1. Relations and the Coordinate System (1-6) Objective: Represent relations. Interpret graphs as relations.

2. Represent a Relation A coordinate system is formed by the intersection of two number lines, the horizontal axis and the vertical axis. (2, 3) The vertical axis is also called the y-axis. The horizontal axis is also called the x-axis. The origin, at (0, 0), is the point where the axes meet. The plane containing the x- and y-axes is the coordinate plane. Each point is named by an ordered pair.

3. Represent a Relation A point is represented on a graph using ordered pairs. An ordered pair is a set of numbers, or coordinates, written in the form (x, y). The x-value, called the x-coordinate, represents the horizontal placement of the point. The y-value, or y-coordinate, represents the vertical placement of the point.

4. Represent a Relation A set of ordered pairs is called a relation. A relation can be depicted in several different ways. An equation can represent a relation as well as graphs, tables, and mappings.

5. Represent a Relation A mapping illustrates how each element of the domain is paired with an element in the range. The set of first numbers of the ordered pairs is the domain. The set of second numbers of the ordered pairs is the range of the relation.

6. Represent a Relation This mapping represents the ordered pairs (-2, 4), (-1, 4), (0, 6), (1, 8), and (2, 8). Domain -2 0 1 2 Range 468468

7. Different Representations of Relations Ordered Pairs (1, 2) (-2, 4) (0, -3) XY 1 4 0 Table 2 -2 -3 Graph Mapping Domain Range 1 -2 0 2 4 -3

8. Represent a Relation The x-values of a relation are members of the domain. The y-values of a relation are members of the range.

9. Example 1 a.Express {(4, 3), (-2, -1), (2, -4), (0, -4)} as a table, a graph, and a mapping. XY 4 3 -2 2 -4 0 4 -2 2 0 3 -4 DomainRange

10. Example 1 b.Determine the domain and range of the relation. D = {-2, 0, 2, 4} R = {-4, -1, 3}

11. Represent a Relation In a relation, the value of the variable that determines the output is called the independent variable. The variable with a value that is dependent on the value of the independent variable is called the dependent variable. The domain contains the values of the independent variable. The range contains the values of the dependent variable.

12. Example 2 Identify the independent and the dependent variables for each relation. a.In warm climates, the average amount of electricity used rises as the daily average temperature increases and falls as the daily average temperature decreases. Independent Variable = temperature Dependent Variable = amount of electricity used

13. Example 2 Identify the independent and the dependent variables for each relation. b.The number of calories you burn increases as the number of minutes that you walk increases. Independent Variable = minutes you walk Dependent Variable = number of calories burned

14. Graphs of a Relation A relation can be graphed without a scale on either axis. These graphs can be interpreted by analyzing their shape.

15. Example 3 The graph represents the temperature in Ms. Ling’s classroom on a winter day. Describe what happens in the graph. The temperature increases as the heat is turned on. The thermostat then makes the temperature fluctuate slightly up and down. Finally, the temperature drops when the heat is turned off.

16. Check Your Progress Choose the best answer for the following. A.Express the relation {(3, -2), (4, 6), (5, 2), (-1, 3)} as a mapping. XY 3-2 46 52 3 A. B.C. X Y 3 4 5 -2 6 2 3 D. X Y -2 2 6 3 4 5

17. Check Your Progress Choose the best answer for the following. 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. Check Your Progress Choose the best answer for the following. In a particular club, as membership dues increase, the number of new members decreases. Identify the independent and dependent variable in this function. A.The number of new members is the independent variable. The dues is the dependent variable. B.Membership dues is the independent variable. The number of new members is the dependent variable. C.X is the independent variable. Y is the dependent variable. D.Both variables are independent.

19. Check Your Progress Choose the best answer for the following. The area of a square increases as the length of a side increases. Identify the independent and dependent variable in this function. A.The length of a side is independent, and the area of the square is dependent. B.The area is independent, and the side length is dependent. C.Both variables are independent. D.Both variables are dependent.

20. Check Your Progress Choose the best answer for the following. The graph represents Macy’s speed as she swims laps in a pool. Describe what is happening in the graph. A.Macy is doing bobs. B.Macy’s speed increases as she crosses the length of the pool, but then decreases to zero when she turns around at the end of each lap. C.Macy is swimming at a constant speed. D.Macy’s speed continues to decrease.