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Holt McDougal Algebra 1 4-1 Graphing Relationships Match simple graphs with situations. Graph a relationship. Objectives.

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1 Holt McDougal Algebra Graphing Relationships Match simple graphs with situations. Graph a relationship. Objectives

2 Holt McDougal Algebra Graphing Relationships continuous graph discrete graph Vocabulary

3 Holt McDougal Algebra Graphing Relationships Graphs can be used to illustrate many different situations. For example, trends shown on a cardiograph can help a doctor see how a patients heart is functioning. To relate a graph to a given situation, use key words in the description.

4 Holt McDougal Algebra Graphing Relationships Example 1: Relating Graphs to Situations Each day several leaves fall from a tree. One day a gust of wind blows off many leaves. Eventually, there are no more leaves on the tree. Choose the graph that best represents the situation. Step 1 Read the graphs from left to right to show time passing.

5 Holt McDougal Algebra Graphing Relationships Step 2 List key words in order and decide which graph shows them. Key Words Segment DescriptionGraphs… Each day several leaves fall Wind blows off many leaves Eventually no more leaves Slanting downward rapidly Graphs A, B, and C Never horizontal Graph B Slanting downward until reaches zero Graphs A, B, and C Example 1 Continued

6 Holt McDougal Algebra Graphing Relationships Never horizontal Slanting downward rapidly Slanting downward until it reaches zero

7 Holt McDougal Algebra Graphing Relationships Check It Out! Example 1 The air temperature increased steadily for several hours and then remained constant. At the end of the day, the temperature increased slightly before dropping sharply. Choose the graph that best represents this situation. Step 1 Read the graphs from left to right to show time passing.

8 Holt McDougal Algebra Graphing Relationships Step 2 List key words in order and decide which graph shows them. Key Words Segment Description Graphs… Increased steadily Remained constant Increased slightly before dropping sharply Slanting upwardGraph C Horizontal Graphs A, B, and C Slanting upward and then steeply downward Graphs B and C Step 3 Pick the graph that shows all the key phrases in order. The correct graph is graph C. Check It Out! Example 1 Continued

9 Holt McDougal Algebra Graphing Relationships As seen in Example 1, some graphs are connected lines or curves called continuous graphs. Some graphs are only distinct points. They are called discrete graphs The graph on theme park attendance is an example of a discrete graph. It consists of distinct points because each year is distinct and people are counted in whole numbers only. The values between whole numbers are not included, since they have no meaning for the situation.

10 Holt McDougal Algebra Graphing Relationships Example 2A: Sketching Graphs for Situations Sketch a graph for the situation. Tell whether the graph is continuous or discrete. A truck driver enters a street, drives at a constant speed, stops at a light, and then continues. The graph is continuous. As time passes during the trip (moving left to right along the x -axis) the truck's speed ( y -axis) does the following: initially increases remains constant decreases to a stop increases remains constant

11 Holt McDougal Algebra Graphing Relationships When sketching or interpreting a graph, pay close attention to the labels on each axis. Helpful Hint

12 Holt McDougal Algebra Graphing Relationships Example 2B: Sketching Graphs for Situations Sketch a graph for the situation. Tell whether the graph is continuous or discrete. A small bookstore sold between 5 and 8 books each day for 7 days. The graph is discrete. The number of books sold (y-axis) varies for each day (x-axis). Since the bookstore can only sell whole numbers of books, the graph is 7 distinct points.

13 Holt McDougal Algebra Graphing Relationships Check It Out! Example 2a Sketch a graph for the situation. Tell whether the graph is continuous or discrete. Jamie is taking an 8-week keyboarding class. At the end of each week, she takes a test to find the number of words she can type per minute. She improves each week. The graph is discrete. Each week (x-axis) her typing speed is measured. She gets a separate score (y-axis) for each test. Since each test score is a whole number, the graph consists of 8 distinct points.

14 Holt McDougal Algebra Graphing Relationships Sketch a graph for the situation. Tell whether the graph is continuous or discrete. Check It Out! Example 2b Henry begins to drain a water tank by opening a valve. Then he opens another valve. Then he closes the first valve. He leaves the second valve open until the tank is empty. As time passes while draining the tank (moving left to right along the x - axis) the water level ( y -axis) does the following: initially declines decline more rapidly and then the decline slows down. The graph is continuous.

15 Holt McDougal Algebra Graphing Relationships Both graphs show a relationship about a child going down a slide. Graph A represents the child s distance from the ground related to time. Graph B represents the child s Speed related to time.

16 Holt McDougal Algebra Graphing Relationships Example 3: Writing Situations for Graphs Write a possible situation for the given graph. A car approaching traffic slows down, drives at a constant speed, and then slows down until coming to a complete stop. Step 1 Identify labels. x - axis: time y -axis: speed Step 2 Analyze sections. over time, the speed: initially decreases, remains constant, and then decreases to zero. Possible Situation:

17 Holt McDougal Algebra Graphing Relationships Check It Out! Example 3 Write a possible situation for the given graph Possible Situation: When the number of students reaches a certain point, the number of pizzas bought increases. Step 1 Identify labels. x - axis: students y -axis: pizzas Step 2 Analyze sections. As students increase, the pizzas do the following: initially remains constant, and then increases to a new constant.

18 Holt McDougal Algebra Relations and Functions Identify functions. Find the domain and range of relations and functions. Objectives

19 Holt McDougal Algebra Relations and Functions relation domain range function Vocabulary

20 Holt McDougal Algebra Relations and Functions In Lesson 4-1 you saw relationships represented by graphs. Relationships can also be represented by a set of ordered pairs called a relation. In the scoring systems of some track meets, for first place you get 5 points, for second place you get 3 points, for third place you get 2 points, and for fourth place you get 1 point. This scoring system is a relation, so it can be shown by ordered pairs. {(1, 5), (2, 3), (3, 2) (4, 1)}. You can also show relations in other ways, such as tables, graphs, or mapping diagrams.

21 Holt McDougal Algebra Relations and Functions Example 1: Showing Multiple Representations of Relations Express the relation {(2, 3), (4, 7), (6, 8)} as a table, as a graph, and as a mapping diagram. Write all x-values under x and all y-values under y x y Table

22 Holt McDougal Algebra Relations and Functions Example 1 Continued Express the relation {(2, 3), (4, 7), (6, 8)} as a table, as a graph, and as a mapping diagram. Use the x- and y-values to plot the ordered pairs. Graph

23 Holt McDougal Algebra Relations and Functions Mapping Diagram x y Write all x-values under x and all y- values under y. Draw an arrow from each x-value to its corresponding y- value. Example 1 Continued Express the relation {(2, 3), (4, 7), (6, 8)} as a table, as a graph, and as a mapping diagram.

24 Holt McDougal Algebra Relations and Functions Check It Out! Example 1 Express the relation {(1, 3), (2, 4), (3, 5)} as a table, as a graph, and as a mapping diagram. x y Table Write all x-values under x and all y- values under y.

25 Holt McDougal Algebra Relations and Functions Graph Use the x- and y-values to plot the ordered pairs. Check It Out! Example 1 Continued Express the relation {(1, 3), (2, 4), (3, 5)} as a table, as a graph, and as a mapping diagram.

26 Holt McDougal Algebra Relations and Functions Mapping Diagram x y Write all x-values under x and all y- values under y. Draw an arrow from each x-value to its corresponding y- value. Check It Out! Example 1 Continued Express the relation {(1, 3), (2, 4), (3, 5)} as a table, as a graph, and as a mapping diagram.

27 Holt McDougal Algebra Relations and Functions The domain of a relation is the set of first coordinates (or x -values) of the ordered pairs. The range of a relation is the set of second coordinates (or y -values) of the ordered pairs. The domain of the track meet scoring system is {1, 2, 3, 4}. The range is {5, 3, 2, 1}.

28 Holt McDougal Algebra Relations and Functions Example 2: Finding the Domain and Range of a Relation Give the domain and range of the relation. Domain: 1 x 5 Range: 3 y 4 The domain value is all x-values from 1 through 5, inclusive. The range value is all y-values from 3 through 4, inclusive.

29 Holt McDougal Algebra Relations and Functions Check It Out! Example 2a Give the domain and range of the relation. –4 – Domain: {6, 5, 2, 1} Range: {–4, –1, 0} The domain values are all x-values 1, 2, 5 and 6. The range values are y-values 0, –1 and –4.

30 Holt McDougal Algebra Relations and Functions Check It Out! Example 2b Give the domain and range of the relation. x y Domain: {1, 4, 8} Range: {1, 4} The domain values are all x-values 1, 4, and 8. The range values are y- values 1 and 4.

31 Holt McDougal Algebra Relations and Functions A function is a special type of relation that pairs each domain value with exactly one range value.

32 Holt McDougal Algebra Relations and Functions Example 3A: Identifying Functions Give the domain and range of the relation. Tell whether the relation is a function. Explain. {(3, –2), (5, –1), (4, 0), (3, 1)} R: {–2, –1, 0, 1} D: {3, 5, 4} Even though 3 is in the domain twice, it is written only once when you are giving the domain. The relation is not a function. Each domain value does not have exactly one range value. The domain value 3 is paired with the range values –2 and 1.

33 Holt McDougal Algebra Relations and Functions –4 – D: {–4, –8, 4, 5} R: {2, 1} Use the arrows to determine which domain values correspond to each range value. This relation is a function. Each domain value is paired with exactly one range value. Example 3B: Identifying Functions Give the domain and range of the relation. Tell whether the relation is a function. Explain.

34 Holt McDougal Algebra Relations and Functions Example 3C: Identifying Functions Give the domain and range of the relation. Tell whether the relation is a function. Explain. Draw lines to see the domain and range values. D: –5 x 3 R: –2 y 1 Range Domain The relation is not a function. Nearly all domain values have more than one range value.

35 Holt McDougal Algebra Relations and Functions Give the domain and range of each relation. Tell whether the relation is a function and explain. Check It Out! Example 3 a. {(8, 2), (–4, 1), (–6, 2),(1, 9)} b. The relation is not a function. The domain value 2 is paired with both –5 and –4. D: {–6, –4, 1, 8} R: {1, 2, 9} The relation is a function. Each domain value is paired with exactly one range value. D: {2, 3, 4} R: {–5, –4, –3}

36 Holt McDougal Algebra Writing Functions Identify independent and dependent variables. Write an equation in function notation and evaluate a function for given input values. Objectives

37 Holt McDougal Algebra Writing Functions independent variable dependent variable function rule function notation Vocabulary

38 Holt McDougal Algebra Writing Functions Example 1: Using a Table to Write an Equation Determine a relationship between the x- and y-values. Write an equation. x y Step 1 List possible relationships between the first x or y -values. 5 – 4 = 1 or

39 Holt McDougal Algebra Writing Functions Example 1 Continued Step 2 Determine if one relationship works for the remaining values. Step 3 Write an equation. or The value of y is one-fifth of x. The value of y is one-fifth,, of x. 15 – 4 3 and 20 – 4 4 and 10 – 4 2 and

40 Holt McDougal Algebra Writing Functions Check It Out! Example 1 Determine a relationship between the x- and y-values. Write an equation. {(1, 3), (2, 6), (3, 9), (4, 12)} x y Step 1 List possible relationships between the first x - or y -values. 1 3 = 3 or = 3

41 Holt McDougal Algebra Writing Functions y = 3 x Check It Out! Example 1 Continued Step 2 Determine if one relationship works for the remaining values. 2 3 = = = The value of y is 3 times x. Step 3 Write an equation. The value of y is 3 times x.

42 Holt McDougal Algebra Writing Functions The equation in Example 1 describes a function because for each x -value (input), there is only one y -value (output).

43 Holt McDougal Algebra Writing Functions The input of a function is the independent variable. The output of a function is the dependent variable. The value of the dependent variable depends on, or is a function of, the value of the independent variable.

44 Holt McDougal Algebra Writing Functions Example 2A: Identifying Independent and Dependent Variables Identify the independent and dependent variables in the situation. A painter must measure a room before deciding how much paint to buy. The amount of paint depends on the measurement of a room. Dependent: amount of paint Independent: measurement of the room

45 Holt McDougal Algebra Writing Functions Identify the independent and dependent variables in the situation. The height of a candle decreases d centimeters for every hour it burns. Dependent : height of candle Independent : time The height of a candle depends on the number of hours it burns. Example 2B: Identifying Independent and Dependent Variables

46 Holt McDougal Algebra Writing Functions A veterinarian must weigh an animal before determining the amount of medication. The amount of medication depends on the weight of an animal. Dependent: amount of medication Independent: weight of animal Identify the independent and dependent variables in the situation. Example 2C: Identifying Independent and Dependent Variables

47 Holt McDougal Algebra Writing Functions Helpful Hint There are several different ways to describe the variables of a function. Independent Variable Dependent Variable x- values y- values DomainRange InputOutput x f(x)f(x)

48 Holt McDougal Algebra Writing Functions Check It Out! Example 2a A company charges $10 per hour to rent a jackhammer. Identify the independent and dependent variable in the situation. The cost to rent a jackhammer depends on the length of time it is rented. Dependent variable: cost Independent variable: time

49 Holt McDougal Algebra Writing Functions Identify the independent and dependent variable in the situation. Check It Out! Example 2b Apples cost $0.99 per pound. The cost of apples depends on the number of pounds bought. Dependent variable: cost Independent variable: pounds

50 Holt McDougal Algebra Writing Functions An algebraic expression that defines a function is a function rule. Suppose Tasha earns $5 for each hour she baby-sits. Then 5 x is a function rule that models her earnings. If x is the independent variable and y is the dependent variable, then function notation for y is f ( x ), readf of x, where f names the function. When an equation in two variables describes a function, you can use function notation to write it.

51 Holt McDougal Algebra Writing Functions The dependent variable is a function of the independent variable. y is a function of x. y = f (x)

52 Holt McDougal Algebra Writing Functions Identify the independent and dependent variables. Write an equation in function notation for the situation. A math tutor charges $35 per hour. The function for the amount a math tutor charges is f ( h ) = 35 h. Example 3A: Writing Functions The fee a math tutor charges depends on number of hours. Dependent: fee Independent: hours Let h represent the number of hours of tutoring.

53 Holt McDougal Algebra Writing Functions A fitness center charges a $100 initiation fee plus $40 per month. The function for the amount the fitness center charges is f ( m ) = 40 m Example 3B: Writing Functions Identify the independent and dependent variables. Write an equation in function notation for the situation. The total cost depends on the number of months, plus $100. Dependent: total cost Independent: number of months Let m represent the number of months

54 Holt McDougal Algebra Writing Functions Check It Out! Example 3a Identify the independent and dependent variables. Write an equation in function notation for the situation. Steven buys lettuce that costs $1.69/lb. The function for the total cost of the lettuce is f ( x ) = 1.69 x. The total cost depends on how many pounds of lettuce Steven buys. Dependent: total cost Independent: pounds Let x represent the number of pounds Steven buys.

55 Holt McDougal Algebra Writing Functions Check It Out! Example 3b Identify the independent and dependent variables. Write an equation in function notation for the situation. An amusement park charges a $6.00 parking fee plus $29.99 per person. The function for the total park cost is f ( x ) = x + 6. The total cost depends on the number of persons in the car, plus $6. Dependent: total cost Independent: number of persons in the car Let x represent the number of persons in the car.

56 Holt McDougal Algebra Writing Functions You can think of a function as an input-output machine. For Tashas earnings, f(x) = 5 x. If you input a value x, the output is 5 x. input 10 x function f(x)=5x output 5x5x

57 Holt McDougal Algebra Writing Functions Example 4A: Evaluating Functions Evaluate the function for the given input values. For f(x) = 3x + 2, find f(x) when x = 7 and when x = –4. = f (7) = 3(7) + 2 Substitute 7 for x. f ( x ) = 3( x ) + 2 = 23 Simplify. f ( x ) = 3( x ) + 2 f (–4) = 3(–4) + 2 Substitute –4 for x. Simplify. = – = –10

58 Holt McDougal Algebra Writing Functions Example 4B: Evaluating Functions Evaluate the function for the given input values. For g(t) = 1.5t – 5, find g(t) when t = 6 and when t = –2. g ( t ) = 1.5 t – 5 g (6) = 1.5(6) – 5 = 9 – 5 = 4 g (–2) = 1.5(–2) – 5 = –3 – 5 = –8

59 Holt McDougal Algebra Writing Functions Example 4C: Evaluating Functions Evaluate the function for the given input values. For, find h(r) when r = 600 and when r = –12. = 202= – 2

60 Holt McDougal Algebra Writing Functions Check It Out! Example 4a Evaluate the function for the given input values. For h(c) = 2c – 1, find h(c) when c = 1 and when c = – 3. h ( c ) = 2 c – 1 h (1) = 2(1) – 1 = 2 – 1 = 1 h ( c ) = 2 c – 1 h (–3) = 2( – 3) – 1 = –6 – 1 = – 7

61 Holt McDougal Algebra Writing Functions Check It Out! Example 4b Evaluate the function for the given input values. For g(t) =, find g(t) when t = –24 and when t = 400. = – 5 = 101

62 Holt McDougal Algebra Writing Functions When a function describes a real-world situation, every real number is not always reasonable for the domain and range. For example, a number representing the length of an object cannot be negative, and only whole numbers can represent a number of people.

63 Holt McDougal Algebra Writing Functions Example 5: Finding the Reasonable Domain and Range of a Function Write a function to describe the situation. Find the reasonable domain and range of the function. Joe has enough money to buy 1, 2, or 3 DVDs at $15.00 each, if he buys any at all. Money spentis $15.00 for each DVD. f ( x ) = $15.00 x If Joe buys x DVDs, he will spend f ( x ) = 15 x dollars. Joe only has enough money to purchase 1, 2, or 3 DVDs. A reasonable domain is {0, 1, 2, 3}.

64 Holt McDougal Algebra Writing Functions Example 5 Continued Substitute the domain values into the function rule to find the range values. A reasonable range for this situation is {$0, $15, $30, $45}. x1 2 3 f(x)f(x) 15(1) = 1515(2) = 3015(3) = (0) = 0

65 Holt McDougal Algebra Writing Functions Check It Out! Example 5 The settings on a space heater are the whole numbers from 0 to 3. The total number of watts used for each setting is 500 times the setting number. Write a function to describe the number of watts used for each setting. Find the reasonable domain and range for the function. Number of watts used is 500 times the setting #. watts f ( x ) = 500 x For each setting, the number of watts is f ( x ) = 500 x watts.

66 Holt McDougal Algebra Writing Functions x f(x) (0) = 0 500(1) = (2) = 1, (3) = 1,500 There are 4 possible settings 0, 1, 2, and 3, so a reasonable domain would be {0, 1, 2, 3}. Check It Out! Example 5 Substitute these values into the function rule to find the range values. The reasonable range for this situation is {0, 500, 1,000, 1,500} watts.

67 Holt McDougal Algebra Graphing Functions Graph functions given a limited domain. Graph functions given a domain of all real numbers. Objectives

68 Holt McDougal Algebra Graphing Functions Scientists can use a function to make conclusions about the rising sea level. Sea level is rising at an approximate rate of 2.5 millimeters per year. If this rate continues, the function y = 2.5 x can describe how many millimeters y sea level will rise in the next x years. One way to understand functions such as the one above is to graph them. You can graph a function by finding ordered pairs that satisfy the function.

69 Holt McDougal Algebra Graphing Functions Example 1A: Graphing Solutions Given a Domain Graph the function for the given domain. x – 3y = –6; D: {–3, 0, 3, 6} Step 1 Solve for y since you are given values of the domain, or x. –x–x –x–x – 3 y = –x – 6 Subtract x from both sides. Since y is multiplied by –3, divide both sides by –3. Simplify. x – 3 y = – 6

70 Holt McDougal Algebra Graphing Functions Example 1A Continued Step 2 Substitute the given value of the domain for x and find values of y. x (x, y) –3 ( – 3, 1) 0(0, 2) 3(3, 3) 6 (6, 4) Graph the function for the given domain.

71 Holt McDougal Algebra Graphing Functions y x Step 3 Graph the ordered pairs. Example 1A Continued Graph the function for the given domain.

72 Holt McDougal Algebra Graphing Functions Graph the function for the given domain. f(x) = x 2 – 3; D: {–2, –1, 0, 1, 2} Example 1B: Graphing Solutions Given a Domain Step 1 Use the given values of the domain to find values of f ( x ). f(x) = x 2 – 3 (x, f(x)) x –2 – f ( x ) = (–2) 2 – 3 = 1 f ( x ) = (–1) 2 – 3 = –2 f ( x ) = 0 2 – 3 = –3 f ( x ) = 1 2 – 3 = –2 f ( x ) = 2 2 – 3 = 1 (–2, 1) (–1, –2) (0, –3) (1, –2) (2, 1)

73 Holt McDougal Algebra Graphing Functions y x Step 2 Graph the ordered pairs. Graph the function for the given domain. f(x) = x 2 – 3; D: {–2, –1, 0, 1, 2} Example 1B Continued

74 Holt McDougal Algebra Graphing Functions Check It Out! Example 1a Graph the function for the given domain. –2x + y = 3; D: {–5, –3, 1, 4} Step 1 Solve for y since you are given values of the domain, or x. – 2 x + y = 3 +2 x y = 2 x + 3 Add 2x to both sides.

75 Holt McDougal Algebra Graphing Functions Check It Out! Example 1a Continued Graph the function for the given domain. –2x + y = 3; D: {–5, –3, 1, 4} y = 2x + 3x ( x, y ) y = 2(–5) + 3 = –7–5 (–5, –7) y = 2(1) + 3 = 51 (1, 5) y = 2(4) + 3 = 11 4 (4, 11) y = 2(–3) + 3 = –3–3 (–3, –3) Step 2 Substitute the given values of the domain for x and find values of y.

76 Holt McDougal Algebra Graphing Functions Step 3 Graph the ordered pairs. Check It Out! Example 1a Continued Graph the function for the given domain. –2x + y = 3; D: {–5, –3, 1, 4}

77 Holt McDougal Algebra Graphing Functions Graph the function for the given domain. f(x) = x 2 + 2; D: {–3, –1, 0, 1, 3} f(x) = x x (x, f(x)) f ( x ) = (–3 2 ) + 2= 11–3 (–3, 11) f ( x ) = = 2 0 (0, 2) f ( x ) = =3 1 (1, 3) Check It Out! Example 1b f ( x ) = (–1 2 ) + 2= 3 –1 (–1, 3) 3 f ( x ) = =11 (3, 11) Step 1 Use the given values of the domain to find the values of f ( x ).

78 Holt McDougal Algebra Graphing Functions Step 2 Graph the ordered pairs. Graph the function for the given domain. f(x) = x 2 + 2; D: {–3, –1, 0, 1, 3} Check It Out! Example 1b

79 Holt McDougal Algebra Graphing Functions If the domain of a function is all real numbers, any number can be used as an input value. This process will produce an infinite number of ordered pairs that satisfy the function. Therefore, arrowheads are drawn at both ends of a smooth line or curve to represent the infinite number of ordered pairs. If a domain is not given, assume that the domain is all real numbers.

80 Holt McDougal Algebra Graphing Functions Graphing Functions Using a Domain of All Real Numbers Step 1 Use the function to generate ordered pairs by choosing several values for x. Step 2 Step 3 Plot enough points to see a pattern for the graph. Connect the points with a line or smooth curve.

81 Holt McDougal Algebra Graphing Functions x –3x + 2 = y(x, y) Example 2A: Graphing Functions Graph the function –3x + 2 = y. –3(1) + 2 = –11 (1, –1) 0–3(0) + 2 = 2 (0, 2) Step 1 Choose several values of x and generate ordered pairs. –1 (–1, 5) –3(–1) + 2 = 5 –3(2) + 2 = –4 2 (2, –4) 3 –3(3) + 2 = –7 (3, –7) –3(–2) + 2 = 8 –2 (–2, 8)

82 Holt McDougal Algebra Graphing Functions Step 2 Plot enough points to see a pattern. Example 2A Continued Graph the function –3x + 2 = y.

83 Holt McDougal Algebra Graphing Functions Step 3 The ordered pairs appear to form a line. Draw a line through all the points to show all the ordered pairs that satisfy the function. Draw arrowheads on both ends of the line. Example 2A Continued Graph the function –3x + 2 = y.

84 Holt McDougal Algebra Graphing Functions Example 2B: Graphing Functions g ( x ) = |–2| + 2= 4 –2 (–2, 4) g ( x ) = |1| + 2= 3 1 (1, 3) 0 g ( x ) = |0| + 2= 2 (0, 2) Step 1 Choose several values of x and generate ordered pairs. g ( x ) = |–1| + 2= 3 –1 (–1, 3) g ( x ) = |2| + 2= 4 2 (2, 4) g ( x ) = |3| + 2= 5 3 (3, 5) Graph the function g(x) = |x| + 2. xg(x) = |x| + 2(x, g(x))

85 Holt McDougal Algebra Graphing Functions Step 2 Plot enough points to see a pattern. Example 2B Continued Graph the function g(x) = |x| + 2.

86 Holt McDougal Algebra Graphing Functions Step 3 The ordered pairs appear to form a v-shaped graph. Draw lines through all the points to show all the ordered pairs that satisfy the function. Draw arrowheads on the ends of the V. Example 2B Continued Graph the function g(x) = |x| + 2.

87 Holt McDougal Algebra Graphing Functions Check If the graph is correct, any point on it will satisfy the function. Choose an ordered pair on the graph that was not in your table. (4, 6) is on the graph. Check whether it satisfies g ( x )= | x | + 2. g ( x ) = | x | |4| Substitute the values for x and y into the function. Simplify. The ordered pair (4, 6) satisfies the function. Example 2B Continued Graph the function g(x) = |x| + 2.

88 Holt McDougal Algebra Graphing Functions Check It Out! Example 2a Graph the function f(x) = 3x – 2. f ( x ) = 3(–2) – 2 = –8 –2 (–2, –8) f ( x ) = 3(1) – 2 = 1 1(1, 1) 0 f ( x ) = 3(0) – 2 = –2 (0, –2) Step 1 Choose several values of x and generate ordered pairs. f ( x ) = 3(–1) – 2 = –5–1 (–1, –5) f ( x ) = 3(2) – 2 = 4 2(2, 4) f ( x ) = 3(3) – 2 = 73 (3, 7) xf(x) = 3x – 2(x, f(x))

89 Holt McDougal Algebra Graphing Functions Step 2 Plot enough points to see a pattern. Check It Out! Example 2a Continued Graph the function f(x) = 3x – 2.

90 Holt McDougal Algebra Graphing Functions Step 3 The ordered pairs appear to form a line. Draw a line through all the points to show all the ordered pairs that satisfy the function. Draw arrowheads on both ends of the line. Check It Out! Example 2a Continued Graph the function f(x) = 3x – 2.

91 Holt McDougal Algebra Graphing Functions Graph the function y = |x – 1|. y = | – 2 – 1| = 3 –2 (–2, 3) y = |1 – 1| = 0 1 (1, 0) 0 y = |0 – 1| = 1 (0, 1) Step 1 Choose several values of x and generate ordered pairs. y = |–1 – 1| = 2–1(–1, 2) y = |2 – 1| = 12(2, 1) Check It Out! Example 2b

92 Holt McDougal Algebra Graphing Functions Step 2 Plot enough points to see a pattern. Graph the function y = |x – 1|. Check It Out! Example 2b Continued

93 Holt McDougal Algebra Graphing Functions Step 3 The ordered pairs appear to form a v- shaped graph. Draw lines through the points to show all the ordered pairs that satisfy the function. Draw arrowheads on both ends of the V. Graph the function y = |x – 1|. Check It Out! Example 2b Continued

94 Holt McDougal Algebra Graphing Functions Check If the graph is correct, any point on the graph will satisfy the function. Choose an ordered pair on the graph that is not in your table. (3, 2) is on the graph. Check whether it satisfies y = | x – 1|. y = | x – 1| 2 | 3 – 1| 2 |2| 2 Substitute the values for x and y into the function. Simplify. The ordered pair (3, 2) satisfies the function. Graph the function y = |x – 1|. Check It Out! Example 2b Continued

95 Holt McDougal Algebra Graphing Functions Example 3: Finding Values Using Graphs Use a graph of the function to find the value of f(x) when x = –4. Check your answer. Locate –4 on the x -axis. Move up to the graph of the function. Then move right to the y- axis to find the corresponding value of y. f ( – 4) = 6

96 Holt McDougal Algebra Graphing Functions Check Use substitution. Example 3 Continued Substitute the values for x and y into the function. Simplify. The ordered pair (–4, 6) satisfies the function. 6 Use a graph of the function to find the value of f(x) when x = –4. Check your answer. f ( – 4) = 6

97 Holt McDougal Algebra Graphing Functions Check It Out! Example 3 Use the graph of to find the value of x when f(x) = 3. Check your answer. Locate 3 on the y- axis. Move right to the graph of the function. Then move down to the x- axis to find the corresponding value of x. f (3) = 3

98 Holt McDougal Algebra Graphing Functions Check Use substitution. Substitute the values for x and y into the function. Simplify. The ordered pair (3, 3) satisfies the function. Check It Out! Example 3 Continued Use the graph of to find the value of x when f(x) = 3. Check your answer. f(3) = 3 3

99 Holt McDougal Algebra Graphing Functions Recall that in real-world situations you may have to limit the domain to make answers reasonable. For example, quantities such as time, distance, and number of people can be represented using only nonnegative values. When both the domain and the range are limited to nonnegative values, the function is graphed only in Quadrant I.

100 Holt McDougal Algebra Graphing Functions Example 4: Problem-Solving Application A mouse can run 3.5 meters per second. The function y = 3.5x describes the distance in meters the mouse can run in x seconds. Graph the function. Use the graph to estimate how many meters a mouse can run in 2.5 seconds.

101 Holt McDougal Algebra Graphing Functions Example 4 Continued Understand the Problem 1 The answer is a graph that can be used to find the value of y when x is 2.5. List the important information: The function y = 3.5 x describes how many meters the mouse can run.

102 Holt McDougal Algebra Graphing Functions Think: What values should I use to graph this function? Both the number of seconds the mouse runs and the distance the mouse runs cannot be negative. Use only nonnegative values for both the domain and the range. The function will be graphed in Quadrant I. 2 Make a Plan Example 4 Continued

103 Holt McDougal Algebra Graphing Functions Solve 3 Choose several nonnegative values of x to find values of y. y = 3.5xx (x, y) y = 3.5(1) = 3.5 1(1, 3.5) y = 3.5(2) = 72 (2, 7) 3 y = 3.5(3) = 10.5 (3, 10.5) y = 3.5(0) = 0 0(0, 0) Example 4 Continued

104 Holt McDougal Algebra Graphing Functions Graph the ordered pairs. Draw a line through the points to show all the ordered pairs that satisfy this function. Use the graph to estimate the y-value when x is 2.5. A mouse can run about 8.75 meters in 2.5 seconds. Solve 3 Example 4 Continued

105 Holt McDougal Algebra Graphing Functions Look Back4 As time increases, the distance traveled also increases, so the graph is reasonable. When x is between 2 and 3, y is between 7 and Since 2.5 is between 2 and 3, it is reasonable to estimate y to be 8.75 when x is 2.5. Example 4 Continued

106 Holt McDougal Algebra Graphing Functions Check It Out! Example 4 The fastest recorded Hawaiian lava flow moved at an average speed of 6 miles per hour. The function y = 6x describes the distance y the lava moved on average in x hours. Graph the function. Use the graph to estimate how many miles the lava moved after 5.5 hours.

107 Holt McDougal Algebra Graphing Functions Check It Out! Example 4 Continued Understand the Problem 1 The answer is a graph that can be used to find the value of y when x is 5.5. List the important information: The function y = 6 x describes how many miles the lava can flow.

108 Holt McDougal Algebra Graphing Functions Think: What values should I use to graph this function? Both the speed of the lava and the number of hours it flows cannot be negative. Use only nonnegative values for both the domain and the range. The function will be graphed in Quadrant I. 2 Make a Plan Check It Out! Example 4 Continued

109 Holt McDougal Algebra Graphing Functions Solve 3 Choose several nonnegative values of x to find values of y. y = 6xx(x, y) y = 6(1) = 6 1 (1, 6) y = 6(3) = 183 (3, 18) 5 y = 6(5) = 30 (5, 30) Check It Out! Example 4 Continued

110 Holt McDougal Algebra Graphing Functions Solve 3 Draw a line through the points to show all the ordered pairs that satisfy this function. Use the graph to estimate the y-value when x is 5.5. The lava will travel about 32.5 meters in 5.5 seconds. Check It Out! Example 4 Continued Graph the ordered pairs.

111 Holt McDougal Algebra Graphing Functions Look Back4 As the amount of time increases, the distance traveled by the lava also increases, so the graph is reasonable. When x is between 5 and 6, y is between 30 and 36. Since 5.5 is between 5 and 6, it is reasonable to estimate y to be 32.5 when x is 5.5. Check It Out! Example 4 Continued

112 Holt McDougal Algebra Scatter Plots and Trend Lines Create and interpret scatter plots. Use trend lines to make predictions. Objectives

113 Holt McDougal Algebra Scatter Plots and Trend Lines scatter plot correlation positive correlation negative correlation no correlation trend line Vocabulary

114 Holt McDougal Algebra Scatter Plots and Trend Lines In this chapter you have examined relationships between sets of ordered pairs or data. Displaying data visually can help you see relationships. A scatter plot is a graph with points plotted to show a possible relationship between two sets of data. A scatter plot is an effective way to display some types of data.

115 Holt McDougal Algebra Scatter Plots and Trend Lines Example 1: Graphing a Scatter Plot from Given Data The table shows the number of cookies in a jar from the time since they were baked. Graph a scatter plot using the given data. Use the table to make ordered pairs for the scatter plot. The x-value represents the time since the cookies were baked and the y-value represents the number of cookies left in the jar. Plot the ordered pairs.

116 Holt McDougal Algebra Scatter Plots and Trend Lines Check It Out! Example 1 The table shows the number of points scored by a high school football team in the first four games of a season. Graph a scatter plot using the given data. Use the table to make ordered pairs for the scatter plot. The x-value represents the individual games and the y-value represents the points scored in each game. Plot the ordered pairs. Game 1234 Score

117 Holt McDougal Algebra Scatter Plots and Trend Lines A correlation describes a relationship between two data sets. A graph may show the correlation between data. The correlation can help you analyze trends and make predictions. There are three types of correlations between data.

118 Holt McDougal Algebra Scatter Plots and Trend Lines

119 Holt McDougal Algebra Scatter Plots and Trend Lines Example 2: Describing Correlations from Scatter Plots Describe the correlation illustrated by the scatter plot. There is a positive correlation between the two data sets. As the average daily temperature increased, the number of visitors increased.

120 Holt McDougal Algebra Scatter Plots and Trend Lines Check It Out! Example 2 Describe the correlation illustrated by the scatter plot. There is a positive correlation between the two data sets. As the years passed, the number of participants in the snowboarding competition increased.

121 Holt McDougal Algebra Scatter Plots and Trend Lines Example 3A: Identifying Correlations the average temperature in a city and the number of speeding tickets given in the city You would expect to see no correlation. The number of speeding tickets has nothing to do with the temperature. Identify the correlation you would expect to see between the pair of data sets. Explain.

122 Holt McDougal Algebra Scatter Plots and Trend Lines the number of people in an audience and ticket sales You would expect to see a positive correlation. As ticket sales increase, the number of people in the audience increases. Example 3B: Identifying Correlations Identify the correlation you would expect to see between the pair of data sets. Explain.

123 Holt McDougal Algebra Scatter Plots and Trend Lines a runners time and the distance to the finish line You would expect to see a negative correlation. As time increases, the distance to the finish line decreases. Example 3C: Identifying Correlations Identify the correlation you would expect to see between the pair of data sets. Explain.

124 Holt McDougal Algebra Scatter Plots and Trend Lines Check It Out! Example 3a Identify the type of correlation you would expect to see between the pair of data sets. Explain. the temperature in Houston and the number of cars sold in Boston You would except to see no correlation. The temperature in Houston has nothing to do with the number of cars sold in Boston.

125 Holt McDougal Algebra Scatter Plots and Trend Lines the number of members in a family and the size of the familys grocery bill You would expect to see a positive correlation. As the number of members in a family increases, the size of the grocery bill increases. Check It Out! Example 3b Identify the type of correlation you would expect to see between the pair of data sets. Explain.

126 Holt McDougal Algebra Scatter Plots and Trend Lines the number of times you sharpen your pencil and the length of your pencil You would expect to see a negative correlation. As the number of times you sharpen your pencil increases, the length of your pencil decreases. Check It Out! Example 3c Identify the type of correlation you would expect to see between the pair of data sets. Explain.

127 Holt McDougal Algebra Scatter Plots and Trend Lines Example 4: Matching Scatter Plots to Situations Choose the scatter plot that best represents the relationship between the age of a car and the amount of money spent each year on repairs. Explain. Graph A Graph B Graph C

128 Holt McDougal Algebra Scatter Plots and Trend Lines Example 4 Continued Choose the scatter plot that best represents the relationship between the age of a car and the amount of money spent each year on repairs. Explain. Graph A The age of the car cannot be negative.

129 Holt McDougal Algebra Scatter Plots and Trend Lines Example 4 Continued Graph B This graph shows all positive values and a positive correlation, so it could represent the data set. Choose the scatter plot that best represents the relationship between the age of a car and the amount of money spent each year on repairs. Explain.

130 Holt McDougal Algebra Scatter Plots and Trend Lines Example 4 Continued Graph C There will be a positive correlation between the amount spent on repairs and the age of the car. Choose the scatter plot that best represents the relationship between the age of a car and the amount of money spent each year on repairs. Explain.

131 Holt McDougal Algebra Scatter Plots and Trend Lines Graph A Graph B Graph C Example 4 Continued Graph A shows negative values, so it is incorrect. Graph C shows negative correlation, so it is incorrect. Graph B is the correct scatter plot. Choose the scatter plot that best represents the relationship between the age of a car and the amount of money spent each year on repairs. Explain.

132 Holt McDougal Algebra Scatter Plots and Trend Lines Check It Out! Example 4 Choose the scatter plot that best represents the relationship between the number of minutes since a pie has been taken out of the oven and the temperature of the pie. Explain. Graph A Graph BGraph C

133 Holt McDougal Algebra Scatter Plots and Trend Lines Check It Out! Example 4 Continued Choose the scatter plot that best represents the relationship between the number of minutes since a pie has been taken out of the oven and the temperature of the pie. Explain. Graph A The pie is cooling steadily after it is taken from the oven.

134 Holt McDougal Algebra Scatter Plots and Trend Lines Choose the scatter plot that best represents the relationship between the number of minutes since a pie has been taken out of the oven and the temperature of the pie. Explain. Graph B The pie has started cooling before it is taken from the oven. Check It Out! Example 4 Continued

135 Holt McDougal Algebra Scatter Plots and Trend Lines Check It Out! Example 4 Continued Choose the scatter plot that best represents the relationship between the number of minutes since a pie has been taken out of the oven and the temperature of the pie. Explain. Graph C The temperature of the pie is increasing after it is taken from the oven.

136 Holt McDougal Algebra Scatter Plots and Trend Lines Check It Out! Example 4 Choose the scatter plot that best represents the relationship between the number of minutes since a pie has been taken out of the oven and the temperature of the pie. Explain. Graph A Graph BGraph C Graph B shows the pie cooling while it is in the oven, so it is incorrect. Graph C shows the temperature of the pie increasing, so it is incorrect. Graph A is the correct answer.

137 Holt McDougal Algebra Scatter Plots and Trend Lines You can graph a function on a scatter plot to help show a relationship in the data. Sometimes the function is a straight line. This line, called a trend line, helps show the correlation between data sets more clearly. It can also be helpful when making predictions based on the data.

138 Holt McDougal Algebra Scatter Plots and Trend Lines Example 5: Fund-Raising Application The scatter plot shows a relationship between the total amount of money collected at the concession stand and the total number of tickets sold at a movie theater. Based on this relationship, predict how much money will be collected at the concession stand when 150 tickets have been sold. Draw a trend line and use it to make a prediction. Draw a line that has about the same number of points above and below it. Your line may or may not go through data points. Find the point on the line whose x-value is 150. The corresponding y-value is 750. Based on the data, $750 is a reasonable prediction of how much money will be collected when 150 tickets have been sold.

139 Holt McDougal Algebra Scatter Plots and Trend Lines Check It Out! Example 5 Based on the trend line, predict how many wrapping paper rolls need to be sold to raise $500. Find the point on the line whose y-value is 500. The corresponding x-value is about 75. Based on the data, about 75 wrapping paper rolls is a reasonable prediction of how many rolls need to be sold to raise $500.

140 Holt McDougal Algebra Arithmetic Sequences Recognize and extend an arithmetic sequence. Find a given term of an arithmetic sequence. Objectives

141 Holt McDougal Algebra Arithmetic Sequences sequence term arithmetic sequence common difference Vocabulary

142 Holt McDougal Algebra Arithmetic Sequences During a thunderstorm, you can estimate your distance from a lightning strike by counting the number of seconds from the time you see the lightning until you hear the thunder. When you list the times and distances in order, each list forms a sequence. A sequence is a list of numbers that often forms a pattern. Each number in a sequence is a term.

143 Holt McDougal Algebra Arithmetic Sequences Distance (mi) Time (s) +0.2 In the distance sequence, each distance is 0.2 mi greater than the previous distance. When the terms of a sequence differ by the same nonzero number d, the sequence is an arithmetic sequence and d is the common difference. The distances in the table form an arithmetic sequence with d = 0.2. Time (s) Distance (mi)

144 Holt McDougal Algebra Arithmetic Sequences The variable a is often used to represent terms in a sequence. The variable a 9, read a sub 9, is the ninth term in a sequence. To designate any term, or the n th term, in a sequence, you write a n, where n can be any number. To find a term in an arithmetic sequence, add d to the previous term.

145 Holt McDougal Algebra Arithmetic Sequences Example 1A: Identifying Arithmetic Sequences Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms. 9, 13, 17, 21,… Step 1 Find the difference between successive terms. You add 4 to each term to find the next term. The common difference is 4. 9, 13, 17, 21,… +4

146 Holt McDougal Algebra Arithmetic Sequences Step 2 Use the common difference to find the next 3 terms. 9, 13, 17, 21, +4 The sequence appears to be an arithmetic sequence with a common difference of 4. The next three terms are 25, 29, 33. Example 1A Continued Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms. 9, 13, 17, 21,… 25, 29, 33,… a n = a n- 1 + d

147 Holt McDougal Algebra Arithmetic Sequences Reading Math The three dots at the end of a sequence are called an ellipsis. They mean that the sequence continues and can read as and so on.

148 Holt McDougal Algebra Arithmetic Sequences Example 1B: Identifying Arithmetic Sequences Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms. 10, 8, 5, 1,… Find the difference between successive terms. The difference between successive terms is not the same. This sequence is not an arithmetic sequence. 10, 8, 5, 1,… –2–3 –4

149 Holt McDougal Algebra Arithmetic Sequences Check It Out! Example 1a Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms. Step 1 Find the difference between successive terms. You add to each term to find the next term. The common difference is.

150 Holt McDougal Algebra Arithmetic Sequences Check It Out! Example 1a Continued Step 2 Use the common difference to find the next 3 terms. The sequence appears to be an arithmetic sequence with a common difference of. The next three terms are,. Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms.

151 Holt McDougal Algebra Arithmetic Sequences Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms. Check It Out! Example 1b –4, –2, 1, 5,… Step 1 Find the difference between successive terms. –4, –2, 1, 5,… The difference between successive terms is not the same. This sequence is not an arithmetic sequence.

152 Holt McDougal Algebra Arithmetic Sequences To find the n th term of an arithmetic sequence when n is a large number, you need an equation or rule. Look for a pattern to find a rule for the sequence below … n Position The sequence starts with 3. The common difference d is 2. You can use the first term and the common difference to write a rule for finding a n. 3, 5, 7, 9… Term a 1 a 2 a 3 a 4 a n

153 Holt McDougal Algebra Arithmetic Sequences The pattern in the table shows that to find the n th term, add the first term to the product of ( n – 1) and the common difference.

154 Holt McDougal Algebra Arithmetic Sequences

155 Holt McDougal Algebra Arithmetic Sequences Example 2A: Finding the nth Term of an Arithmetic Sequence Find the indicated term of the arithmetic sequence. 16th term: 4, 8, 12, 16, … Step 1 Find the common difference. 4, 8, 12, 16,… The common difference is 4. Step 2 Write a rule to find the 16th term. The 16th term is 64. Write a rule to find the nth term. Simplify the expression in parentheses. Multiply. Add. Substitute 4 for a 1,16 for n, and 4 for d. a n = a 1 + ( n – 1) d a 16 = 4 + (16 – 1)(4) = 4 + (15)(4) = = 64

156 Holt McDougal Algebra Arithmetic Sequences Example 2B: Finding the nth Term of an Arithmetic Sequence Find the indicated term of the arithmetic sequence. The 25th term: a 1 = –5; d = –2 Write a rule to find the nth term. Simplify the expression in parentheses. Multiply. Add. The 25th term is –53. Substitute –5 for a 1, 25 for n, and –2 for d. a n = a 1 + ( n – 1) d a 25 = –5 + (25 – 1)(–2) = –5 + (24)(–2) = –5 + (–48) = –53

157 Holt McDougal Algebra Arithmetic Sequences Check It Out! Example 2a Find the indicated term of the arithmetic sequence. 60th term: 11, 5, –1, –7, … Step 1 Find the common difference. 11, 5, –1, –7,… –6 –6 –6 The common difference is –6. Step 2 Write a rule to find the 60th term. The 60th term is –343. Write a rule to find the nth term. Simplify the expression in parentheses. Multiply. Add. Substitute 11 for a 1, 60 for n, and –6 for d. a n = a 1 + ( n – 1) d a 60 = 11 + (60 – 1)(–6) = 11 + (59)(–6) = 11 + (–354) = –343 +2

158 Holt McDougal Algebra Arithmetic Sequences Check It Out! Example 2b Find the indicated term of the arithmetic sequence. 12th term: a 1 = 4.2; d = 1.4 Write a rule to find the nth term. Simplify the expression in parentheses. Multiply. Add. The 12th term is Substitute 4.2 for a 1,12 for n, and 1.4 for d. a n = a 1 + ( n – 1) d a 12 = (12 – 1)(1.4) = (11)(1.4) = (15.4) = 19.6

159 Holt McDougal Algebra Arithmetic Sequences Example 3: Application A bag of cat food weighs 18 pounds at the beginning of day 1. Each day, the cats are fed 0.5 pound of food. How much does the bag of cat food weigh at the beginning of day 30? Notice that the sequence for the situation is arithmetic with d = –0.5 because the amount of cat food decreases by 0.5 pound each day. Since the bag weighs 18 pounds to start, a 1 = 18. Since you want to find the weight of the bag on day 30, you will need to find the 30th term of the sequence, so n = 30.

160 Holt McDougal Algebra Arithmetic Sequences A bag of cat food weighs 18 pounds at the beginning of day 1. Each day, the cats are fed 0.5 pound of food. How much does the bag of cat food weigh at the beginning of day 30? Example 3 Continued There will be 3.5 pounds of cat food remaining at the beginning of day 30. Write the rule to find the nth term. Simplify the expression in parentheses. Multiply. Add. Substitute 18 for a 1, –0.5 for d, and 30 for n. a n = a 1 + ( n – 1) d a 31 = 18 + (30 – 1)(–0.5) = 18 + (29)(–0.5) = 18 + (–14.5) = 3.5

161 Holt McDougal Algebra Arithmetic Sequences Check It Out! Example 3 Each time a truck stops, it drops off 250 pounds of cargo. After stop 1, its cargo weighed 2000 pounds. How much does the load weigh after stop 6? Notice that the sequence for the situation is arithmetic because the load decreases by 250 pounds at each stop. Since the load will be decreasing by 250 pounds at each stop, d = –250. Since the load is 2000 pounds, a 1 = Since you want to find the load after the 6th stop, you will need to find the 6th term of the sequence, so n = 6.

162 Holt McDougal Algebra Arithmetic Sequences There will be 750 pounds of cargo remaining after stop 6. Write the rule to find the nth term. Simplify the expression in parentheses. Multiply. Add. Substitute 2000 for a 1, –250 for d, and 6 for n. Check It Out! Example 3 Continued a n = a 1 + ( n – 1) d a 6 = (6 – 1)(–250) = (5)(–250) = (–1250) = 750 Each time a truck stops, it drops off 250 pounds of cargo. After stop 1, its cargo weighed 2000 pounds. How much does the load weigh after stop 6?


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