Absolute Value Functions

Transcription

1 Absolute Value Functions 15. Provide an opportunity for each group to share the mind maps with the larger group. Discuss similarities, differences, and key points brought forth by participants. 16. Distribute the vocabulary organizer template to each participant. Ask participants to construct a vocabulary model for the term absolute value functions. Maximizing Algebra II Performance Explore/Explain/Elaborate 3 185

2 Absolute Value Functions Maximizing Algebra II Performance Explore/Explain/Elaborate 3 195

3 Absolute Value Functions Maximizing Algebra II Performance Explore/Explain/Elaborate 3 196

4 Absolute Value Functions Maximizing Algebra II Performance Explore/Explain/Elaborate 3 197

5 Grade level/course: Algebra II; Regular Dates: 11/2, 3 Topic: Absolute Value functions Essential question: What are the critical attributes of absolute value functions? How does changing the parameters of the function affect the graph? How do you solve absolute value functions? Focus TEKS: 2A.1A, 2A.1, 2A.2A, 2A.4A, 2A.4B Prerequisite knowledge: Graph linear equations Basic understanding of Absolute Value Engage: What factors might affect the cost of homeowner's insurance? Does the distance to a fire station or fire hydrant matter to your insurance rates? Why? How far do you think your home i from the fire station? MTC Fire Station Problem 1-12: Activate Prior Knowledge. Determine the businesses that are on either side of the Fire station. Apply that to distance and direction. These problems can be worked independently or in partners in order to formatively assess prior knowledge about linear equations, domain, range and data. Explore: MTC Fire Station Problem Create a deeper understanding of limiting domains and how they effect equations. Explain: MTC Fire Station Problem: Discuss questions24-28: Discuss the differences between the Algebra I concept of absolute value as an expression of distance. how to regress each line from the data points; graph the equations on the calculator and determine similarities and differences. Elaborate/Extend: MTC Fire Station Problem: problem 10 and discussion on questions 11 & 12: Adjusting the window and the relation to the data points. Evaluate: HW: MTC Fire Station Problem: Part 2 question 1-5 (attached after next section) Knowledge check Materials/Resources: MTC Fire Station Problem houses Graphing Calculators Reflection: Best Practices : Consider Student choice x Technology x Hands on activity x Differentiated instruction x Small group Multiple intelligence x Use of visuals x Higher order thinking skills x Literacy strategies x Real world connections Weaves in interventions

6 Meaningful assessments

7 The Fire Station Problem A fire station is located at Main Street and has buildings at every block to the right and to the left. You will investigate the relationship between the address number on a building and its distance from the fire station. 1. Complete the table below that relates the address of a building (x) with its distance in blocks from the fire station (y). Address Number (x) Distance in Blocks from the Fire Station (y) Where are these building in relationship to the Fire Station? 3. Draw a scatterplots that represents the data in the table. 5 Distance in Blocks from the Fire

8 Address Number 4. Make a scatterplot of your data using your graphing calculator, Describe your viewing window. 5. What function or functions might you use to describe the scatterplot? 6. Find the linear functions that pass through the data points. What is the slope? What is the y-intercept? 7. Complete the table below that relates the address of a building (x) with its distance in blocks from the fire station (y). Address Number (x) Distance in Blocks from the Fire Station (y) Where are these building in relationship to the Fire Station?

9 9. Draw a scatterplots that represents the data in the table. 5 Distance in Blocks from the Fire Address Number 10. Make a scatterplot of your data using your graphing calculator and the given viewing window. 11. What function or functions might you use to describe the scatterplot? 12. Find the linear functions that pass through the data points. What is the slope? What is the y-intercept?

10 13. Using your data from the two previous tables, complete the following table. Address Number (x) Distance in Blocks from the Fire Station (y) 14. Draw a scatterplots that represents the data in the table. 5 Distance in Blocks from the Fire Address Number 15. Describe your graph.

11 16. Graph the equations on your calculator. How are they similar? How are they different? 17. Where do the equations fit the graph of the data points? Where do the equations not fit the graph of the data points? 18. What is the domain and range of each of the linear functions that model the fire station problem? Linear Equation Domain Range Y1= Y2= 19. What is the domain and range of the data set? 20. How do the domains and ranges compare? 21. How do the linear equations compare to each other? 22. At what point do the graphs of the lines intersect? 23. Which parts of the graphs of the lines model our data set? Which parts do not? Why?

12 Numerically, when we consider the distance of a number form 0 on a number line, we call the distance the absolute value of the number. For example 7 =7, since -7 has a distance of 7 from 0. Similarly, 7 =7 since 7 has a distance of 7 from 0. The x-coordinate below represent locations on a number line. The y-coordinates represent the distance that location is from 0. For the given x-values, use the number line to find the corresponding y-values. 24. Make a scatterplot of y versus x on your graphing Calculator. Describe your window. Sketch your scatterplot. 25. What is the shape of the scatterplot? 26. Find two linear functions that model this situation. Graph these functions.

13 27. Restrict the domain so that the model is an even better fit. Graph the restricted functions. 28. Now try 3=. How does this function compare to the two linear functions? h <0 Algebraically, the definition of the absolute value function is = h = is a new parent function to add to your list of parent functions. With which other parent functions are you familiar? 30. What are some characteristics of the absolute value function?

14 The Fire Station Problem KEY A fire station is located at Main Street and has buildings at every block to the right and to the left. You will investigate the relationship between the address number on a building and its distance from the fire station. 1. Complete the table below that relates the address of a building (x) with its distance in blocks from the fire station (y). Address Number (x) Distance in Blocks from the Fire Station (y) Where are these building in relationship to the Fire Station? The buildings are to the right of the fire station. 3. Draw a scatterplots that represents the data in the table. 5 Distance in Blocks from the Fire

15 Address Number 4. Make a scatterplot of your data using your graphing calculator and the given viewing window. Check calculators. 5. What function or functions might you use to describe the scatterplot? A linear equation 6. Find the linear functions that pass through the data points. What is the slope? What is the y-intercept? = Complete the table below that relates the address of a building (x) with its distance in blocks from the fire station (y). Address Number (x) Distance in Blocks from the Fire Station (y)

16 8. Where are these building in relationship to the Fire Station? To the right of the fire station. 9. Draw a scatterplots that represents the data in the table. 5 Distance in Blocks from the Fire Address Number 10. Make a scatterplot of your data using your graphing calculator and the given viewing window. 11. What function or functions might you use to describe the scatterplot? (Answers may Vary) Parts of two different lines. 12. Find the linear functions that pass through the data points. What is the slope? What is the y-intercept? = +12

17 13. Using your data from the two previous tables, complete the following table. Address Number (x) Distance in Blocks from the Fire Station (y) Draw a scatterplots that represents the data in the table. 5 Distance in Blocks from the Fire Address Number 15. Describe your graph. (Answers vary) Looks like a V one side is the reflection of the other side.

18 16. Graph the equations on your calculator. How are they similar? How are they different? (Answers vary) The lines are the same as the data above the x axis there is no data below the axis. 17. Where do the equations fit the graph of the data points? Where do the equations not fit the graph of the data points? Y1 is the same as the data when 1200, but only the equation exists when x<1200. Y2 is the same as the data when x <1200, but only the equation exists when What is the domain and range of each of the linear functions that model the fire station problem? Linear Equation Domain Range Y1=.01x - 12 All real numbers All real numbers Y2= -.01x + 12 All real numbers All real numbers 19. What is the domain and range of the data set? D={ 800, 900, 1000, 1100, 1200, 1300, 1400, 1500, 1600} R ={0, 1, 2, 3, 4} 20. How do the domains and ranges compare? (Answers Vary) The domain and range of the linear equations are not restricted, the data set is restricted, discrete so must be in roster form. 21. How do the linear equations compare to each other? Y 1 = -(Y 2 ) 22. At what point do the graphs of the lines intersect? (1200, 0) 23. Which parts of the graphs of the lines model our data set? Which parts do not? Why? Y1 is the same when x 0. Y2 is the same when x < 0.

19 Numerically, when we consider the distance of a number form 0 on a number line, we call the distance the absolute value of the number. For example 7 =7, since -7 has a distance of 7 from 0. Similarly, 7 =7 since 7 has a distance of 7 from 0. The x-coordinate below represent locations on a number line. The y-coordinates represent the distance that location is from 0. For the given x-values, use the number line to find the corresponding y-values Make a scatterplot of y versus x on your graphing Calculator. Describe your window. Sketch your scatterplot. 25. What is the shape of the scatterplot? It makes a V 26. Find two linear functions that model this situation. Graph these functions. Y = -x and Y = x

20 27. Restrict the domain so that the model is an even better fit. Graph the restricted functions. Y = -x when x < 0 and Y = x when x > 0 or = <0 = 0 in the calculator 28. Now try 3=. How does this function compare to the two linear functions? Exactly the same h <0 Algebraically, the definition of the absolute value function is = h = is a new parent function to add to your list of parent functions. With which other parent functions are you familiar? Answers Vary 30. What are some characteristics of the absolute value function? Domain All Real Numbers, Range y>0, the fact that two pieces of the equations are linear.