ࡱ>  xx dbjbj[[ 99% 8TH|D*nJ"&p" |"y{{{{{{{{{{{{{F{{"!|J"""{{JD|+++"y{+"y{++2Z%_ ױ56&A\.e{Z|0|o\|'|\%_|%_@""+"""""{{{{+"""|""""|""""""""" : Math Objectives For the graphs of  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4 , students will: Infer why the conditions EMBED Equation.DSMT4  are necessary for the function to be logarithmic. Determine that for  EMBED Equation.DSMT4  the function is increasing and for  EMBED Equation.DSMT4  the function is decreasing. Determine the x-intercept, y-intercept, domain, range, and asymptotes. Determine that for a >1 the function approaches " as x approaches " and that for 0 < a < 1 the function approaches  " as x approaches ". Students will construct viable arguments & critique the reasoning of others (CCSS Mathematical Practice). Vocabulary logarithm function end behavior intercepts domain and range asymptotes increasing and decreasing functions extraneous solution About the Lesson The time varies for this activity depending on whether students create the TI-Nspire document file or use the pre-constructed file. Students will investigate the graphs of the family of logarithm functions  EMBED Equation.DSMT4 , by changing the a-value over the interval EMBED Equation.DSMT4 . As a result, students will: Infer why the conditions EMBED Equation.DSMT4  are necessary. Determine how the value of a affects the increasing or decreasing behavior of the function. Determine the x-intercept, domain, range, and asymptotes. Describe the end behavior. TI-Nspire Navigator System Use Live Presenter to demonstrate how to use sliders. Use Screen Capture to examine patterns that emerge. Use Quick Poll to assess students understanding throughout the activity.   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 Graph_Logarithms_Student.pdf Graph_Logarithms_Student.doc Optional Materials: Graph_Logarithms _Create.doc Graph_Logarithms _Create.pdf TI-Nspire document Graph_Logarithms.tns Visit  HYPERLINK "http://www.mathnspired.com" www.mathnspired.com for lesson updates and tech tip videos. Discussion Points and Possible Answers   Move to page 1.2. 1. Explore several different a-values by clicking " or ". TI-Nspire Navigator Opportunity: Live Presenter See Note 1 at the end of this lesson. a. Set a = 1. Describe the graph. Answer: It is nonexistent. b. By definition, for the logarithmic function EMBED Equation.DSMT4 , a cannot equal 1. What mathematical reason can you give for this restriction? Answer: When you have  EMBED Equation.DSMT4 this is equivalent to  EMBED Equation.DSMT4 , and 1 to any power is 1. Therefore, the possible solution is the vertical line x = 1; since this is not a function, x = 1 is an extraneous solution. c. Set a = 0. Describe the graph. Answer: It is nonexistent. d. By definition, for the logarithmic function  EMBED Equation.DSMT4 , a cannot equal 0. What mathematical reason can you give for this restriction? Answer: When you have  EMBED Equation.DSMT4 this is equivalent to  EMBED Equation.DSMT4  , and 0 to any power is 0. Therefore, the possible solution is the vertical line x = 0; since this is not a function, x = 0 is an extraneous solution. 2. Explore several different a-values by clicking " or ". a. For what a-values is the function increasing? Why? Answer: When a > 1,  EMBED Equation.DSMT4  is increasing. This is because if  EMBED Equation.DSMT4  then  EMBED Equation.DSMT4 . When a > 1, as the power increases the resulting value increases. For example, EMBED Equation.DSMT4  because  EMBED Equation.DSMT4 .  b. For what a-values is the function decreasing? Why? Answer: When 0 < a < 1,  EMBED Equation.DSMT4  is decreasing. This is because if  EMBED Equation.DSMT4 then EMBED Equation.DSMT4 . When 0 < a < 1, as the power decreases the resulting value decreases. For example,  EMBED Equation.DSMT4  because  EMBED Equation.DSMT4 . TI-Nspire Navigator Opportunity: Screen Capture See Note 2 at the end of this lesson. 3. Explore several different a-values by clicking " or ". a. For each a-value, identify the x-intercept of the function. Interpret your results. Answer: The x-intercept is always (1, 0) because  EMBED Equation.DSMT4  for all a, which can be written in exponential form  EMBED Equation.DSMT4 . b. When a > 0, why is there no y-intercept? Answer: For there to be a y-intercept, then x would be 0. This implies that  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 , which is impossible. c. For each a-value, what part of point  EMBED Equation.DSMT4 remains the same? Interpret your results. Answer: For all a-values, the point  EMBED Equation.DSMT4 has the coordinates (a, 1) since  EMBED Equation.DSMT4  can be written in exponential form  EMBED Equation.DSMT4 . TI-Nspire Navigator Opportunity: Screen Capture See Note 3 at the end of this lesson. 4. Explore several different a-values by clicking " or ", such that a > 1. a. What does f(x) approach as x approaches  EMBED Equation.DSMT4 ? Explain. Answer: EMBED Equation.DSMT4 ; As you input larger and larger positive values for x, f(x) gets larger and larger. b. What does f(x) approach as x approaches 0? Explain. Answer:  EMBED Equation.DSMT4 ; As you input positive x-values that approach 0, f(x) gets smaller and smaller. c. What is the equation of the vertical asymptote? Answer:  EMBED Equation.DSMT4  TI-Nspire Navigator Opportunity: Screen Capture See Note 4 at the end of this lesson. 5. Explore several different a-values by clicking " or ", such that 0 < a < 1. a. What does f(x) approach as x approaches EMBED Equation.DSMT4 ? Explain. Answer:  EMBED Equation.DSMT4 ; As you input larger and larger positive values for x, f(x) get smaller and smaller. b. What does f(x) approach as x approaches 0? Explain. Answer:  EMBED Equation.DSMT4 ; As you input positive x-values that approach 0, f(x) gets larger and larger. c. What is the equation of the vertical asymptote? Answer:  EMBED Equation.DSMT4  TI-Nspire Navigator Opportunity: Screen Capture See Note 5 at the end of this lesson. 6. Find the domain and range for the family of logarithmic functions  EMBED Equation.DSMT4  where a > 0 and a `" 1. Answer: The domain is  EMBED Equation.DSMT4 , and the range is  EMBED Equation.DSMT4 . TI-Nspire Navigator Opportunity: Quick Poll and Live Presenter See Note 6 at the end of this lesson. 7. Gail believes  EMBED Equation.DSMT4  will eventually intersect the y-axis. Is she correct? Why or why not? Answer: Gail is incorrect. The function will never cross the y-axis. It will only approach the y-axis.TI-Nspire Navigator Opportunity: Quick Poll See Note 7 at the end of this lesson. 8. Judy believes  EMBED Equation.DSMT4  has a horizontal asymptote. Is she correct? Why or why not? Answer: Judy is incorrect. When EMBED Equation.DSMT4 , the function decreases very slowly when 0 < a < 1 and increases very slowly when a > 1 . TI-Nspire Navigator Opportunity: Quick Poll See Note 8 at the end of this lesson. Wrap Up Upon completion of the discussion, the teacher should ensure that students understand that for the graph of  EMBED Equation.DSMT4 : The conditions EMBED Equation.DSMT4  are necessary. When a > 1, the function is increasing and when  EMBED Equation.DSMT4 , the function is decreasing. The x-intercept is always (1, 0) and there is never a y-intercept. When a > 1, the function approaches  EMBED Equation.DSMT4  as x approaches  EMBED Equation.DSMT4 , and the function approaches  EMBED Equation.DSMT4  as x approaches 0. When 0 < a < 1, the function approaches  EMBED Equation.DSMT4  as x approaches  EMBED Equation.DSMT4 , and the function approaches  EMBED Equation.DSMT4  as x approaches 0. The domain is  EMBED Equation.DSMT4  and the range is  EMBED Equation.DSMT4 . The function has a vertical asymptote of EMBED Equation.DSMT4 . 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hM)h=*wCJEHOJQJUV^JaJ$hM)h=*wCJEHOJQJ^JaJ-jhM)h=*wCJEHOJQJU^JaJresenter: You may want to demonstrate how to change the a-values by clicking the arrows using Live Presenter. Note 2 Question 2a and 2b, Screen Capture: Take a Screen Capture of page 1.2 where students are on different a-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 where students are on different a-values. As a class, discuss the various cases that occur. Note 4 Question 4a4c, Screen Capture: Take a Screen Capture of page 1.2 where students are on different a-values. As a class, discuss the various cases that occur. Note 5 Question 5a5c, Screen Capture: Take a Screen Capture of page 1.2 where students are on different a-values. As a class, discuss the various cases that occur. Note 6 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 of the graphs, the possible x- and y-values are similar. Note 7 Question 7, Quick Poll (Open Response): Send an Open Response Quick Poll, asking students to submit their answer to question 7.For students having difficulty, use Live Presenter and change the a-value to 0.9 and the y-max to 200. Repeatedly zoom in on the y-intercept. Note 8 Question 8, Quick Poll (Open Response): Send an Open Response Quick Poll, asking students to submit their answer to question 8.      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com.apple.print.ticket.client com.apple.print.pm.PostScript com.apple.print.ticket.modDate 2003-07-01T17:49:36Z com.apple.print.ticket.stateFlag 1 com.apple.print.ticket.APIVersion 00.20 com.apple.print.ticket.privateLock com.apple.print.ticket.type com.apple.print.PaperInfoTicket com.apple.print.ticket.APIVersion 00.20 com.apple.print.ticket.privateLock com.apple.print.ticket.type com.apple.print.PageFormatTicket 8BIMxHH@Rg(HH(dh 8BIM,,8BIM&?8BIM 8BIM8BIM 8BIM 8BIM' 8BIM5-8BIM8BIM@@8BIM8BIMUPnTI Logo 2-line-KnPnullboundsObjcRct1Top longLeftlongBtomlongPRghtlongnslicesVlLsObjcslicesliceIDlonggroupIDlongoriginenum ESliceOrigin autoGeneratedTypeenum ESliceTypeImg boundsObjcRct1Top longLeftlongBtomlongPRghtlongnurlTEXTnullTEXTMsgeTEXTaltTagTEXTcellTextIsHTMLboolcellTextTEXT horzAlignenumESliceHorzAligndefault vertAlignenumESliceVertAligndefault bgColorTypeenumESliceBGColorTypeNone topOutsetlong leftOutsetlong bottomOutsetlong rightOutsetlong8BIM( ?8BIM8BIM )$ JFIFHH Adobe_CMAdobed         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