Graphing Exponential Functions
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1 Graphing Eponential Functions Math Objectives Students will infer why the conditions b0 and b 1 are necessary for the function to be eponential. Students will determine that for b 1 the function is increasing and for 0b 1 the function is decreasing. Students will determine that the y-intercept is always 0,1 and there is no -intercept. Students will determine that for b 1 the function approaches as approaches and that for 0b 1 the function approaches as approaches., and the range as 0,. Students will identify the domain as Students will identify the equation of the function s horizontal asymptote as y 0. Students will construct viable arguments & critique the reasoning of others ( CCSS Mathematical Practice). Vocabulary eponential function end behavior intercepts domain and range asymptotes increasing and decreasing functions About the Lesson Students will investigate the graphs of the family of eponential functions f ( ) b. As a result, students will: Infer why the conditions b0 and b 1 are necessary. Determine how the value of b affects the increasing or decreasing behavior of the function. Determine the y-intercept, domain, and range. Describe the end behavior. State the equation of the asymptote. TI-Nspire Navigator System Use Live Presenter to demonstrate how to use sliders. Use Screen Capture to eamine patterns that emerge. TI-Nspire Technology Skills: Download a TI-Nspire document Open a document Move between pages Use a minimized slider Tech Tips: Make sure the font size on your TI-Nspire handheld is set to Medium. You can hide the entry line by pressing /G. Lesson Materials: Student Activity Graphing_Eponential_ Functions_Student.pdf Graphing_Eponential_ Functions_Student.doc Optional Materials: Graphing_Eponential_ Functions_Create.doc Graphing_Eponential_ Functions_Create.pdf TI-Nspire document Graphing_Eponential_ Function.tns Visit for lesson updates and tech tip videos Teas Instruments Incorporated 1 education.ti.com
2 Graphing Eponential Functions Use Quick Poll to assess students understanding throughout the activity. Discussion Points and Possible Answers TI-Nspire Navigator Opportunity: Live Presenter See Note 1 at the end of this lesson. Teacher Tip: Students can either use the premade file or use the create activity to create the TI-Nspire document file first. Move to page Eplore several different b-values by dragging the slider. a. Set b = 1. Describe the graph. Answer: It is the horizontal line y = 1. b. By definition, for the eponential function f ( ) b, b cannot equal 1. What mathematical reason can you give for this restriction? Answer: When b = 1, the function becomes a linear relationship and is no longer an eponential function. c. Set b = 0. Describe the graph. Answer: It is a horizontal line for 0. It is the line y 0. d. By definition, for the eponential function f ( ) b, b cannot equal 0. What mathematical reason can you give for this restriction? Answer: When b = 0, the function becomes a linear relationship and is no longer an eponential function. 2. Eplore several different b-values by dragging the slider. a. For what b-values is the function increasing? Why is this true? 2011 Teas Instruments Incorporated 2 education.ti.com
3 Graphing Eponential Functions n m Answer: When b 1, f ( ) b is increasing because b b when n m. b. For what b-values is the function decreasing? Why is this true? n m Answer: When 0b 1, f ( ) b is decreasing because b b when n m. TI-Nspire Navigator Opportunity: Screen Capture See Note 2 at the end of this lesson. 3. Eplore several different b-values by dragging the slider. a. For each b-value, identify the y-intercept of the function. Interpret your results. 0 Answer: The y-intercept is always 0, 1 because b 1for all b. b. When b 0, why is there no -intercept? Answer: There is no eponent for b that would result in an answer of 0. c. Another special point is when = 1. Describe the general point regardless of the value of b. Eplain your answer. Answer: When = 1, the point will always be (1, b) because the base raised to the power of 1 will be the value of b. TI-Nspire Navigator Opportunity: Screen Capture: See Note 3 at the end of this lesson. Teacher Tip: If students have not encountered asymptotes before, there may need to be some discussion before they answer the net questions. 4. Drag the slider to eplore several different b-values where b 1. a. What does f( ) approach as approaches? Eplain. Answer: ; As you choose larger and larger positive eponents for b, the result will be greater and greater. b. What does f( ) approach as approaches? Eplain Teas Instruments Incorporated 3 education.ti.com
4 Graphing Eponential Functions Answer: 0; As you choose smaller and smaller negative eponents for b, the result will get closer and closer to 0 without ever reaching 0. c. What is the equation of the horizontal asymptote? Answer: y 0 5. Drag the slider to eplore several different b-values where 0b 1. a. What does f( ) approach as approaches? Eplain. Answer: ; As you choose smaller and smaller negative eponents for b, the result will get larger and larger. b. What does f( ) approach as approaches? Eplain. Answer: 0; As you choose larger and larger eponents for b, the result will get closer and closer to 0 without ever reaching 0. c. What is the equation of the horizontal asymptote? Answer: y 0 6. Find the domain and range for the family of eponential functions f ( ) b where b 0 and b 1. Answer: The domain is, and the range is 0,. TI-Nspire Navigator Opportunity: Quick Poll See Note 5 at the end of this lesson. 7. Wade believes the function f ( ) b will eventually intersect the -ais. Is he correct? Why or why not? Answer: Wade is incorrect. The function will never cross the -ais. It will only approach the -ais. TI-Nspire Navigator Opportunity: Quick Poll See Note 6 at the end of this lesson Teas Instruments Incorporated 4 education.ti.com
5 Graphing Eponential Functions 8. Eric believes that for b 1 the function f ( ) b increases on only one side of the y-ais. Is he correct? Why or why not? Answer: Eric is incorrect. The function increases more quickly on one side of the y-ais, but it increases over the entire domain. TI-Nspire Navigator Opportunity: Live Presenter See Note 7 at the end of this lesson. Etension: Students have been considering functions of the form f( ) b, where b 0, b 1. Ask students to make conjectures about the characteristics and behavior of the function when b 0. Wrap Up Upon completion of the discussion, the teacher should ensure that students understand that for graphs of f ( ) b : The conditions b0 and b 1 are necessary. For b 1 the function is increasing, and for 0b 1 the function is decreasing. The y-intercept is always (0, 1) and there is never an -intercept. For b >1 the function approaches as approaches. For 0b 1 the function approaches as approaches. The domain is, and the range is 0,. The function has a horizontal asymptote of y 0. TI-Nspire Navigator Note 1 Before the lesson, Live Presenter: You may want to demonstrate or have a student use Live Presenter to show how to change the b-values by dragging the slider. Note 2 Question 2a and 2b, Screen Capture: Take a Screen Capture of page 1.2 when students are on different b-values. As a class, discuss the various cases that occur. Note 3 Question 3a and 3b, Screen Capture: Take a Screen Capture of page 1.2 when students are on different b-values. As a class, discuss the various cases that occur Teas Instruments Incorporated 5 education.ti.com
6 Graphing Eponential Functions Note 4 Question 4a 4f, Screen Capture: Take a Screen Capture of page 1.2 when students are on different b-values. As a class, discuss the various cases that occur. Note 5 Question 6, Quick Poll (Open Response): Send two Open Response Quick Polls, asking students to submit their domain and range. If students struggle to identify the domain and range, consider taking a Screen Capture and discussing how, for all graphs, the possible - and.y-values are similar. Note 6 Question 7, Quick Poll (Open Response): Send an Open Response Quick Poll, asking students to submit their answer to Question 7. Note 7 Question 8, Quick Poll (Open Response): Send an Open Response Quick Poll, asking students to submit their answer to Question Teas Instruments Incorporated 6 education.ti.com
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