7/7/2016 Unit 4: Linear Relations Grade 9 Mathematics

Transcription

1 Rene Descartes, a mathematician who lived during the 17 th century, developed a system for graphing ordered pairs on a grid. This system is called the Cartesian Coordinate System. 1

2 In this system, ordered pairs (x, y) are graphed on a grid made up of 2 perpendicular number lines. These lines meet at a point (0,0), called the ORIGIN. The horizontal line is called the x-axis. The vertical line is called the y-axis. The x y axis divide the plane into 4 quadrants. 2

3 (+, +) (-, +) (-, -) (+, -) Quadrant I Quadrant II Quadrant III Quadrant IV y II I III IV x 3

4 Example 1: Plot A( -3, 5), B(2, -4), C(3, 0) Which quadrant are these points located? Solution: The first number in the ordered pair is the x-c0rdinate and tells how far you move left or right on the horizontal axis. The second number in the ordered pair is the y-coordinate and tells how far you move up or down on the vertical axis. A (-3, 5) is located in Quadrant II B(2, -4) is located in Quadrant IV C(3,0) is located in neither quadrant, it is on the x-axis. 4

5 Worksheet #4.2 (1): #1 to #6. Worksheet #4.2 (2): #1 to #10. Worksheet #4.2 (3): #1 and #2. A relation is a set of ordered pairs. A relation can be described as: - table of values - in words, or - by an equation 5

6 FOCUS: Analyze the graph of a linear relation. LINEAR RELATION: When the graph of the relation is a straight line, we have a linear relation. In a linear relation, a constant change in one quantity produces a constant change in a related quantity. 6

7 Example 1: (Graphing Relations) The table of values below shows how 2n + 1 relates to n. Input, n Output, 2n Use the data in the table to graph the relation. 7

8 Solution: Graph the relation on grid paper! **Join the points with a line. The points lie on a straight line. This is a linear relation. The graph shows how 2n + 1 relates to n. On the graph, we see that each time the input increases by 1, the output increases by 2. Note: The change in the input is CONSTANT. The change in the output is CONSTANT. 8

9 Example 2: HMV, in St. John s, is selling DVD s according to the given table of values. # of DVDs purchased, d Cost, C ($)

10 a) Graph the data. b) Should the points be joined? Why or Why not? c) Is the relation linear? Explain. d) Describe the patterns in the table. How are these patterns shown on the graph? 10

11 Solution: a) Graph the data! See Graph Board!! b) The points on the graph should NOT be joined because you cannot buy part of a DVD. This is called discrete data, when there is NOT an infinite number of points between the given points. 11

12 Solution (con t): c) The variable C depends on the value of the variable d. When two variables are related in this way, they form a relation. This is a linear relation since the points on the graph lie on a straight line. d) The table of values shows that: the number of DVDs purchased increases by 1 each time. The cost increases by $5 each time. 12

13 Example 3: Graph y = 3 x, using the given table of values. x y

14 (a) Complete the table of values. (b) Graph the relation represented by the data in the table of values. (c) Describe the patterns in the graph and in the table. (d) Is the relation linear? 14

15 Solution: a) Ordered pairs for the graph are: x y

16 (a) Write ordered pairs as: (-3, 6), (-2,5), (-1,4), (0,3), (1,2), (2,1), (3,0) (b) Since we can substitute any value for x, the points can be joined with a line. (c) Each point on the graph is 1 unit right and 1 unit down. In the table, when x decreases by 1, y decreases by 1. d) This is a linear relation because its graph is a straight line. 16

17 IMPORTANT NOTE: If the numbers in the first column increase by the same amount, and If the differences between consecutive numbers in the second column are CONSTANT, the relationship is LINEAR. 17

18 NOTE: For any equation, for example, y = 2x + 4 The value of the variable y depends on the value of the variable x. 18

19 We say that y is the DEPENDENT variable and we plot it on the y-axis (the output of a relation). We say that x is the INDEPENDENT variable and we plot it on the x-axis (the input of a relation). This two variables are related so we have a relation. 19

20 One way to remember dependent and independent variables on the graph is: d Called: did! id 20

21 HOMEWORK: Textbook: page : #4, 5, 6, 7, 8, 9, 10, 12, 13, 14, and 15. Extra Practice 2: #1 to #4. Worksheet #4-2 (4): #1 to #10. Workbook: Practice, page 154: #1 to #6. 21