Mastering Graphing Techniques in the Cartesian Plane

1. Math 374 Graphs

2. Topics • Cartesian Plane • Methods of Graphing • Intercept Slope • Scale • First Quadrant • Inequality Graphs • Region

3. Cartesian Plane • Named after Rene Deo Cartes a french mathematician • Also a philosopher “I think therefore I am” • His goal was to create a “picture” that could show a relationship between two variables. We have one for one variable – the number line.

4. Notes • We recall - 2 -1 0 1 2 3

5. Some Facts • We only need two points to draw a straight line • The point where a graph crosses or touches the x axis is called the x intercept • It is found by substituting y = 0 • The point where a graph crosses or touches the y axis is called the y intercept • It is found by substituting x = 0

6. Intercept Method • Calculate both intercepts. Place on graph and join • Example #1: y = 2x – 6 • X intercept (y = 0) 0 = 2x – 6 • -2x = - 6 • x = 3

7. Intercept Method • Now find Y intercept • Example #1: y = 2x – 6 • Y intercept (x = 0) y = 2 (0) – 6 • y = -6

8. Finding X and Y Intercept • Example #2: 5x – 3y = 15 • x int (y = 0)  5x – 3(0) = 15 • 5x = 15 • x = 3 • y Int (x = 0)  5(0) – 3y = 15 • -3y = 15 • y = -5

9. Drawing on Graph • Now that you know the x & y intercept, you have two points and now can draw the straight line… do it! • Practice plotting with other points…

10. Plotting Q2 (-,+) Q1 (+,+) A (3,1) B (-4,2) C (-4, -4) D (2, -2) . . B (-4, 2) (A 3, 1) . Q4 (+,-) Q3 (-,-) . D (2, -2) . C (-4, -4)

11. Standard Form Method • All straight lines have a y intercept and a slant called a slope. • If the relationship is in standard form we can write it… • y = m x + b Slope Y intercept

12. Identifying Slant and Slope Y Int Slant Slant Y Int

13. Standard Form • Recall y = mx + b Dependent Variable (DV) Y Intercept or Starting Value Slope Independent Variable (IV)

14. Relationship of y & b • It is easy to see how b is the y intercept; we substitute x = 0 • x = 0  y = m(0) + b • y = b

15. Rise, Run & Slope Slope Slope Rise Rise Run Run

16. Understanding the Slope • If m or the slope is 2 this means a rise of 2 and a run of 1 (2 can be written as 2 ) 1 • If m = - 5, this means a rise of -5 and right 1 • If m= -2 this means rise of -2 right 3 3

17. Understanding the Slope • Consider m = -3 4 • What is the rise and what is the run? • Suggest to put the negative sign on the top to clarify (rise of -3) • Numerator always rise (could go up or down) • Denominator always run (right only) Rise Run

18. Consider y = 2x + 3 • What is the slope, rise, run and y intercept? • We have a slope 2 • 2 can be written as 2 1 Rise of 2 Run of 1 y intercept of 3 (y = b) Plot on graph paper the following…

19. Ex#1: y=2x+3 Question: Draw this line (1,5) Where can you plot the y intercept? 0,3 What is the y intercept? What is the slope What does the slope mean? Up 2, Right 1

20. Example #2 y = -5 x + 1 7 • What is the y intercept, slope? Rise and run? • Y intercept is 1 • Slope is -5/7 • Rise is – 5 • Run is 7 • Plot on graph (put it on graph paper)

21. Example #3 y = x • What is the y intercept, slope, rise and run? • y intercept = 0 (y int let x = 0) • Slope = 1 • Rise of 1 • Run of 1 • Plot on graph

22. Example #4: 3x – 4y = 12 • What is the y intercept, slope, rise and run? • Must put in standard form • -4y = - 3x + 12 • y = 3x – 3 4 • y intercept = -3 • Slope ¾ • Rise of 3, run of 4 • Plot on graph

23. Graphing with Scale • Scale is mostly used to make sure your graph can be seen • Consider y = 2x + 100 3

24. Ex#5 y=2x+100 3 y x 500 How will you measure m = 2/3? Y intercept? (300,300) Slope? (0,100) You can put 500 along the x axis which means each hash mark is 100 Note slope is a ratio so scale does not effect it

25. Ex. #5 200x + 300y = 120000 • 300y = - 200x + 120000 • Y = -2x + 400 3 • Plot it • Do #4 on stencil use form C

26. 1st Quadrant • There will be times when you will need to put the graph only in the 1st quadrant • The problem only exists when the y intercept is negative • In that case, work with the x intercept (sub y = 0)

27. Consider y = 2x – 5 3 • Show how the graph intersects in the 1st quadrant • Notice that b is negative. • In those cases, work with x int (let y = 0) • 0 = 2x – 5 3 • 0 = 2x – 15 • -2x = -15 • x = 7.5 Stencil: Do #5

28. Inequality Graphs • The straight line of the graph divides the plane into two regions • One side will be greater than, one side less than

29. The Trick in Standard Form • If greater then shade above > • If less then shade below < • If equal then solid line • If not equal then dotted line

30. Ex y > x + 3 y x 5 Step 1: Draw Line Y intercept? Slope? m = 1 (up 1, right 1) Dotted Line or solid? Shade above or below?

31. Ex y < x + 3 y x 5 Step 1: Draw Line Y intercept? Slope? m = 1 (up 1, right 1) Dotted Line or solid? Shade above or below?

32. Ex 5x - 10y < 30 y x 5 Step 1: Put in Standard Form • 10y < - 5x + 30 • y >1x – 3 • 2 y intercept? Slope? m = 1 (up 1, right 2) Dotted Line or solidline? Do #6 in C Shade above or below?

33. Point of Intersection • If we have two graphs, we create four regions 1 2 4 3

34. Consider y > 3x – 5 y < -2x + 5 Draw lines… one at a time y x 5 2nd line… y intercept of 1st? y int? Slope? Slope? Dotted / solid? m = 3 (3 up, right 1) Above or Below? Dotted Line or solid? Use arrows Shade above or below? Hint… with 2 lines, use arrows at first instead of shading Shade where they intersect!

35. Find POI (Point of Intersection), you can also use equations • y = 3x – 5 • y = -2x + 5 • 3x – 5 = -2x + 5 • 5x = 10 • x = 2 • x = 2  y = 3(x) – 5 • y = 3 (2) – 5 • y = 1 • POI (2, 1) Do 7 in E Finish Study Guide