential function. f () - 4_ The -intercept of the function is located at (0, 4), so a = 4. To find b, divide the -coordinate of (, 0) b the -coordinate of (0, 4). b = 0 4 = 5 f () = 4 (5) INTEGRATE TECHNOLOGY Have students use a graphing calculator to graph both equations on the same graph. Compare the two graphs. 00 Lesson Performance Task A pumpkin is being grown for a contest at the state fair. Its growth can be modeled b the equation P = 5 (.56) n, where P is the weight of the pumpkin in pounds and n is the number of weeks the pumpkin has been growing. B what percentage does the pumpkin grow ever week? After how man weeks will the pumpkin be 80 pounds? After the pumpkin grows to 80 pounds, it grows more slowl. From then on, its growth can be modeled b P = 5 (.3) n, where n is the number of weeks since the pumpkin reached 80 pounds. Estimate when the pumpkin will reach 50 pounds. The pumpkin grows b 56% ever week until it reaches 80 pounds, then it grows b 3%. Using a graphing calculator and the graph of f () = 5 (.56), ou can find that when the graph reaches a -value of 80, is approimatel.6, so.6. The pumpkin will be 80 pounds after.6 weeks. It onl needs to grow = 70 more pounds. Let P = 70 and solve the new equation. Add the solution to this equation to the original estimate to find the total time it will take to grow to 50 pounds. 70 = 5 (.3) n Using a graphing calculator and the graph of f () = 5 (.3), ou can find that when the graph reaches a -value of 70, is approimatel 4.9, so = 7.5 It will take the pumpkin about 7.5 weeks to grow to 50 pounds. Houghton Mifflin Harcourt Publishing Compan INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. Have students describe how the pumpkin grew over the period of time until it reached 50 pounds. The could sa that the pumpkin grew to a weight of 80 pounds in about.6 weeks, while it took about 4.9 weeks to gain another 70 pounds. The could also sa that the rate provided in the first equation is 56% per week, while the second equation shows the rate declined to 3%. Module Lesson 4 EXTENSION ACTIVITY Have students research how the weight of a giant pumpkin can be estimated before it is picked. Have students do an Internet search for over the top (OTT) weight tables. There are three measurements that must be found and added together to get the over-the-top measurement. Have students use the data the find to determine what the relationship appears to be between the over-the-top measurement in inches and the weight of a giant pumpkin in pounds. A graph of these values is a scatter plot that could be represented b an eponential function. Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. point: Student shows good understanding of the problem but does not full solve or eplain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Graphing Eponential Functions 690