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New Jersey Center for Teaching and Learning Progressive Mathematics Initiative.

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  1. New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org
  2. 8th Grade Functions 2012-11-19 www.njctl.org
  3. Setting the PowerPoint View Use Normal View for the Interactive Elements To use the interactive elements in this presentation, do not select the Slide Show view. Instead, select Normal view and follow these steps to set the view as large as possible: On the View menu, select Normal. Close the Slides tab on the left. In the upper right corner next to the Help button, click the ^ to minimize the ribbon at the top of the screen.  On the View menu, confirm that Ruler is deselected. On the View tab, click Fit to Window. Use Slide Show View to Administer Assessment Items To administer the numbered assessment items in this presentation, use the Slide Show view. (See Slide 13 for an example.)
  4. Table of Contents click on the topic to go to that section Relationships and Functions Domain and Range Vertical Line Test Linear Vs. Non-Linear Functions Interpreting with Functions Analyzing a Graph Comparing Different Representations of Functions
  5. Relationships and Functions Return to Table of Contents
  6. A relation is a set of ordered pairs. A function is a relation where every input (x) has exactly one output (y). Let's look at a few examples of relations. Some are functions and some are not.
  7. Let's look at these a little closer This is a function. Each state (input) has only one capital (output). This is not a function. Each person (input) has more than one grandparent. (output).
  8. Each number in the x (input) column has only one y (output) as an answer so it is a function. Notice that 4 has an output of both 3 and 6. Therefore it cannot be a function.
  9. Let's compare the last problem with another one. How do they differ? Are either of them functions? B A click to reveal In table A, there is one x (4) that has two different y's (3 & 6). This is NOT a function. In table B, there are two different x's (3 & 5) that have the same y (8). This is a function.
  10. Consider this: Two people in a class (input) may have the same birthday (output). However, any one person (input) does not have more than one birthday (output). Sarah John Dwayne Alan Jada April 9 Jan. 23 Sept. 14 Feb. 23 May 5 July 19 Sarah John Dwayne Alan Jada June 8 Oct. 28 March 18 Oct. 28 Dec. 29 John and Alan have the same birthday. That can happen. It is a function. Dwayne's birthday is February 23 and September 14. This is not possible. It is not a function.
  11. Notice that there are different ways to express functions. They may be in words, a table, ordered pairs, a graph, or like the last slide, a function map. In a function map one side is the input and the other is output. An arrow extends from each input element to its matching output element. Example: Input Output 1 2 3 4 A B C D
  12. Functions can also be written as sets of ordered pairs. All of the x coordinates are the input and all of the y coordinates are the output. To figure out if sets of ordered pairs are functions, make a function map or table first. {(2, 6), (3, 7), (4, 8), (5, 9)} This is a function because every input has exactly one output.
  13. Is the following a function? 1 1 2 3 4 5 4 3 2 A Yes B No
  14. 2 Is the following a function? A Yes B No
  15. Is the following a function? 3 A person's social security number A Yes B No
  16. Is the following a function? 4 A Yes B No
  17. 5 Is the following a function? Any number times 3. A Yes B No
  18. Is the following a function? 6 1 2 3 4 5 6 7 8 9 10 11 A Yes B No
  19. 7 Is the following a function? {(1, 2), (2, 3), (3, 4), (4, 5)} A Yes B No
  20. 8 Is the following a function? {(1, 2), (1, 3), (2, 4), (3, 4), (3, 5)} A Yes B No
  21. Is the following a function? 9 A Yes B No
  22. Domain and Range Return to Table of Contents
  23. Any set of numbers that is the input of a relation or function is called the domain. Any set of numbers that is the output of a relation or function is called the range. Domain: {1, 2, 3, 4} Range: {42, 98, 106, 125}
  24. Which numbers are part of the domain and which are part of the range? Rule: x + 5 = y 8 1 7 2 4 9 3 6 Domain Range
  25. How about when we have ordered pairs? Which numbers are in the domain? Which are in the range? {(1,8), (3, 10), (5, 12), (7, 14), (9, 16)} 16 10 1 Domain Range 8 9 3 7 12 14 5
  26. Which numbers are in the domain? 10 A 2 10 20 30 40 50 2 4 6 8 10 B 10 C 12 D 8 E 20
  27. Which numbers are in the range? 11 A 2 10 20 30 40 50 2 4 6 8 10 B 10 C 12 D 8 E 20
  28. Which numbers are part of the domain? 12 {(24, 12), (22, 11), (20, 10), (18, 9)} A 12 B 20 C 24 D 11 E 9
  29. Which numbers are part of the range? 13 {(24, 12), (22, 11), (20, 10), (18, 9)} A 12 B 20 C 24 D 11 E 9
  30. Vertical Line Test Return to Table of Contents
  31. A function can also be expressed as a graph.
  32. In order to verify that a graph is a function you can use the Vertical Line Test. In order to be a function, there may be only one point that goes through any given vertical line on the graph. Function Function Not a Function Not a Function
  33. A linear equation with a slope of zero is a function. Prove that a linear equation with an undefined slope cannot be a function. Undefined slope Slope = 0
  34. Is this a function? 14 A Yes B No
  35. Is this a function? 15 A Yes B No
  36. Is this a function? 16 A Yes B No
  37. Is this a function? 17 A Yes B No
  38. Is this a function? 18 A Yes B No
  39. Is this a function? 19 A Yes B No
  40. Is this a function? 20 A Yes B No
  41. Linear vs. Non-Linear Functions Return to Table of Contents
  42. A Linear Function is a graph that is represented by a straight line. As an equation it is written as y=mx+b (see Graphing Equations Unit). Linear functions will never have an exponent larger than 1 in the equation. y = -1/4x +4 y = 2x - 3
  43. Linear Functions are incremental. y = 2x +3 +1 +3 +1 +3 +1 +3 +1 +3
  44. A Non-linear Function is any function that is not represented as a straight line. Non-linear functions often have exponents in their equation. y = x2 y = x3 + 2x2 - x
  45. Non-linear functions are not incremental. y = x2 + 2 +1 -3 +1 -1 +1 +1 +1 +3
  46. 21 The following is a linear function. A True B False
  47. The following is a linear function. 22 A True B False
  48. The following is a linear function. 23 A True B False
  49. 24 The following is a linear function. y = 5x + 7 A True B False
  50. The following is a linear function. 25 A True B False
  51. The following is a linear function 26 A True B False
  52. The following is a linear function. 27 A True B False
  53. The following is a function. 28 y = 2x2 + 3 A True B False
  54. 29 The following is a function. A True B False
  55. Interpreting with Functions Return to Table of Contents
  56. Any function can be written as a table, graph, verbal model, or equation. We can find the rate of change and the y-intercept from any of these representations. Review Remember, to find slope we can use the formula: Slope = y2 - y1 x2 - x1 To find the y-intercept (initial value) we look to where x = 0
  57. Look at the given table. We can use any two values to determine slope. We can find where x = 0 to determine the y-intercept. y2 - y1 7 - 4 3 x2 - x1 = Slope = = = 3 2 - 1 1 y-intercept = 1 since when x = 0, y = 1
  58. Sometimes a table will not show the x-coordinate of zero. In that case you need to figure it out. There are a few ways to do it. y = 3x - 4 There are two ways to find the y-intercept here.
  59. y = 3x - 4 0 +3 -3 +3 +3 We can simply continue the table so that we can find y when x = 0. We see that the y's are moving at intervals of +3. In order to work backwards to where x = 0 we need to subtract 3 from y. -3 - (-1) = -4 so when x = 0, y = -4. Note: This technique works best when the table is close to x = 0
  60. y = 3x - 4 Another technique would be to substitute for x and solve for y using the equation. y = 3x - 4 y = 3(0) - 4 y = 0 - 4 y = -4 Note: This technique only works if you have the equation.
  61. What is the slope of the following table? 30
  62. What is the y-intercept of the following table? 31
  63. What is the slope of this table? 32
  64. What is the y-intercept of this table? 33 y = x - 6
  65. What is the y-intercept of this table? 34
  66. Carla puts away a certain amount of money per week. She started out with a certain amount. How could we figure out what she started out with and what she puts in per week?
  67. If we look at the slope or rate of change we can figure out how much she puts in each week. So what is the slope of this table? $8 So, Carla puts away $8 per week.
  68. If we look at the initial value or y-intercept we can figure out how much Carla started with because the y-intercept is where Carla started without any weeks of savings. The initial value is 74 Therefore, Carla started with $74 in her account.
  69. Try one. Rob is training for a marathon. He is increasing his run each week. Based on the table: How much more will he run each week? How many miles was he running before training? How many weeks until he runs a full marathon 26.2 miles?
  70. Look at the given scenario. Luke wants to buy cookies online for $12 per box with a flat fee for shipping and handling of $4.00. Notice in this scenario that the initial value would be $4.00. However, in a real-life scenario, one does not pay for shipping and handling if nothing is being bought. BE CAREFUL with your interpretation so that it makes sense with the scenario.
  71. 35 Evan is buying pizza from a store that sells each pie for a given price and each topping for set price as well. Use the table to identify the cost of each topping.
  72. 36 Evan is buying pizza from a store that sells each pie for a given price and each topping for set price as well. How much is a pie without any toppings.
  73. 37 Evan is buying pizza from a store that sells each pie for a given price and each topping for set price as well. The toppings would be the: Rate of Change A B Initial Value
  74. 38 Evan is buying pizza from a store that sells each pie for a given price and each topping for set price as well. The pizza would be the: Rate of Change A B Initial Value
  75. Look at the given graph. What is the slope? What is the y-intercept? +2 Slope is often referred to as . To find the rise count the number up or down to the next point. To find the run count the number left or right to the next point. The rise is +2 and therunis2. rise run +2 rise run 2 2 1 = = So, the slope is 1.
  76. Now let's look at the y-intercept. All you need to do for this is find the point that crosses the y-axis and that is the y-intercept. In this case, the y-intercept is -4. Hint: put your finger on the y-axis and follow it until you reach the line
  77. 39 What is the slope of this graph?
  78. 40 What is the y-intercept of this graph?
  79. Jorge is emptying his pool. According to this graph, how many gallons were in the pool to start? How quickly is the pool draining?
  80. Jorge is emptying his pool. According to this graph, how many gallons were in the pool to start? The initial value is where we start at the beginning of the scenario. Notice that when x = 0 there are 2000 gallons in the pool. Therefore Jorge started with 2000 gallons.
  81. Jorge is emptying his pool. How quickly is the pool draining? To find this we can find the rate of change or the slope. Find two well plotted points and find the rise over run. 2000 1750 1500 1250 1000 750 500 250 0 -1000 + 9 Gallons -1000 9 = -111 approx. 3 6 9 12 15 18 21 24 This can be interpreted as emptying 1000 gallons every 9 hours or approximately 111 gallons per hour. Time in hours
  82. 2000 1750 1500 1250 1000 750 500 250 0 Jorge is emptying his pool. How quickly is the pool draining? To find this we can find the rate of change or the slope. Another way to find this would be to make ordered pairs and use the slope formula. (0, 2000) (18, 0) 0 - 2000-2000 18-0 18 -1000 + 9 Gallons Notice the slope is negative because the pool is being emptied. 3 6 9 12 15 18 21 24 Time in hours -1000 9 = -111 approx. = =
  83. Sandra is going to a buffet. The meal is a fixed price but she has to pay for each soda she drinks. What is the initial value? Be prepared to explain how it relates to the scenario. 41
  84. 42 Sandra is going to a buffet. The meal is a fixed price but she has to pay for each soda she drinks. What is the slope? Use the points (0, 15) and (6, 25). Be prepared to explain how it relates to the scenario.
  85. 43 Nora is selling hoodies for a fundraiser at $15 per item. She pays a total of $225 for them. How many does she have to sell to break even?
  86. What does the circled coordinate mean? 44 This tree grows 4 feet every year A This tree was planted when it was 4 feet B C There are 4 trees planted The tree was planted when it was 4 years old. D
  87. How many feet does the tree grow every year? 45 A 1 foot B 2 feet C 4 feet 12 feet D
  88. Sometimes you will be given a scenario. Turning it into an equation may make it easier to analyze. In a linear equation, we use the format of y=mx+b where m is the slope (rate of change) and b is the y-intercept (initial value). This is called slope intercept form. In a description, the slope usually pertains to something that will change and is unknown. The y-intercept (initial value) will be a number that stays the same in the scenario, like a flat fee.
  89. y = mx + b Mica is having a pool party. The cost to rent the pool is $325 and $7.00 per person attending the party. Notice that regardless of how many people come, Mica will have to pay $325. This is the initial value, the y-intercept, the "b", also known as the constant. Also notice that it costs $7.00 per person. This amount will change as the number of guests changes. This will be the slope, the rate of change, or the "m". So the equation of this problem becomes: y = 7x + 325
  90. Try one!! Raul is at the gas station. He is filling up his gas tank at $3.45 per gallon and is also buying $12 worth of food from the convenience store. Write an equation to show this scenario. y = 3.45x + 12
  91. 46 Sandy charges $45 per necklace that she makes and charges a flat fee of $9 for shipping. Which equation would best fit this scenario? A y = 45x + 9 B y = 9x + 45 C 45 = 9x 9 = 45x D
  92. What does the cost of the necklace represent? 47 A The slope B The y-intercept C The total cost The initial value D Sandy charges $45 per necklace that she makes and charges a flat fee of $9 for shipping.
  93. 48 What does the cost of the shipping represent? The rate of change A B The y-intercept C The slope D The range Sandy charges $45 per necklace that she makes and charges a flat fee of $9 for shipping.
  94. How much does it cost to buy 5 necklaces? 49 $9 $45 $90 $234 A B C D Sandy charges $45 per necklace that she makes and charges a flat fee of $9 for shipping.
  95. How many necklaces can a person buy with $377? 50 A 4 6 8 10 B C D Sandy charges $45 per necklace that she makes and charges a flat fee of $9 for shipping.
  96. Analyzing a Graph Return to Table of Contents
  97. Sometimes you will be given a graph and be asked to describe a situation that would relate to that graph. For example, look at the graph below: Jack is going to school. He walks and takes two buses. Describe his trip to school
  98. 1. Jack walks to the bus stop. 2. He waits for the bus. 3. He takes the first bus. 4. He waits for the second bus. 5. He takes that bus directly to school. 5 4 3 2 1
  99. Did he have a longer wait for the first bus or the second bus? The first because it is a longer line along the x-axis. Was the first or second bus a longer trip? The first because it went further on the y-axis than the second bus. 4 5 3 2 1
  100. Let's look at another graph. This graph relates to studying for a test and actual test scores. Match the scenario with the point on the graph. Joanna didn't study but did fine on her test. e 2. William studied a long time but did poorly on his test. a 3. Alfredo studied some and passed the test. c 4. Ingrid studied hard and did well on the test. d Make your own scenario for the fifth point. a Study Time d c e b Test score
  101. This graph could show a comparison between: 51 Height and Age A B Sugar Used and Amount of Cookies Baked C Daylight and Time of Year Water Level Before, During, and After a Bath D
  102. 52 This graph could show a comparison between: Height and Age A Sugar Used and Amount of Cookies Baked B C Daylight and Time of Year D Water Level Before, During, and After a Bath
  103. 53 This graph could show a comparison between: Height and Age A B Sugar Used and Amount of Cookies Baked C Daylight and Time of Year D Water Level Before, During, and After a Bath
  104. This graph could show a comparison between: 54 Height and Age A B Sugar Used and Amount of Cookies Baked C Daylight and Time of Year D Water Level Before, During, and After a Bath
  105. 55 Which point shows no weight lost over a long time? a b A Weight Lost B C D c d Time
  106. 56 Which point shows a lot of weight lost over a long time? a b A Weight Lost B C D c d Time
  107. 57 Which point shows a lot of weight loss over a short period of time? a b A Weight Lost B C D c d Time
  108. Sometimes you will be given a graph and will have to explain the scenario, and other times you will be given a scenario and will have to create the graph. In order to do that you need to decide which variable will go where. There are two types of variables for a graph. Independent Variables go on the x-axis. These are variables which will happen regardless of anything else. Dependent Variables go on the y-axis. These are variables which rely on one or more other variables.
  109. Let's look at the example of Jack going to school. We compared distance and time. The distance that Jack could travel would be based on the amount of time that had elapsed. So it is the dependent variable. Time would occur regardless of whether Jack traveled to school or not so it is the independent variable. Dependent Variable = Distance Independent Variable = Time
  110. Dependent Variable = Distance Independent Variable = Time Notice in the graph that Distance is on the y-axis and Time is on the x-axis.
  111. Now you try! With your group figure out the independent and dependent variable. Click each rectangle to reveal. Cost of a vehicle and time Cost Dependent Time Independent Distance from a restaurant and time it takes to get there Distance Independent Time Dependent Amount of sleep and energy the next morning Sleep Independent Energy Dependent
  112. In the case of the speed of a ball rolling down a hill and the slope of the hill, the speed is independent. 58 A True B False
  113. In the case of a number of cookies needed and the number of people coming to the party, the number of people is independent. 59 A True B False
  114. In the case of the speed a ball rolls down a hill and the ball's weight, the speed is independent. 60 True A B False
  115. In the case of the charge remaining on a phone battery and the amount the phone is used, the phone usage is independent. 61 A True B False
  116. So, in order to make a graph, we must first decide on the independent and dependent variable and thus on which axis it will lie. Let's take the example of the distance from a restaurant and the time it takes to get there. We know that the time it takes to get to a restaurant relies (depends) on the distance you are from that restaurant (independent). In this case we would put distance on the x-axis and time on the y-axis.
  117. In this case we would put distance on the x-axis and time on the y-axis. Time Distance
  118. If two species interact as predator and prey there is often an interesting interaction between the two. Sharks are predators and fish are their prey. We can use a graph to show the relationship between these two creatures. On the next slide, make a graph comparing the sharks to the fish using the descriptions given. Keep in mind that sharks rely on the fish to live as you label your graph. Adapted from Shell Centre for Mathematical Education, University of Nottingham, 1985 http://www.primas-project.eu/artikel/en/1200/The+language+of+functions+and+graphs+/view.do?lang=en
  119. Due to the absence of sharks, there is an abundance of fish. b. Since there are so many fish, sharks come for the food. c. The large quantity of sharks eat many of the fish. d. There is a very low population of fish because the sharks have eaten so many of them. e. The sharks leave because there are too few fish. f. The population of fish increase because there are so few sharks to eat them. a Fish Adapted from Shell Centre for Mathematical Education, University of Nottingham, 1985 http://www.primas-project.eu/artikel/en/1200/The+language+of+functions+and+graphs+/view.do?lang=en
  120. Notice on the last graph there are no numbers. The shark and fish graph is really just a sketch which is an approximation. If we have specific numbers, it is sometimes easier to graph. Example: You are renting a pool for a swim party. The pool rental fee is $125 and you must pay $10 per person. Represent this in graph form.
  121. You are renting a pool for a swim party. The pool rental fee is $125 and you must pay $10 per person. Represent this in graph form. First you must decide which is the dependent and independent variable. Independent = People Dependent = Cost
  122. You are renting a pool for a swim party. The pool rental fee is $125 and you must pay $10 per person. Represent this in graph form. Cost Next, because there are specific numbers you need to create numbers for your graph. These numbers must be incremental - increasing at a regular rate. People
  123. You are renting a pool for a swim party. The pool rental fee is $125 and you must pay $10 per person. Represent this in graph form. Cost The people can go up by 1s. The cost is a little more tricky. It should go up by 10s but it would be a very long graph if we started at 10 since our first number on the cost side is 125. 2 3 4 5 6 7 8 9 10 People
  124. You are renting a pool for a swim party. The pool rental fee is $125 and you must pay $10 per person. Represent this in graph form. We can start at 125 (because that's our lowest number) but because we are not going up by 125s we must put an axis break. This is a symbol to show that there is a in the increment. This can be done by either a zig-zag line or a space in the y-axis.
  125. You are renting a pool for a swim party. The pool rental fee is $125 and you must pay $10 per person. Represent this in graph form. We can now start at 125 and go up by 10s. Next, we need to plot some points. Remembering back to earlier in this unit, we can make a table.
  126. You are renting a pool for a swim party. The pool rental fee is $125 and you must pay $10 per person. Represent this in graph form. If nobody comes we still need to pay $125 so our y-intercept is 125. From there it costs $10 more for each person.
  127. You are renting a pool for a swim party. The pool rental fee is $125 and you must pay $10 per person. Represent this in graph form. We can now plot our points and create a line.
  128. Now try one! Remember you will not always need an axis break. Chocolate chip cookies call for 3 cups of flour for each 2 dozen cookies. Represent this in a graph.
  129. Tom and Mara are going for a drive. Their car gets 32 miles to the gallon which means they use one gallon of gas every 32 miles. Represent this situation with a graph.
  130. The average infant weighs 7 lbs at birth and then gains approximately 1 lb every 3 weeks until 4 months old. Represent this in a graph.
  131. Brad rolled a bowling ball down a lane at the rate of 60 feet per 4 seconds. Does the graph represent this scenario? 62 A Yes B No
  132. A plane flies 600 mile per hour. Does the graph represent the scenario? 63 A Yes B No
  133. Ellen is selling lemonade. She is charging $1 per cup and spends $10 in supplies. She wants to graph her earnings. Did she do it right? 64 A Yes B No
  134. 65 Kisha downloaded some songs from the internet. She was able to download 5 songs in 2 minutes. Which of the following graphs represents this scenario? B A D C
  135. The recycling club has gained 3 members every two weeks. Which of the graphs represents this scenario? 66 A B C D
  136. One way to analyze a function is to assess if the rate of change (slope) is positive or negative. A positive rate or change will show a ratio that continues upward so both the x and y will be progressing in a positive manner or both will progress in a negative manner. (Remember, a negative divided by a negative is a positive.) With a negative rate of change, there will be a ratio that continues downward so either the x or the y will be progressing in a negative manner.
  137. It is quite simple to identify a positive or negative rate of change, but is different with each kind of representation of a function.
  138. With an equation, we look at the slope. If the slope is positive number, the function has a positive rate of change. If the slope is a negative number, the function has a negative rate of change. Example: y = 3x + 4 y = -2x + 6 Because the 3 is the slope Because -2 is the slope and 3 is positive, the rate and -2 is negative, the of change is positive. rate of change is negative.
  139. 67 Which functions have a positive rate of change? A y = 2x + 5 B y = -2x + 5 C y = 2x - 5 y = -2x - 5 D E y = 7x + 4 F y = 7x - 4
  140. 68 Which functions have a negative rate of change? A y = -4x + 3 B y = -4x - 3 C y = 4x + 3 y = 4x - 3 D E y = -6x + 3 F y = -6x - 3
  141. 69 When looking of for a positive rate of change: Look for a slope with one negative and one positive A B Look for a positive constant C Look for a positive coefficient Look for a slope with two positives D E Look for a slope with two negatives F C, D, and E
  142. 70 When looking for a negative slope: A Look for a negative coefficient B Look for a negative constant C Look for a slope with two negatives D Look for a slope with one negative and one positive E B and C F A and D
  143. With ordered pairs or a table, we can either use the slope formula to find the slope and then identify it as either positive or negative like an equation, or we can simply identify how the x-coordinate and y-coordinate are moving. Remember, two positives or two negatives make a positive rate of change, and one positive and one negative make a negative rate of change.
  144. +1 +1 +1 +1 +1 +1 Example: +3 +3 +3 +3 +3 +3 In the above case, both the x and y coordinates are moving in a positive direction so it is a positive rate of change. -1 -1 -1 -1 -1 -1 In this case both the x and y coordinates are moving in a negative direction so it is also a positive rate of change. -3 -3 -3 -3 -3 -3
  145. Example: 6 - 3 3 2 - 1 1 = = 3 Also note that if we use the slope formula the slope (rate of change) is positive in either case. 7 - 10-3 4 - 5 -1 = = 3 In this case both the x and y coordinates are moving in a negative direction so it is also a positive rate of change.
  146. Example: Look at the following ordered pairs. {(1,4), (2, 2), (3, 0), (4, -2), (5, -4)} Notice that while the x value goes up, the y value goes down. This is an indication that there is a negative rate of change because one value is going up while the other is going down. We can also use the slope formula to figure out if it is a negative rate of change. 2 - 4-2 2 - 1 1 = = -2
  147. 71 The following ordered pairs have a positive rate of change. {(4, 8), (5, 10), (6, 12), (7, 14), (8, 16)} A True B False
  148. 72 The following table has a positive rate of change. A True B False
  149. Which one of the following sets of ordered pairs has a negative rate of change? 73 A {(-2, 5), (-1, 7), (0, 9), (1, 11)} B {(3, 1), (6, 3), (9, 5), (12, 7)} C {(-20, -5), (-15, -4), (-10, -3), (-5, -2)} D {(1, -1), (2, -2), (3, -3), (4, -4)}
  150. Which one of the following set of ordered pairs has a positive rate of change? 74 A {(3, -7), (4, -9), (5, -11), (6, -13)} {(-6, -2), (-3, -1), (3, 1), (6, 2)} B C {(12, 5), (14, 4), (16, 3), (18, 2)} {(-5, 1), (-6, 2), (-7, 3), (-8, 4)} D
  151. Comparing Different Representations of a Function Return to Table of Contents
  152. We have learned how to represent a function several ways: Table/Ordered Pairs Graph Equation Verbal Description (Scenario) Next we will compare two different models to each other. We will look at the relationship between the two models in terms of the rate of change.
  153. In order to compare the rate of change of two different types of representations of functions we simply find the rate of change of each and compare them. The higher the absolute value of the rate of change, the bigger it is. For example, if a graph has a slope of -4 and an equation has a slope of 3, the slope of the graph is steeper because the absolute value of -4 = 4 and the absolute value of 3 = 3. 4 > 3 so The graph has a bigger slope, or rate of change.
  154. Let's try one! Which has a greater rate of change? A B y = -5x +6 Slope = -5 1 Slope = 3-12 2-1 1 = 2 = 2 absolute value of -5 = 5 and absolute value of 2 = 2 5>2 so A had a greater rate of change than B. 3
  155. Let's try to compare a table and a verbal model. Chris and Shari are going to have a bowling party. It costs $10 to rent a lane and $2 per pair of shoes. A B Which has the greater rate of change?
  156. Chris and Shari are going to have a bowling party. It costs $10 to rent a lane and $2 per pair of shoes. A We can turn this into an equation. 10 is a constant fee. 2 changes depending on the amount of people at the party. So the equation is y = 2x + 10. The rate of change = 2
  157. B To find the rate of change we can use the slope formula. 13 - 76 4 - 2 2 = = 3 The rate of change is 3. Which has the greater rate of change?
  158. Let's try to compare a table and a verbal model. Chris and Shari are going to have a bowling party. It costs $10 to rent a lane and $2 per pair of shoes. A B Which has the greater rate of change? Rate of change of A = 2 Rate of change of B = 3 Therefore, B has the greater rate of change.
  159. 75 Which has the greater rate of change? A {(1, 4), (2, 6), (3, 8), (4, 10), (5, 12)} B
  160. 76 Which has the greater rate of change? A y = 1/3x + 5 Victoria and Sandy were selling cookies. They charged $1 for 2 cookies. B
  161. 77 Which has the greater rate of change? Schuyler and Craig were doing dishes at a rate of 3 dishes per minute. A B
  162. 78 Which has the greater rate of change? A y = x - 4 B
  163. 79 Which has the greatest rate of change? A {(1, 3), (2, 4), (3, 5), (4, 6), (5, 7)} Ryan and Andrew jump down the stairs 3 steps at a time. B C y = 1/8x - 2 D
  164. 80 Which has the greatest rate of change? Emily and Gavin are making banana pancakes. They slice up 2 bananas for every dozen pancakes. A B y = 5x + 6 C {(9, 3), (6, 2), (3, 1), (0, 0), (-3, -1)} D
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