Teacher Notes 1/16/13(c) Evergreen Public Schools 0 You might want to update prices at Office MaxOffice Max PRACTICES: 1 Problem solving & perseverance, 3 Construct viable arguments and critique the reasoning of others, 4 Model with mathematics, 6 Attend to precision. The slides for DUPLEX laser printer and Precision at the end of the presentation are optional.

Linear Functions 1/16/13 1 (c) Evergreen Public Schools

What could you shop for by price? 1/16/13(c) Evergreen Public Schools 2

The least expensive way to make copies is through the district Print Shop. They charge us $0.014 for one single-sided black and white copy. Would it cost the math department less money for one single sided copy if we bought a black and white laser printer? 1/16/13 3 (c) Evergreen Public Schools

1/16/13(c) Evergreen Public Schools 4 Linear Function 2a I can represent a situation using tables, graphs, equations and in words. What does it mean to “represent” a function?

Need 1 of toner cartridge High Yield Black Toner Cartridge Yields up to 2,600 pages Price: $70 Black Toner Cartridge Yields up to 1,200 pages Price: $45 Need both Drum Cartridge Yields up to 12,000 page Price: $105 Multipurpose Paper 5,000/Case Price:$50 1/16/13 5 (c) Evergreen Public Schools

1/16/13(c) Evergreen Public Schools 6 Mr. Benson manages the district Print Shop. Write a letter to Mr. Benson to convince him to whether or not to buy the math department laser printers. Support your answer with mathematical information in your letter. Listen to the presentations of others. - Write one complement about their mathematics. - Write one question about their mathematics.

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1/16/13(c) Evergreen Public Schools 8 The graph shows the estimated cost of buying and using an inkjet printer.

The estimated cost of buying and using a home laser printer can be found with the equation L(x) = x where x is the number of pages. 1.Use the graph to decide whether it is more cost effective to use the laser printer or the school copier for 10, 100, 1000 or 5000 copies. 2.How many pages do you need to print for the laser printer to be more cost effective? 3.Describe the domain and range of each function. 1/16/13(c) Evergreen Public Schools 9

1/16/13(c) Evergreen Public Schools 10 LF2b I can identify a function and its domain and range, and use function notation to find quantities in the domain or range.domain and rangefunction notation LF3cI can compare two functions or one function represented two different ways. Find the inverse function LF4 I can solve a system of equations with a table or by graphing.

Laser Printer Function of Pages The district Print Shop charges $0.014for single-sided white paper. Write the equation for The Print Shop costs are determined by the function, where the cost, d(x), is a function of the number of pages printed, x. Use your equation to find: a) Find d(10) b) Find d(150) C) Find d(800) 6.Use your equation to solve for x. a) d(x) = 1.00 b) d(x) = 5.00d c) (x) = Write a function to find the number of pages printed, p(x), as a function of the cost, x. p(x) = ____________________________

Laser Printer Function of Pages The function p(x) is the inverse function of d(x), because they undo each other. In other words, when d(a) = b, then p(b) = a.inverse function

Laser Printer Function of Pages 8.Use p(x) = to verify your answers to #16. 9.Is the function, p(x), linear? Support your answer with words, numbers and/or diagrams. 10.Find the value of x where d(x) = p(x). 11.Describe the domain and range of each function. Be Prepared to Present Your Solution to the Class domainrange d(x)d(x) p(x)p(x)

Laser Printer Function of Pages As you already found the estimated cost as a function of the number of pages printed is L(x) = x For L(x), _____________ is the input and ________________ is the out put. Use your equation to solve for x. a)L(x) = 110 b)L(x) = 150 c)L(x) = 10

Laser Printer Function of Pages 14.Now find N(x), the inverse function of L(x). 15.For N(x), For L(x), _____________ is the input and ________________ is the out put. 16.Use p(x) = to verify your answers to # Is the function, N(x), linear? Support your answer with words, numbers and/or diagrams.

Laser Printer Function of Pages 18.Find the value of x where L(x) = N(x). 19.Describe the domain and range of each function. domainrange d(x)d(x) p(x)p(x)

Laser Printer Function of Pages 20.In #18 and #23, you were asked to find the value of x where the function and its inverse function were equal to each other. What do your answers to #18 and #23 have in common? 21.In #19 and #24, you were asked to find the domain and range of function and its inverse function. What do your answers to #19 and #24 have in common? Be Prepared to Present Your Solution to the Class

Linear Functions 3a: I can identify the characteristics of linear functions and use function notation. 1/16/13(c) Evergreen Public Schools 18

Printer Speeds Mr. Aho wants to know if the laser printer is faster than the inkjet printer he already has. He used his inkjet printer to print pages and recorded the pages and times in the table. He looked up the specifications for the laser printer he wants to buy and made a graph to determine the time it would take to print pages. 1/16/13(c) Evergreen Public Schools 19

Printer Speeds inkjet printerlaser printer 1/16/13(c) Evergreen Public Schools 20 Pages Time (sec)

6.Describe the input and output of the laser printer function, L(x). 7. Was the input or the output changed in #5? Support your answer with words, numbers and/or diagrams. 8.Was the input or the output changed in #6? Support your answer with words, numbers and/or diagrams. Be Prepared to Present Your Solution 1/16/13(c) Evergreen Public Schools 21

9.Mr. Blake liked Mrs. Krsak’s idea of buying a laser printer and decided to buy a laser printer for his home at the beginning of second semester. He has one Algebra 1 class fall semester, and three Algebra 1 classes next semester. How does this change the graph of the function L(x)? 10.Mrs. Doss also decided to buy a laser printer for her home at the beginning of second semester. Mrs. Doss’s schedule is changing also. She has 70 fewer students in Algebra 1 classes next semester because she’s teaching more Geometry classes. How does this change the graph of the function L(x)? 1/16/13(c) Evergreen Public Schools 22

11. Describe the input and output of the laser printer function, L(x). 12. Was the input or the output changed in #9? Support your answer with words, numbers and/or diagrams. 13. Was the input or the output changed in #10? Support your answer with words, numbers and/or diagrams. Be Prepared to Present Your Solution 1/16/13(c) Evergreen Public Schools 23

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The district Print Shop charges $0.014for single- sided white paper. 14. Write the equation for The Print Shop costs are determined by the function, where the cost, d(x), is a function of the number of pages printed, x. 15. Use your equation to find d(10), d(150), and d(800) 16. Use your equation to solve for x: d(x) = 1.00, d(x) = 5.00, and d(x) = Write a function to find the number of pages printed, p(x), as a function of the cost, x. 1/16/13(c) Evergreen Public Schools 25

The function p(x) is the inverse function of d(x), because they undo each other. In other words, when d(a) = b, then p(b) = a.inverse function 18. Use p(x) = to verify your answers to # Is the function, p(x), linear? Support your answer with words, numbers and/or diagrams. 1/16/13(c) Evergreen Public Schools 26

20.Find the value of x where d(x) = p(x). 21.Describe the domain and range of each function. Be Prepared to Present Your Solution 1/16/13(c) Evergreen Public Schools 27

As you already found the estimated cost as a function of the number of pages printed is L(x) = x For L(x), _______ is the input and _______ is the out put. 23. Use your equation to solve for x. L(x) = 110 L(x) = 150 L(x) = 10 1/16/13(c) Evergreen Public Schools 28

24. Now find N(x), the inverse function of L(x). 25. For N(x), _______ is the input and _______ is the out put. 26. Use p(x) = to verify your answers to #23. 1/16/13(c) Evergreen Public Schools 29

27. Is the function, N(x), linear? Support your answer with words, numbers and/or diagrams. 28.Find the value of x where L(x) = N(x). 29.Describe the domain and range of each function. 1/16/13(c) Evergreen Public Schools 30

30. In #18 and #23, you were asked to find the value of x where the function and its inverse function were equal to each other. What do your answers to #18 and #23 have in common? 31.In #19 and #24, you were asked to find the domain and range of function and its inverse function. What do your answers to #19 and #24 have in common? Be Prepared to Present Your Solution 1/16/13(c) Evergreen Public Schools 31

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Mr. Aho wants to know if the laser printer is faster than the inkjet printer he already has. He used his inkjet printer to print pages and recorded the pages and times in the table. He looked up the specifications for the laser printer he wants to buy and made a graph to determine the time it would take to print pages. 32.Use the laser printer table and the inkjet printer graph to answer Mr. Aho’s question. 33.Is Mr. Aho’s inkjet printer function, represented by a table, linear? Support your answer with words, numbers and/or diagrams. Be Prepared to Present Your Solution 1/16/13(c) Evergreen Public Schools 33

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The school district is trying to save money. They put all teachers on a $900 budget for what they order from the Print Shop. At the end of first semester Mr. Kaden had ordered 48,963 copies (both one-sided and two-sided, and white and color paper) from the Print Shop and was charged $ /16/13(c) Evergreen Public Schools 35

34. Write a function Mr. Kaden can use to determine what is left in his budget from the number of copies he has made. Be sure to define the variables you use. 35.Is the budget function linear? Support your answer with words, numbers and/or diagrams. 36. Make a graph that Mr. Kaden can refer to and find out how much of his budget remains. Be sure to show in your graph when Mr. Kaden has no money left in his budget. 37. Find the maximum number of copies Mr. Kaden can make. Support your answer with information from your graph. Be Prepared to Present Your Solution 1/16/13(c) Evergreen Public Schools 36

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If we buy paper, toner and drum cartridges, as we need them would the graph really look like this? If you think this is the graph, support your answer with words, numbers and/or diagrams. If you think the graph would be different, describe or sketch the new graph. 1/16/13(c) Evergreen Public Schools 38

Throughout this lesson, we have used the function L(x) = x to estimate the printer costs. 1/16/13(c) Evergreen Public Schools 39

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Mr. Aho made a detailed graph of the costs of the laser printer. Support all of your answers with information from the diagram. 38.Is the graph a function? 39.Is the graph a linear function? 40.Describe the domain and range of the values shown on the graph. 41.Find the start up cost of the printer. What is included in that cost? 1/16/13(c) Evergreen Public Schools 41

Support all of your answers with information from the diagram. 38. Is the graph a function? 39. Is the graph a linear function? 40. Describe the domain and range of the values shown on the graph. 41. Find the start up cost of the printer. What is included in that cost? 42. Pick two different intervals. Describe what is happening in each of the intervals. Be Prepared to Present Your Solution 1/16/13(c) Evergreen Public Schools 42

The least expensive way to make copies is through the district Print Shop. They charge us $0.02 for one single-sided black and white copy. Would it cost the math department less money for one single sided copy if we bought a black and white laser printer? 1/16/13 43 (c) Evergreen Public Schools

How many decimal places did you use when you calculated the cost of toner for 1 page? LIST ANSWERS: How many places are necessary? How can you find a more accurate answer? 1/16/13(c) Evergreen Public Schools 44

HOW TO USE THE MEMORY OF A GRAPHING CALCULATOR Find cost of toner for 1 page. Store this cost in your calculator as “T” STO> T Find cost of the drum for 1 page. Store this cost in your calculator as “D” STO> D Find cost of 1 piece of paper. Store this cost in your calculator as “P” The cost of 1 page T + D + P Find cost of toner for 1 page. Store this cost in your calculator as “T” STO> C 1/16/13(c) Evergreen Public Schools 45