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Splash Screen. Weekly assignment for 10-15 through 10-19 Mon: p146: 8-13, 14-17 Tue: p146: 18-29 Wed: p152: 15-27 Thur:p153:28-38 Fri: p159: 12-23 quiz.

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Presentation on theme: "Splash Screen. Weekly assignment for 10-15 through 10-19 Mon: p146: 8-13, 14-17 Tue: p146: 18-29 Wed: p152: 15-27 Thur:p153:28-38 Fri: p159: 12-23 quiz."— Presentation transcript:

1 Splash Screen

2 Weekly assignment for through Mon: p146: 8-13, Tue: p146: Wed: p152: Thur:p153:28-38 Fri: p159: quiz ch 3.1 through 3.3

3 Chapter Menu Lesson 3-1Lesson 3-1Representing Relations Lesson 3-2Lesson 3-2Representing Functions Lesson 3-3Lesson 3-3Linear Functions Lesson 3-4Lesson 3-4Arithmetic Sequences Lesson 3-5Lesson 3-5Proportional and Nonproportional Relationships

4 Lesson 3-1 Menu Five-Minute Check (over Chapter 2) Main Ideas and Vocabulary California Standards Example 1: Represent a Relation Example 2: Real-World Example Key Concept: Inverse of a Relation Example 3: Inverse Relation

5 warm up 15, october 18, 2012 (over Chapter 2) 5 Min 1-1 A. A B. B C. C D. D Translate three times a number decreased by eight is negative thirteen, into an equation. A.3(8 – n) = –13 B.3n – 8 = –13 C.3(n – 8) = –13 D.3 × 8 – n = –13

6 (over Chapter 2) 5 Min 1-2 A. A B. B C. C D. D Solve –24 + b = –13. A.b = 11 B.b = –37 C.b = –11 D.b = 37

7 (over Chapter 2) 5 Min 1-3 A. A B. B C. C D. D Solve for b.

8 (over Chapter 2) 5 Min 1-4 A. A B. B C. C D. D A stamp collector bought a rare stamp for $16, and sold it a year later for $ Find the percent of change. Round to the nearest whole percent. A.22 percent decrease B.28 percent decrease C.22 percent increase D.28 percent increase

9 (over Chapter 2) 5 Min 1-5 A. A B. B C. C D. D A.3 hours B.4 hours C.6 hours D.7 hours One train traveling at 60 miles per hour and another at 80 miles per hour leave the same location going in opposite directions. In how many hours will these trains be 420 miles apart?

10 Lesson 3-1 Ideas/Vocabulary Represent relations as sets of ordered pairs, tables, mappings, and graphs. mapping Find the inverse of a relation. inverse Notes 15, october 18, 2012

11 Lesson 3-1 CA Preparation for Standard 16.0 Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions. Standard 17.0 Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.

12 Lesson 3-1 Example 1a A. Express the relation {(4, 3), (–2, –1), (–3, 2), (2, –4), (0, –4)} as a table, a graph, and a mapping. Represent a Relation Table List the x-coordinates in the first column and the corresponding y-coordinates in the second column. Animation: Relation

13 Lesson 3-1 Example 1a Graph Graph each ordered pair on a coordinate plane. Represent a Relation

14 Lesson 3-1 Example 1a Mapping List the x-values in set X and the y-values in set Y. Draw an arrow from the x-value to the corresponding y-value. Represent a Relation

15 Lesson 3-1 Example 1b B. Determine the domain and range for the relation {(4, 3), (–2, –1), (–3, 2), (2, –4), (0, –4)}. Represent a Relation Answer: The domain for this relation is {–3, –2, 0, 2, 4}. The range is {–4, –1, 2, 3}.

16 A.A B.B C.C D.D CYP 1-1 A. Express the relation {(3, –2), (4, 6), (5, 2), (–1, 3)} as a mapping. A.C. B.D.

17 B. Determine the domain and range of the relation {(3, –2), (4, 6), (5, 2), (–1, 3)}. A.D = {–1, 3, 4, 5}; R = {–2, 2, 3, 6} B.D = {–2, 2, 3, 6}; R = {–1, 3, 4, 5} C.D = {–1, 3}; R = {–2, 2} D.D = {4}; R = {4}

18 Lesson 3-1 Example 2a A. OPINION POLLS The table shows the percent of students satisfied with their grades at the time of the survey. Use a Relation Determine the domain and range of the relation. Answer: The domain is {1996, 1999, 2002, 2005}. The range is {21, 32, 51, 60}.

19 Lesson 3-1 Example 2b B. Graph the data. The values of the x-axis need to go from 1996 to It is not practical to begin the scale at 0. Begin at 1996 and extend to 2005 to include all of the data. The units can be 1 unit per grid square. Use a Relation The values on the y-axis need to go from 21 to 60. In this case it is possible to begin the scale at 0. Begin at 0 and extend to 70. You can use units of 10.

20 Lesson 3-1 Example 2c C. What conclusions might you make from the graph of the data? Answer: Satisfaction increased from 1996 to 2002, but decreased from 2002 to Use a Relation

21 ENDANGERED SPECIES The table shows the approximate world population of the Indian Rhinoceros from 1986 to Determine the domain and range for the data provided in the table. A.D = {1000, 1700, 1900, 2100, 2400}; R = {1986, 1990, 1994, 1998, 2005} B.D = {5}; R = {4} C.D = {1986, 1990, 1994, 1998, 2005}; R = {1000, 1700, 1900, 2100, 2400} D.D = {2005}; R = {2400}

22 A.A B.B C.C D.D CYP 2-2 B. Graph the data. A.C. B.D.

23 C. What conclusions might you make from the graph of the data? A.The population of the Indian rhinoceros has been decreasing since B.The population of the Indian rhinoceros has been increasing since C.The population of the Indian rhinoceros has stayed the same since D.The Indian rhinoceros has become extinct.

24 Lesson 3-1 Key Concept 1

25 Lesson 3-1 Example 3 Express the relation shown in the mapping as a set of ordered pairs. Then write the inverse of the relation. Inverse Relation Answer: {(1, 5), (2, 7), (–9, 4), (2, 0)} Relation:Notice that both 7 and 0 in the domain are paired with 2 in the range. Answer:{(5, 1), (7, 2), (4, –9), (0, 2)} Inverse:Exchange X and Y in each ordered pair to write the inverse relation.

26 Express the relation shown in the mapping as a set of ordered pairs. Then write the inverse of the relation. A.Relation {(2, 3), (1, –4), (2, 5)} Inverse {(3, 2), (–4, 1), (5, 2)} B.Relation {(3, 2), (–4, 1), (5, 2)} Inverse {(–3, –2), (4, –1), (–5, –2)} C.Relation {(3, 2), (–4, 1), (5, 2)} Inverse {(2, 3), (1, –4), (2, 5)} D.Relation {(3, 2), (–4, 1), (5, 2)}

27 End of Lesson 3-1

28 Lesson 3-2 Menu Five-Minute Check (over Lesson 3-1) Main Ideas and Vocabulary California Standards Key Concept: Functions Example 1:Identify Functions Example 2:Equations as Functions Example 3:Function Values Example 4:Nonlinear Function Values Example 5:Standards Example

29 warm up Number 16 (over Lesson 3-1) A.A B.B C.C D.D 5 Min 2-1 Which option expresses the relation {(–1, 0), (2, –4), (–3, 1), (4, –3)} as a table and graph? A.B. C.D.

30 A. A B. B C. C D. D (over Lesson 3-1) 5 Min 2-2 Write the inverse of the relation {(5, –2), (0, 3), (2, 1), (–4, 3)}. A.{(–2, 5), (0, 3), (1, 2), (3, –4)} B.{(–2, 5), (3, 0), (1, 2), (–4, 3)} C.{(–2, 5), (3, 0), (2, 1), (–4, 3)} D.{(–2, 5), (3, 0), (1, 2), (3, –4)}

31 (over Lesson 3-1) A.A B.B C.C D.D 5 Min 2-3 Jason, a waiter, expressed his customers bills and the tips they left him as the relation {(10, 2), (8, 1.5), (4, 1.25)}. Which of the following options expresses the relation as a table? A.C. B.D.

32 (over Lesson 3-1) 5 Min 2-4 A. A B. B C. C D. D Determine the domain (D) and the range (R) for the relation {(3, –1), (2, 3), (4, 0), (–1, –2)}. A.D = {–1, 2, 3, 4} R = {–2, –1, 3} B.D = {–1, 2, 3, 4} R = {–2, –1, 0, 3} C.D = {–2, –1, 0, 3} R = {–1, 2, 3, 4} D.D = {–1, 2, 3} R = {–2, –1, 3}

33 Lesson 3-2 Ideas/Vocabulary Determine whether a relation is a function. function vertical line test function notation function value Find functional values. Notes 16,

34 Lesson 3-2 CA Standard 16.0 Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions. Standard 18.0 Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion.

35 Lesson 3-2 Key Concepts 1

36 Lesson 3-2 Example 1a A. Determine whether the relation is a function. Explain. Identify Functions Answer: This is a function because the mapping shows each element of the domain paired with exactly one member of the range.

37 Lesson 3-2 Example 1b B. Determine whether the relation is a function. Explain. Identify Functions Answer: This table represents a function because the table shows each element of the domain paired with exactly one element of the range. Animation: Is This a Function?

38 A. Is this relation a function? Explain. A.Yes; for each element of the domain, there is only one corresponding element in the range. B.Yes; because it can be represented by a mapping. C.No; because it has negative x-values. D.No; because both –2 and 2 are in the range.

39 B. Is this relation a function? Explain. A.No; because the element 3 in the domain is paired with both 2 and –1 in the range. B.No; because there are negative values in the range. C.Yes; because it is a line when graphed. D.Yes; because it can be represented in a chart.

40 Lesson 3-2 Example 2 Determine whether x = –2 is a function. Equations as Functions Graph the equation. Since the graph is in the form Ax + By = C, the graph of the equation will be a line. Place your pencil at the left of the graph to represent a vertical line. Slowly move the pencil to the right across the graph. At x = –2 this vertical line passes through more than one point on the graph. Answer: The graph does not pass the vertical line test. Thus, the line does not represent a function.

41 Determine whether 3x + 2y = 12 is a function. A.yes B.no C.not enough information

42 Lesson 3-2 Example 3a A. If f(x) = 3x – 4, find f(4). Function Values f(4)=3(4) – 4Replace x with 4. =12 – 4Multiply. = 8Subtract. Answer:f(4) = 8

43 Lesson 3-2 Example 3b B. If f(x) = 3x – 4, find f(–5). Function Values f(–5)=3(–5) – 4Replace x with –5. =–15 – 4Multiply. = –19Subtract. Answer:f(–5) = –19

44 Lesson 3-2 Example 3CYP-A A. A B. B C. C D. D A.8 B.7 C.6 D.11 A. If f(x) = 2x + 5, find f(3).

45 Lesson 3-2 Example 3CYP-B A. A B. B C. C D. D A.–3 B.–11 C.21 D.–16 B. If f(x) = 2x + 5, find f(–8).

46 Lesson 3-2 Example 4 PHYSICS The function h(t) = 160t + 16t 2 represents the height of an object ejected downward from an airplane at a rate of 160 feet per second. Nonlinear Function Values h(3)=160(3) + 16(3) 2 Replace t with 3. = Multiply. = 624Simplify. Answer:h(3) = 624 A. Find the value h(3).

47 Lesson 3-2 Example 4 PHYSICS The function h(t) = 160t + 16t 2 represents the height of an object ejected downward from an airplane at a rate of 160 feet per second. Nonlinear Function Values h(2z)=160(2z) + 16(2z) 2 Replace t with 2z. =320z + 64z 2 Multiply. Answer:h(2z) = 320z + 64z 2 B. Find the value h(2z).

48 A. Find the value h(2). The function h(t) = 180 – 16t 2 represents the height of a ball thrown from a cliff that is 180 feet above the ground. A.164 ft B.116 ft C.180 ft D.16 ft

49 B. Find the value h(3z). The function h(t) = 180 – 16t 2 represents the height of a ball thrown from a cliff that is 180 feet above the ground. A.180 – 16z 2 ft B.180 ft C.36 ft D.180 – 144z 2 ft

50 Lesson 3-2 Example 5 The algebraic form of a function is m = 5d, where d is the number of dollars customers of Mikes Car Rental donate to a charity and m is the donation made by Mikes Car Rental. Which of the following represents the same function? A.For every $2 dollars donated, Mikes Car Rental donates $7. B.f(d) = 5m C.D.

51 Lesson 3-2 Example 5 Read the ItemThe independent variable is d and the dependent variable is m. Solve the ItemChoice A represents the function m = 3.5d. This is incorrect because it should be m = 5d. Choices B and C represent the function, which is incorrect. In Choice D, the graph represents the function m = 5d, which is correct. The answer is D.

52 Lesson 3-2 CYP 5 A. A B. B C. C D. D The algebraic form of a function is c = 5h, where h is the number of hours a pool is rented and c is the cost to the PTA. Which of the following represents the same function? A.f(h) = 5c B.For every two hours the pool is rented, the cost to the PTA is $10. C.f(h) = 2c D.For every two hours the pool is rented, the cost to the PTA is $5.

53 End of Lesson 3-2

54 Notes 17, october 19

55 Lesson 3-3 Menu Five-Minute Check (over Lesson 3-2) Main Ideas and Vocabulary California Standards Key Concept: Standard Form of a Linear Equation Example 1: Identify Linear Equations Example 2: Real-World Example Example 3: Real-World Example Example 4: Graph by Making a Table Example 5: Graph by Using Intercepts

56 5 Min 3-1 warm up 17 (over Lesson 3-2) October 19, 2012 A.yes B.no Determine whether the relation shown in the mapping is a function.

57 5 Min 3-2 (over Lesson 3-2) A.yes B.no Determine whether the relation shown in the table is a function.

58 5 Min 3-3 A.yes B.no Determine whether the relation {(7, 0), (0, 7), (–7, 0), (0, –7)} is a function. (over Lesson 3-2)

59 5 Min 3-4 A.yes B.no Determine whether the relation y = 6 is a function. (over Lesson 3-2)

60 5 Min 3-5 A. A B. B C. C D. D If g(x) = –2x – 2, find g(–2x). A.g(–2x) = –4x – 2 B.g(–2x) = –2x – 2 C.g(–2x) = 2x – 2 D.g(–2x) = 4x – 2

61 A. A B. B C. C D. D (over Lesson 3-2) 5 Min 3-6 A.25 B.29 C.31 D.32

62 Lesson 3-3 Ideas/Vocabulary Identify linear equations, intercepts, and zeros. linear equation standard form x-intercept y-intercept zero Graph linear equations. Notes 17, october

63 Lesson 3-3 CA Standard 6.0 Students graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4). (Key, CAHSEE) Standard 7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula. (Key, CAHSEE)

64 Lesson 3-3 Key Concept 1

65 First rewrite the equation so that the variables are on the same side of the equation. Lesson 3-3 Example 1a A. Determine whether 5x + 3y = z + 2 is a linear equation. If so, write the equation in standard form. Identify Linear Equations 5x + 3y = z + 2Original equation 5x + 3y – z=z + 2 – zSubtract z from each side. 5x + 3y – z= 2Simplify. Since 5x + 3y – z has three different variables, it cannot be written in the form Ax + By = C. Answer: This is not a linear equation.

66 Rewrite the equation so that both variables are on the same side of the equation. Lesson 3-3 Example 1b Identify Linear Equations Subtract y from each side. Original equation B. Determine whether is a linear equation. If so, write the equation in standard form. Simplify.

67 Lesson 3-3 Example 1b To write the equation with integer coefficients, multiply each term by 4. Identify Linear Equations Answer: This is a linear equation. Original equation Multiply each side of the equation by 4. 3x – 4y=32Simplify. The equation is now in standard form where A = 3, B = –4, and C = 32.

68 A. Determine whether y = 4x – 5 is a linear equation. If so, write the equation in standard form. A.linear equation; y = 4x – 5 B.not a linear equation C.linear equation; 4x – y = 5 D.linear equation; 4x + y = 5

69 B. Determine whether 8y –xy = 7 is a linear equation. If so, write the equation in standard form. A.not a linear equation B.linear equation; 8y – xy = 7 C.linear equation; 8y = 7 + xy D.linear equation; 8y – 7 = xy

70 Lesson 3-3 Example 2a WATER STORAGE A valve on a tank is opened and the water is drained, as shown in the graph. A. Determine the x-intercept, y-intercept, and zero. Answer: x-intercept = 4; y-intercept = 200; zero of the function = 4

71 Lesson 3-3 Example 2b WATER STORAGE A valve on a tank is opened and the water is drained, as shown in the graph. B. Describe what the intercepts mean. Answer: The x-intercept 4 means that after 4 minutes, there are 0 gallons of water left in the tank. The y-intercept of 200 means that at time 0, or before any water was drained, there were 200 gallons of water in the tank.

72 BANKING Janine has money in a checking account. She begins withdrawing a constant amount of money each month as shown in the graph. A. Determine the x-intercept, y-intercept, and zero. A.10; 250; 10 B.10; 10; 250 C.250; 10; 250 D.5; 10; 5

73 B. In relation to the previous problem, describe what the x-intercept of 10 means? A.It is the amount of money before anything was withdrawn. B.That after 10 months, there is no money left in the account. C.It is the amount of money in the account after 6 months. D.This cannot be determined.

74 Lesson 3-3 Example 3a ANALYZE TABLES A box of peanuts is poured into bags a the rate of 4 ounces per second. The table shows the function relating to the weight of the peanuts in the box and the time in seconds the peanuts have been pouring out of the box. A. Determine the x-intercept, y-intercept, and zero of the graph of the function. Answer: x-intercept = 500; y-intercept = 2000; zero of the function = 500

75 Lesson 3-3 Example 3a B. Describe what the intercepts in the previous problem mean. Answer: The x-intercept 500 means that after 500 seconds, there are 0 ounces of peanuts left in the box. The y-intercept of 2000 means that at time 0, or before any peanuts were poured, there were 2000 ounces of peanuts in the box. Interactive Lab: Graphing Relations and Functions

76 ANALYZE TABLES Jules has a food card for Disney World. The table shows the function relating the amount of money on the card and the number of times he has stopped to purchase food. A. Determine the x-intercept, y-intercept, and zero. A.5; 125; 5 B.5; 5; 125 C.125; 5; 125 D.5; 10; 15

77 B. In relation to the previous problem, describe what the y-intercept of 125 means? A.It represents the time when there is no money left on the card. B.It represents the original value of the card. C.At time 0, or before any food stops, there was $125 on the card. D.This cannot be determined.

78 Lesson 3-3 Example 4 Graph y = 2x + 2. Graph by Making a Table Select values from the domain and make a table. Then graph the ordered pairs. The domain is all real numbers, so there are infinite solutions. Draw a line through the points. Answer:

79 Is this graph the correct graph for y = 3x – 4? A.yes B.no C.not enough information to determine

80 Lesson 3-3 Example 5 Graph 4x – y = 4 using the x-intercept and the y-intercept. Graph by Using Intercepts To find the x-intercept, let y = 0. 4x – y =4Original equation 4x – 0 = 4Replace y with 0. 4x=4Divide each side by 4. x=1x=1 To find the y-intercept, let x = 0. 4x – y = 4Original equation 4(0) – y =4Replace x with 0. –y=4Divide each side by –1. y =–4

81 Lesson 3-3 Example 5 Graph 4x – y = 4 using the x-intercept and y-intercept. Graph Using Intercepts Answer: The x-intercept is 1, so the graph intersects the x-axis at (1, 0). The y-intercept is –4, so the graph intersects the y-axis at (0, –4). Plot these points. Then draw a line that connects them. Animation: Graph Linear Equations

82 Is this graph the correct graph for 2x + 5y = 10? A.yes B.no C.not enough information to determine

83 End of Lesson 3-3

84 Lesson 3-4 Menu Five-Minute Check (over Lesson 3-3) Main Ideas and Vocabulary California Standards Key Concept: Arithmetic Sequence Example 1: Identify Arithmetic Sequences Key Concept: Writing Arithmetic Sequences Example 2: Real-World Example Key Concept: nth Term of an Arithmetic Sequence Example 3: Real-World Example

85 (over Lesson 3-3) 5 Min 4-1 A. A B. B C. C D. D Is 2x + y = –9 a linear equation? If so, write the equation in standard form. A.no B.yes; 2x = 9 + y C.yes; 2x + y = –9 D.yes; y = –2x –9

86 (over Lesson 3-3) 5 Min 4-2 A. A B. B C. C D. D Is 3x – xy + 7 = 0 a linear equation? If so, write the equation in standard form. A.no B.yes; 3x – xy = –7 C.yes; D.yes; 3x – xy + 7 = 0

87 (over Lesson 3-3) 5 Min 4-3 A. A B. B C. C D. D Is a linear equation? If so, write the equation in standard form. A.no B.yes; 2x – y = 10 C.yes; 2x = 10 – y D.yes; y = 2x – 10

88 (over Lesson 3-3) A.A B.B C.C D.D 5 Min 4-4 Graph y = –3x + 3. A.C. B.D.

89 5 Min 4-5 (over Lesson 3-3) A.$90.25 B.$90.00 C.$80.25 D.$75.00 Jakes Window Service uses the equation c = 5w to calculate the total charge c based on the number of windows w that are washed. What will be the charge for washing 15 windows?

90 (over Lesson 3-2) 5 Min 4-6 A. A B. B C. C D. D Which of the following is not a linear equation? A.y = 5 B.xy = 1 C. D.y = 1 – x

91 sequence terms arithmetic sequence common difference Lesson 3-4 Ideas/Vocabulary Recognize arithmetic sequences. Extend and write formulas for arithmetic sequences.

92 Lesson 3-4 CA Reinforcement of Standard 7AF3.4 Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities. (Key, CAHSEE)

93 Lesson 3-4 Key Concepts 1

94 Lesson 3-4 Example 1a A. Determine whether –15, –13, –11, –9,... is an arithmetic sequence. Explain. Identify Arithmetic Sequences Answer: This is an arithmetic sequence because the difference between terms is constant.

95 Lesson 3-4 Example 1b Identify Arithmetic Sequences Answer: This is not an arithmetic sequence because the difference between terms is not constant. B. Determine whether is an arithmetic sequence. Explain.

96 A. Determine whether 2, 4, 8, 10, 12, … is an arithmetic sequence. A.cannot be determined B.This is not an arithmetic sequence because the difference between terms is not constant. C.This is an arithmetic sequence because the difference between terms is constant.

97 CYP B. Determine whether … is an arithmetic sequence. A.cannot be determined B.This is not an arithmetic sequence because the difference between terms is not constant. C.This is an arithmetic sequence because the difference between terms is constant.

98 Lesson 3-4 Key Concept 2

99 Lesson 3-4 Example 2 TEMPERATURE The arithmetic sequence –8, –11, –14, –17, … represents the daily low temperature in ºF. Find the next three terms. Find the common difference by subtracting successive terms. The common difference is –3.

100 Lesson 3-4 Example 2 Subtract 3 from the last term of the sequence to get the next term in the sequence. Continue subtracting 3 until the next three terms are found. Answer:The next three terms are –20, –23, –26.

101 Lesson 3-4 Example 2CYP A. A B. B C. C D. D A.78, 83, 88 B.76, 79, 82 C.73, 78, 83 D.83, 88, 93 TEMPERATURE The arithmetic sequence 58, 63, 68, 73, … represents the daily high temperature in ºF. Find the next three terms.

102 Lesson 3-4 Key Concept 3

103 The common difference is +9. Lesson 3-4 Example 3a MONEY The arithmetic sequence 1, 10, 19, 28, … represents the total number of dollars Erin has in her account after her weekly allowance is added. In this sequence, the first term, a 1, is 1. Find the common difference. A. Write an equation for the nth term of the sequence.

104 Use the formula for the nth term to write an equation. Lesson 3-4 Example 3a a n =a 1 + (n –1)dFormula for the nth term a n = 1 + (n –1)(9)a 1 = 1, d = 9 a n = 1 + 9n – 9Distributive Property a n = 9n – 8Simplify.

105 Check: For n = 1, 9(1) – 8 = 1. Lesson 3-4 Example 3a For n = 2, 9(2) – 8 = 10. For n = 3, 9(3) – 8 = 19, and so on. Answer:a n = 9n – 8

106 B. Find the 12th term in the sequence. Lesson 3-4 Example 3b Replace n with 12 in the equation written in part A. a n = 9n – 8Equation for the nth term a 12 = 9(12) – 8Replace n with 12. a 12 = 100Simplify. Answer:100

107 C. Graph the first five terms of the sequence. Lesson 3-4 Example 3c Answer: The points fall on a line. The graph of an arithmetic sequence is linear.

108 MONEY The arithmetic sequence 2, 7, 12, 17, 22, … represents the total number of pencils Claire has in her collection after she goes to her school store each week. A. Write an equation for the nth term of the sequence. A. a n = 2n + 7 B. a n = 5n + 2 C. a n = 2n + 5 D. a n = 5n – 3

109 MONEY The arithmetic sequence 2, 7, 12, 17, 22, … represents the total number of pencils Claire has in her collection after she goes to her school store each week. B. Find the 12th term in the sequence. A.12 B.57 C.52 D.62

110 MONEY The arithmetic sequence 2, 7, 12, 17, 22, … represents the total number of pencils Claire has in her collection after she goes to her school store each week. C. Does this graph show the first five terms of the sequence? A.yes B.no C.cannot determine

111 End of Lesson 3-4

112 Lesson 3-5 Menu Five-Minute Check (over Lesson 3-4) Main Ideas and Vocabulary California Standards Example 1: Proportional Relationships Example 2: Nonproportional Relationships

113 (over Lesson 3-4) 5 Min 5-1 A. A B. B C. C D. D Determine whether the sequence –21, –17, –12, –6, 1, … is an arithmetic sequence. If it is, state the common difference. A.yes; 4 B.yes; 5 C.yes; 6 D.no

114 (over Lesson 3-4) 5 Min 5-2 A. A B. B C. C D. D A.yes; 0.1 B.yes; 1 C.yes; 1.1 D.no Determine whether the sequence 1.1, 2.2, 3.3, 4.4, … is an arithmetic sequence. If it is, state the common difference.

115 (over Lesson 3-4) 5 Min 5-3 A. A B. B C. C D. D A.–2.5, –4.0, –5.5 B.–2.5, –4.0, –4.5 C.–2.5, –3.0, –3.5 D.–1.5, –2.0, –2.5 Find the next three terms of the arithmetic sequence 3.5, 2, 0.5, –1.0, ….

116 (over Lesson 3-4) 5 Min 5-4 A. A B. B C. C D. D A.28 B.26 C.24 D.22 Find the nth term of the sequence described by a 1 = –2, d = 4, n = 7.

117 (over Lesson 3-4) 5 Min 5-5 A. A B. B C. C D. D A.a n = 2n + 21 B.a n = 2n + 19 C.a n = –2n + 21 D.a n = 2n + –19 Write an equation for the nth term of the sequence 19, 17, 15, 13, ….

118 5 Min 5-6 (over Lesson 3-4) A.16 B.4 C.2 D.1 A sequence is formed by writing the perimeter of squares. The side of the first square measures 1 unit. If the measure of the side of the square is increased by 1 unit for each new term in the sequence, what is the common difference between the terms?

119 Lesson 3-5 Ideas/Vocabulary Write an equation for a proportional relationship. Write an equation for a nonproportional relationship. inductive reasoning

120 Lesson 3-5 CA Standard 16.0 Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions.

121 Graph the data. What conclusion can you make about the relationship between the number of hours driving h and the numbers of miles driven m? Lesson 3-5 Example 1a A. ENERGY The table shows the number of miles driven for each hour of driving. Proportional Relationships Answer: The graph shows a linear relationship between the number of hours driving and the number of miles driven.

122 Look at the relationship between the domain and the range to find a pattern that can be described as an equation. Lesson 3-5 Example 1b B. Write an equation to describe this relationship. Proportional Relationships

123 Since this is a linear relationship, the ratio of the range values to the domain values is constant. The difference of the values for h is 1, and the difference of the values for m is 50. This suggests that m = 50h. Check to see if this equation is correct by substituting values of h into the equation. Lesson 3-5 Example 1b Proportional Relationships

124 CheckIf h = 1, then m = 50(1) or 50. Lesson 3-5 Example 1b Proportional Relationships If h = 2, then m = 50(2) or 100. If h = 3, then m = 50(3) or 150. If h = 4, then m = 50(4) or 200. The equation checks. Answer: f(h) = 50h Since this relation is also a function, we can write the equation as f(h) = 50h, where f(h) represents the number of miles driven.

125 A. Graph the data in the table. What conclusion can you make about the relationship between the number of miles walked and the time spent walking? A.There is a linear relationship between the number of miles walked and time spent walking. B.There is a nonlinear relationship between the number of miles walked and time spent walking. C.There is not enough information on the table to determine a relationship. D.There is an inverse relationship between miles walked and time spent walking.

126 Lesson 3-5 Example 1CYP-B A. A B. B C. C D. D A.m = 3h B.m = 2h C.m = 1.5h D.m = 1h B. Write an equation to describe the relationship between hours and miles walked.

127 Lesson 3-5 Example 2 Write an equation in function notation for the relation graphed below. Nonproportional Relationships Make a table of ordered pairs for several points on the graph.

128 Lesson 3-5 Example 2 The difference in the x values is 1, and the difference in the y values is 3. The difference in y values is three times the difference of the x values. This suggests that y = 3x. Check this equation. Nonproportional Relationships Check If x = 1, then y = 3(1) or 3. But the y value for x = 1 is 1. This is a difference of –2. Try some other values in the domain to see if the same difference occurs. y is always 2 less than 3x.

129 Lesson 3-5 Example 2 This pattern suggests that 2 should be subtracted from one side of the equation in order to correctly describe the relation. Check y = 3x – 2. Nonproportional Relationships If x = 2, then y = 3(2) – 2 or 4. If x = 3, then y = 3(3) – 2 or 7. Answer: y = 3x – 2 correctly describes this relation. Since the relation is also a function, we can write the equation in function notation as f(x) = 3x – 2.

130 1.A 2.B 3.C 4.D Lesson 3-5 Example 2CYP A. f(x) = x + 2 B. f(x) = 2x C. f(x) = 2x + 2 D. f(x) = 2x + 1 Write an equation in function notation for the relation that is graphed.

131 End of Lesson 3-5

132 Resources Menu Image Bank Math Tools Animation Menu Graphing Relations and Functions

133 CIM Menu RelationRelation 3-2Is This a Function? 3-3Graph Linear Equations

134 Image Bank 1 To use the images that are on the following three slides in your own presentation: 1.Exit this presentation. 2.Open a chapter presentation using a full installation of Microsoft ® PowerPoint ® in editing mode and scroll to the Image Bank slides. 3.Select an image, copy it, and paste it into your presentation.

135 Image Bank 2

136 Image Bank 3

137 Image Bank 4

138 End of Custom Shows This slide is intentionally blank.


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