Warm Up

1. Warm Up State whether each word or phrase represents an amount that is increasing, decreasing, or constant. 1. stays the same 2. rises 3. drops 4. slows down constant increasing decreasing decreasing

2. Objectives Match simple graphs with situations. Graph a relationship.

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

4. 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. Example 1 Continued Step 2 List key words in order and decide which graph shows them. Never horizontal Graph B Slanting downward rapidly Graphs A, B, and C Slanting downward until reaches zero Graphs A, B, and C Step 3 Pick the graph that shows all the key phrases in order. The correct graph is B.

6. 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 .

7. Check It Out! Example 1 Continued Step 2 List key words in order and decide which graph shows them. Slanting upward Graph C Graphs A, B, and C Horizontal 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.

8. 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.

9. y Speed x Time 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. 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 The graph is continuous.

10. Helpful Hint When sketching or interpreting a graph, pay close attention to the labels on each axis.

11. 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 number of books sold (y-axis) varies for each day (x-axis). Since the bookstore accounts for the number of books sold at the end of each day, the graph is 7 distinct points. The graph is discrete.

12. 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. Each week (x-axis) her typing speed is measured. She gets a separate score (y-axis) for each test. Since each score is separate, the graph consists of distinct units. The graph is discrete.

13. Water tank Water Level Time Check It Out! Example 2b Sketch a graph for the situation. Tell whether the graph is continuous or discrete. 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.

14. 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.

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

16. Check It Out! Example 3 Write a possible situation for the given graph 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. Possible Situation: When the number of students reaches a certain point, the number of pizzas bought increases.

17. Warm-Up 1. Write a possible situation for the given graph. 2. A pet store is selling puppies for $50 each. It has 8 puppies to sell. Sketch a graph for this situation.

18. Warm-Up Write a possible situation for the given graph. 1. 2.

19. Objectives Identify functions. Find the domain and range of relations and functions.

20. In Lesson 4-1 you saw relationships represented by graphs. Relationships can also be represented by a set of ordered pairs called arelation. 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. Table x y 2 3 4 7 6 8 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”.

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

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

24. 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}.

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

26. 6 –4 5 –1 2 0 1 Check It Out! Example 2a Give the domain and range of the relation. The domain values are all x-values 1, 2, 5 and 6. The range values are y-values 0, –1 and –4. Domain: {6, 5, 2, 1} Range: {–4, –1, 0}

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

28. A function is a special type of relation that pairs each domain value with exactly one range value.

29. 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)} Even though 3 is in the domain twice, it is written only once when you are giving the domain. D: {3, 5, 4} R: {–2, –1, 0, 1} 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.

30. Example 3B: Identifying Functions Give the domain and range of the relation. Tell whether the relation is a function. Explain. –4 2 –8 1 4 5 D: {–4, –8, 4, 5} R: {2, 1} This relation is a function. Each domain value is paired with exactly one range value.

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

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

33. Warm-Up 1. Express the relation {(–2, 5), (–1, 4), (1, 3), (2, 4)} as a table, as a graph, and as a mapping diagram. 2. Give the domain and range of the relation.

34. Objectives Identify independent and dependent variables. Write an equation in function notation and evaluate a function for given input values.

35. 5 – 4 = 1 and Example 1: Using a Table to Write an Equation Determine a relationship between the x- and y-values. Write an equation. 5 10 15 20 1 3 2 4 Step 1 List possible relationships between the first x and y-values.

36. 10 – 4  2 and 15 – 4  3 and 20 – 4  4 and The value of y is one-fifth, , of x. or Example 1 Continued Step 2 Determine which relationship works for the other x- and y- values. Step 3 Write an equation. The value of y is one-fifth of x.

37. 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 4 1 2 3 y 3 6 9 12 Step 1 List possible relationships between the first x- and y-values. 1  3 = 3 and 1 + 2 = 3

38. 2 • 3= 6 2 + 2  6 3 • 3 = 9 3 + 2  9 4 • 3 = 12 4 + 2  12 Check It Out! Example 1 Continued Step 2 Determine which relationship works for the other x- and y- values. The value of y is 3 times x. Step 3 Write an equation. y = 3x The value of y is 3 times x.

39. The equation in Example 1 describes a function because for each x-value (input), there is only one y-value (output).

40. The inputof 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.

41. 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 paintdependson the measurement of a room. Dependent: amount of paint Independent: measurement of the room

42. Example 2B: Identifying Independent and Dependent Variables Identify the independent and dependent variables in the situation. The height of a candle decrease d centimeters for every hour it burns. The height of a candledepends on the number of hours it burns. Dependent: height of candle Independent: time

43. Example 2C: Identifying Independent and Dependent Variables Identify the independent and dependent variables in the situation. A veterinarian must weight an animal before determining the amount of medication. The amount of medicationdepends on the weight of an animal. Dependent: amount of medication Independent: weight of animal

44. Helpful Hint There are several different ways to describe the variables of a function. Independent Variable Dependent Variable y-values x-values Domain Range Input Output x f(x)

45. Check It Out! Example 2a Identify the independent and dependent variable in the situation. A company charges $10 per hour to rent a jackhammer. The cost to rent a jackhammerdependson the length of time it is rented. Dependent variable: cost Independent variable: time

46. Check It Out! Example 2b Identify the independent and dependent variable in the situation. Camryn buys p pounds of apples at $0.99 per pound. The cost of applesdepends on the number of pounds bought. Dependent variable: cost Independent variable: pounds

47. An algebraic expression that defines a function is a function rule. If x is the independent variable and y is the dependent variable, then function notation for y is f(x), read “f of x,” where f names the function. When an equation in two variables describes a function, you can use function notation to write it.

48. The dependent variableisa function ofthe independent variable. yisa function ofx. y=f(x) y = f(x)

49. Example 3A: Writing Functions Identify the independent and dependent variables. Write a rule in function notation for the situation. A math tutor charges $35 per hour. The amount a math tutor charges depends on number of hours. Dependent: charges Independent: hours Let h represent the number of hours of tutoring. The function for the amount a math tutor charges isf(h) = 35h.

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