2. Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

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4. Contents Lesson 2-1Relations and Functions Lesson 2-2Linear Equations Lesson 2-3Slope Lesson 2-4Writing Linear Equations Lesson 2-5Modeling Real-World Data: Using Scatter Plots Lesson 2-6Special Functions Lesson 2-7Graphing Inequalities

5. Lesson 1 Contents Example 1Domain and Range Example 2Vertical Line Test Example 3Graph Is a Line Example 4Graph Is a Curve Example 5Evaluate a Function

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8. Relation A relation is a set of inputs and outputs, often written as ordered pairs (input, output). We can also represent a relation as a mapping diagram or a graph. For example, the relation can be represented as: Mapping Diagram of Relation

9. Domain & Range The first components in the ordered pairs (x- coordinate) make up the domain. The second components (y- coordinate or f(x) coordinate) make up the range.

10. Domain & Range For example, the group of ordered pairs (1, 2), (3, 5), and (7, 10) is a relation. The Domain of this relation consists of the x-coordinates of the ordered pairs and the range consists of the y-coordinates. Domain: (1, 3, 7} Range: {2, 5, 10}

11. Function A function is a relation in which each input has only one output. For example, the relation {(1, 2), (3, 5), and (7, 10)} is a function since no x values produce more than one y value. The relation {(1, 2), (3, 4), and (1, 5)} would not be considered a function since the x value 1 produces two different y values, 2 and 5.

12. Functions The diagram on the right shows the relation {(-1, 2), (0, 2), (1, 3), (2, -2), (8, 3)}. This relation is a function since each x value produces one and only one y value.

13. The Vertical Line Test To determine if a relation is a function you can use the vertical line test. A relation is a function if there are no vertical lines that intersect the graph at more than one point.verticallinesgraphpoint The graph at the right is not a function since it would fail the vertical line test.

14. Function Not a Function

15. Example 1-1a State the domain and range of the relation shown in the graph. Is the relation a function? The relation is {(1, 2), (3, 3), (0, –2), (–4, 0), (–3, 1)}. Answer: The domain is {–4, –3, 0, 1, 3}. The range is {–2, 0, 1, 2, 3}. Each member of the domain is paired with exactly one member of the range, so this relation is a function.

16. Example 1-1b State the domain and range of the relation shown in the graph. Is the relation a function? Answer: The domain is {–3, 0, 2, 3}. The range is {–2, –1, 0, 1}. Yes, the relation is a function.

17. Example 1-2a Transportation The table shows the average fuel efficiency in miles per gallon for light trucks for several years. Graph this information and determine whether it represents a function. Year Fuel Efficiency (mi/gal) 199520.5 199620.8 199720.6 199820.9 199920.5 200020.5 200120.4

18. Example 1-2b Year Fuel Efficiency (mi/gal) 199520.5 199620.8 199720.6 199820.9 199920.5 200020.5 200120.4 Use the vertical line test. Notice that no vertical line can be drawn that contains more than one of the data points.

19. Example 1-2c Answer: Yes, this relation is a function.

20. Example 1-2d Health The table shows the average weight of a baby for several months during the first year. Graph this information and determine whether it represents a function. Age (months) Weight (pounds) 1 12.5 216 422 624 925 1226

21. Example 1-2e Answer: Yes, this relation is a function.

22. 2 1 0 –1 yx Example 1-3a Graph the relation represented by Make a table of values to find ordered pairs that satisfy the equation. Choose values for x and find the corresponding values for y. 5 2 –1 –4 Then graph the ordered pairs. (–1, –4)(0, –1) (1, 2) (2, 5)

23. Example 1-3b Find the domain and range. Since x can be any real number, there is an infinite number of ordered pairs that can be graphed. All of them lie on the line shown. Notice that every real number is the x -coordinate of some point on the line. Also, every real number is the y -coordinate of some point on the line. Answer:The domain and range are both all real numbers. (–1, –4) (0, –1) (1, 2) (2, 5)

24. Example 1-3c Determine whether the relation is a function. This graph passes the vertical line test. For each x value, there is exactly one y value. (–1, –4) (0, –1) (1, 2) (2, 5) Answer:Yes, the equation represents a function.

25. Example 1-3d a.Graph b.Find the domain and range. c.Determine whether the relation is a function. Answer: Answer:The domain and range are both all real numbers. Answer:Yes, the equation is a function.

26. 1 2 yx 0 –1 –2 Example 1-4a Make a table. In this case, it is easier to choose y values and then find the corresponding values for x. 2 1 2 5 (1, 0) (2, –1) (5, –2) Graph the relation represented by 5 (5, 2) (2, 1) Then sketch the graph, connecting the points with a smooth curve.

27. Example 1-4b Find the domain and range. Every real number is the y -coordinate of some point on the graph, so the range is all real numbers. But, only real numbers that are greater than or equal to 1 are x -coordinates of points on the graph. (1, 0) (2, –1) (5, –2) (5, 2) (2, 1) Answer: The domain is. The range is all real numbers.

28. You can see from the table and the vertical line test that there are two y values for each x value except x = 1. Example 1-4c Determine whether the relation is a function. (1, 0) (2, –1) (5, –2) (5, 2) (2, 1) 1 2 yx 0 –1 –2 2 1 2 5 5

29. Example 1-4d Answer:The equation does not represent a function.

30. Example 1-4e a.Graph b.Find the domain and range. c.Determine whether the relation is a function. Answer:The domain is {x | x  –3}. The range is all real numbers. Answer: Answer:No, the equationdoes not represent a function.

31. Example 1-5a Given, find Answer: Original function Substitute. Simplify.

32. Example 1-5b Givenfind Original function Substitute. Multiply. Answer: Simplify.

33. Example 1-5c Given, find Original function Substitute. Answer:

34. Example 1-5d Givenand find each value. a. b. c. Answer: 6 Answer: 0.625 Answer:

35. Class Work: Page 60-61 in your textbooks #18, 20, 22, 28, 30, 35-41

36. Assignment: Page 60-61 #24, 32, 46, 47

37. End of Lesson 1

38. Lesson 2 Contents Example 1Identify Linear Functions Example 2Evaluate a Linear Function Example 3Standard Form Example 4Use Intercepts to Graph a Line

39. Linear Equations Looks like:x + 5 = 10 x + y = 1 3x – 5y = 12 A linear equation has no operations other than addition, subtraction, and multiplication of a variable by a constant. The variables may not be multiplied together or appear in a denominator. A linear equation does not contain variables with exponents other than 1. The graph of a linear equation is always a straight line.

40. Linear Functions A function whose ordered pairs satisfy a linear equation. Linear functions look like:f(x) = x + 3 f (x) = 5x + 1 f(x) = 2x – 4

41. Linear Functions Look at the linear function f(x) = x – 7. This means that you may plug in some value for x, and subtract 7 from it to get f(x). Note: f(x) acts as y.

42. Example 2-1a State whether is a linear function. Explain. Answer: This is a linear function because it is in the form

43. Example 2-1b Answer:This is not a linear function because x has an exponent other than 1. State whether is a linear function. Explain.

44. Example 2-1c State whether is a linear function. Explain. Answer:This is a linear function because it can be written as

45. Example 2-1d State whether each function is a linear function. Explain. a. b. c. Answer: yes; Answer:No; x has an exponent other than 1. Answer:No; two variables are multiplied together.

46. Try These State whether each function is linear. Write yes or no. If no, explain your reasoning. h(x) = 2x 3 –4x 2 + 5 f(x) = 6x - 19

47. Try These State whether each function is linear. Write yes or no. If no, explain your reasoning. h(x) = 2x 3 –4x 2 + 5 f(x) = 6x - 19 No, a linear function cannot have a variable in the denominator. No, a linear function cannot have variables raised to powers greater than 1. Yes No, a linear function cannot have square roots.

48. Example 2-2a Meteorology The linear function can be used to find the number of degrees Fahrenheit, f (C), that are equivalent to a given number of degrees Celsius, C. On the Celsius scale, normal body temperature is 37  C. What is normal body temperature in degrees Fahrenheit? Original function Substitute. Simplify. Answer:Normal body temperature, in degrees Fahrenheit, is 98.6  F.

49. Example 2-2b There are 100 Celsius degrees between the freezing and boiling points of water and 180 Fahrenheit degrees between these two points. How many Fahrenheit degrees equal 1 Celsius degree? Answer: 1.8  F = 1  C Divide 180 Fahrenheit degrees by 100 Celsius degrees.

50. Example 2-2c Meteorology The linear functioncan be used to find the distance d (s) in miles from a storm, based on the number of seconds s that it takes to hear thunder after seeing lightning. a.If you hear thunder 10 seconds after seeing lightning, how far away is the storm? b.If the storm is 3 miles away, how long will it take to hear thunder after seeing lightning? Answer: 2 miles Answer: 15 seconds

51. Try These When a sound travels through water, the distance y in meters that the sounds travels in x seconds is given by the equation y = 1440x. a)How far does sounds travel underwater in 5 seconds? b) In air, the equation is y = 343x. Does sound travel faster in air or water? Explain.

52. Try These When a sound travels through water, the distance y in meters that the sounds travels in x seconds is given by the equation y = 1440x. a)How far does sounds travel underwater in 5 seconds? b) In air, the equation is y = 343x. Does sound travel faster in air or water? Explain. 7200 meters Sound travels only 1715 meters in 5 seconds in air, so it travels faster under water.

53. Standard Form of a Linear Equation Ax + By = C Where A, B, and C are integers whose greatest common factor is 1. Example: 5x + 3y = 10

54. Example 2-3a Writein standard form. Identify A, B, and C. Original equation Subtract 3x from each side. Multiply each side by –1 so that A  0. Answer: and

55. Example 2-3b Writein standard form. Identify A, B, and C. Original equation Subtract 2y from each side. Multiply each side by –3 so that the coefficients are all integers. Answer: and

56. Example 2-3c Writein standard form. Identify A, B, and C. Original equation Subtract 4 from each side. Divide each side by 2 so that the coefficients have a GCF of 1. Answer: and

57. Example 2-3d Write each equation in standard form. Identify A, B, and C. a. b. c. Answer: and Answer:and Answer: and

58. Try These Write each equation in standard form. Identify A, B, and C. y = 12xx = 7y + 2 4x = 8y – 12

59. Try These Write each equation in standard form. Identify A, B, and C. y = 12xx = 7y + 2 4x = 8y – 12 12x – y = 0; A = 12, B = -1, C = 0 x – 7y = 2; A = 1, B = -7, C = 2 x – 2y = -3; A = 1, B = -2, C = -3 x – y = -6; A = 1, B = -1, C = -6 9

60. X-Intercept The x-intercept is the place at which a graph crosses the x-axis. At the right is the graph of the linear function of y = 2x + 4. In this graph the x –intercept is –2. In order pair form it is (-2, 0)

61. Y-Intercept The y-intercept is the place at which a graph crosses the y- axis. In this graph the y – intercept is 4. In order pair form it is (0, 4)

62. Finding x- and y-intercepts Since the x-intercept is the place where the graph crosses the x-axis, it is also the place where the y- value is 0. So, to find the x-intercept of an equation, just plug in 0 for y. And to find the y-intercept, plug in 0 for x. Example: Find the x- and y-intercepts of 2x + 3y = 12. x-intercept: 2x + 3(0) = 12 2x = 12 X = 6 Y-intercept: 2(0) + 3y = 12 3y = 12 Y = 4

63. Finding x- and y-intercepts Continued So this means that our graph hits the x-axis at 6, also known as (6, 0) and hits the y-axis at 4, also known as (0, 4). The graph of this equation could easily be drawn now that we know two points: 4 6

64. Example 2-4a Find the x -intercept and the y -intercept of the graph of Then graph the equation. The x -intercept is the value of x when The x -intercept is –2. The graph crosses the x -axis at (–2, 0). Original equation Substitute 0 for y. Add 4 to each side. Divide each side by –2.

65. Example 2-4b The y -intercept is 4. The graph crosses the y -axis at (0, 4). Likewise, the y -intercept is the value of y when Original equation Substitute 0 for x. Add 4 to each side.

66. Example 2-4c Use the ordered pairs to graph this equation. Answer:The x -intercept is –2, and the y -intercept is 4. (0, 4) (–2, 0)

67. Example 2-4d Find the x -intercept and the y -intercept of the graph of Then graph the equation. Answer:The x -intercept is –2, and the y -intercept is 6.

68. Try These Find the x-intercept and y-intercept of the graph of each equation. Then graph the equation. 2x – 6y = 12 y = 4x – 2

69. Try These Find the x-intercept and y-intercept of the graph of each equation. Then graph the equation. 2x – 6y = 12 y = 4x – 2 x-intercept: 6 y-intercept: -2 x-intercept: 1/2 y-intercept: -2

70. Assignment: Page 66-67 #42, 53, 54, 55

71. End of Lesson 2

72. Lesson 3 Contents Example 1Find Slope Example 2Use Slope to Graph a Line Example 3Rate of Change Example 4Parallel Lines Example 5Perpendicular Line

73. Slope of a Line The slope of a line measures the steepness of the line. Most of you are probably familiar with associating slope with "rise over run". Rise means how many units you move up or down from point to point. On the graph that would be a change in the y values. Run means how far left or right you move from point to point. On the graph, that would mean a change of x values.

74. Positive Slope Here are some visuals to help you with this definition: The slope of this line is 2/3.

75. Negative Slope Here is a graph of a line with a negative slope. The slope here is –2/1.

76. Formula for Calculating Slope of a Line To calculate the slope of a line given two points on that line you can use the formula:

77. Example 3-1a Find the slope of the line that passes through (1, 3) and (–2, –3). Then graph the line. Slope formula and Simplify.

78. Example 3-1b Graph the two ordered pairs and draw the line. (1, 3) Use the slope to check your graph by selecting any point on the line. Then go up 2 units and right 1 unit or go down 2 units and left 1 unit. This point should also be on the line. Answer:The slope of the line is 2. (–2, –3)

79. Example 3-1c Find the slope of the line that passes through (2, 3) and (–1, 5). Then graph the line. Answer:The slope of the line is

80. Try These Find the slope of the line that passes through each pair of points. (6, 8), (5, -5)(-2, -3), (0, -5) (4, -1.5), (4, 4.5)

81. Try These Find the slope of the line that passes through each pair of points. (6, 8), (5, -5)(-2, -3), (0, -5) (4, -1.5), (4, 4.5) Slope = 13Slope = -1 Slope is undefined Slope = -5/4

82. Example 3-2a Graph the line passing through (1, –3) with a slope of Graph the ordered pair (1, –3). Then, according to the slope, go down 3 units and right 4 units. Plot the new point at (5, –6). (1, –3) (5, –6) Draw the line containing the points.

83. Example 3-2b Graph the line passing through (2, 5) with a slope of –3. Answer:

84. Try These Graph the line passing through the given point with the given slope. (-3, -1), m = -1/5 (3, -4), m = 2 (2, 2), m = 0

85. Try These Graph the line passing through the given point with the given slope. (-3, -1), m = -1/5 (3, -4), m = 2 (2, 2), m = 0

86. Example 3-3a Communication Refer to the graph. Find the rate of change of the number of radio stations on the air in the United States from 1990 to 1998. Slope formula Substitute.

87. Example 3-3b Simplify. Answer: Between 1990 and 1998, the number of radio stations on the air in the United States increased at an average rate of 0.225(1000) or 225 stations per year.

88. Example 3-3c Computers Refer to the graph. Find the rate of change of the number of households with computers in the United States from 1984 to 1998. Answer: The rate of change is 2.9 million households per year.

89. Slopes of Parallel Lines Parallel lines have the same slopes. This is because the are going in the same direction. They do, however, have different y- intercepts since they do not hit the y-axis at the same place.

90. Slopes of Perpendicular Lines Perpendicular lines have opposite reciprocal slopes. For example, 2/3 and –3/2. The graph at the right shows a pair of perpendicular liens whose slopes are 1/1 and –1/1.

91. Example 3-4a Graph the line through (1, –2) that is parallel to the line with the equation The x -intercept is –2 and the y -intercept is 2. Use the intercepts to graph The line rises 1 unit for every 1 unit it moves to the right, so the slope is 1. Now, use the slope and the point at (1, –2) to graph the line parallel to (1, –2)(2, –1)

92. Example 3-4b Graph the line through (2, 3) that is parallel to the line with the equation Answer:

93. Example 3-5a Graph the line through (2, 1) that is perpendicular to the line with the equation The x -intercept is or 1.5 and the y -intercept is –1. Use the intercepts to graph 2x – 3y = 3 The line rises 1 unit for every 1.5 units it moves to the right, so the slope is or

94. 2x – 3y = 3 Example 3-5b Graph the line through (2, 1) that is perpendicular to the line with the equation The slope of the line perpendicular is the opposite reciprocal of or Start at (2, 1) and go down 3 units and right 2 units. (2, 1) (4, –2) Use this point and (2, 1) to graph the line.

95. Example 3-5c Graph the line through (–3, 1) that is perpendicular to the line with the equation Answer:

96. Try These Graph the line that satisfies each set of conditions. Passes through (2, -2) and is parallel to a line whose slope is –1. Passes through (3, 0) and is perpendicular to a line whose slope is ¾. Passes through the origin and is parallel to the graph of x + y = 10

97. Try These Graph the line that satisfies each set of conditions. Passes through (2, -2) and is parallel to a line whose slope is –1. Passes through (3, 0) and is perpendicular to a line whose slope is ¾. Passes through the origin and is parallel to the graph of x + y = 10

98. Assignment: Page 72-73 #18, 34, 44, 46

99. End of Lesson 3