1. IDENTIFYING AND GRAPHING MARCH, 2011 MS. ADLER Parallel & Perpendicular Lines BEGIN

2. Review: Parallel Lines Parallel lines are two lines that run in the same direction; the two lines lie in the same plane and NEVER intersect. What exactly makes them parallel?

3. REVIEW: SLOPE Two lines are parallel because they have equal slopes. What is slope?

4. SLOPE IS RISE OVER RUN. IT IS THE CHANGE IN Y DIVIDED BY THE CHANGE IN X. REVIEW: SLOPE

5. USING PENCIL AND PAPER, ANSWER THIS REVIEW QUESTION FROM LAST WEEK: WHAT IS THE SLOPE BETWEEN THE POINTS (0,1) AND (2,7)? review 1/451/54 3

6. REMEMBER THAT SLOPE IS RISE OVER RUN. INCORRECT. HINT TRY THIS PROBLEM AGAIN. TRY THIS PROBLEM AGAIN.

7. IF YOU ARE STILL HAVING TROUBLE, REMEMBER THAT WE LEARNED TO USE THIS FORMULA: HINT BACK TO REVIEW QUESTION BACK TO REVIEW QUESTION

8. GREAT JOB! YOU KNOW THAT SLOPE IS EQUAL TO (Y 2 -Y 1 )/(X 2 -X 1 ) CORRECT!

9. Perpendicular Lines This is new material, so pay close attention! Perpendicular lines are the opposite of parallel lines. While parallel lines never cross, perpendicular lines cross and create a right angle at their intersection. parallel perpendicular

10. PERPENDICULAR LINES Take a look at the perpendicular lines shown here. What makes them different from the parallel lines we studied earlier? Pay close attention to the differences in the slope. You can see this in the slope-intercept form.

11. Reciprocals A reciprocal is the “upside down” version of a number. It is the number that you multiply by to get 1. For example, let’s find the reciprocal of 3: 3 can be re-written as 3/1. Now flip it. It’s reciprocal would be 1/3! 3/1 × 1/3 = 1

12. Reciprocals, cont’d A reciprocal is the number you multiply by to get 1. Another way to think of this is the number you get when you divide 1 by the original number. Ex)1=3r(r is for reciprocal) 1/3=r Ex)1=1/4×r 4=r

13. WHAT IS THE RECIPROCAL OF 7? mini quiz! -7-1/7141/141/7

14. TO FIND THE RECIPROCAL, DIVIDE 1 BY THE ORIGINAL NUMBER. INCORRECT. TRY THIS PROBLEM AGAIN TRY THIS PROBLEM AGAIN

15. GREAT JOB! YOU KNOW THAT 1/7 IS THE RECIPROCAL OF 7 BECAUSE 1/7×7=1 CORRECT!

16. Slopes Recall that parallel lines have the same slope. Perpendicular lines have slopes which are NEGATIVE reciprocals of one another. Ex) parallel slopes: 2 and 2 perpendicular slopes: 2 and -1/2

17. WHICH OF THE FOLLOWING PAIRS OF SLOPES COULD BE THOSE OF A PAIR OF PERPENDICULAR LINES? mini quiz! 1/4 and 4 -3 and -3 2 and 1/2 -3/2 and - 3/2 -3/2 and - 3/2 5 and -1/5

18. HINT: RECALL THAT PERPENDICULAR SLOPES ARE NEGATIVE OPPOSITES OF ONE ANOTHER. INCORRECT. TRY THIS PROBLEM AGAIN.

19. HINT: REMEMBER THAT PERPENDICULAR SLOPES ARE RECIPROCALS. INCORRECT. TRY THIS PROBLEM AGAIN!

20. GOOD JOB! YOU REMEMBERED THAT PARALLEL LINES HAVE IDENTICAL SLOPES, BUT PERPENDICULAR LINES HAVE SLOPES THAT ARE NEGATIVE RECIPROCALS OF ONE ANOTHER! CORRECT!

21. Review of equation forms STANDARD EQUATION Ax+By+C=0 SLOPE-INTERCEPT FORM y=mx+b POINT-SLOPE FORM y-y 1 =m(x-x 1 ) Example of each form. All of these examples are equal: -4x+y-2=0 y=4x+2 y-2=4(x-0) With pencil and paper, practice manipulating these equations and discovering how they relate before moving on to the next slide.

22. FIND THE EQUATION OF THE LINE WHICH HAS SLOPE M=-2 AND INCLUDES THE POINT (3,1/2). PUT THE FINAL ANSWER IN STANDARD EQUATION FORM. mini quiz! y=-2x-13/2 y=-2x+13/2 2x+y+13/2=0 2x+y-13/2=0

23. HINT: STANDARD EQUATION FORM IS AX+BY+C=O. SLOPE-INTERCEPT FORM IS Y=MX+B. INCORRECT. TRY THIS PROBLEM AGAIN.

24. HINT: WHEN STARTING THIS PROBLEM, TRY PUTTING IT INTO POINT-SLOPE FORM FIRST. Y-Y 1 =M(X-X 1 ) INCORRECT. TRY THIS PROBLEM AGAIN.

25. GREAT WORK! YOU KNEW TO PLUG THE SLOPE AND X AND Y-VALUES INTO POINT-SLOPE FORM AND MANIPULATE INTO STANDARD EQUATION FORM, AX+BY+C=0. CORRECT.

26. How to Graph We have learned in class how to graph equations, but let’s review and apply it to our current lesson. First, manipulate the equation into point slope form.

27. How to Graph When you have the graph in slope intercept form (y=mx+b), plot the y-intercept. From that point on the y-axis, apply the slope to determine the next point. Once you have the two points, you can use the slope to find the next point, and draw the line.

28. Graphing an Actual Problem For instance, the line y=2x+2 has a y- intercept value of 2, which means that the line crosses the y-axis at point (0,2). Then, since the slope is 2, and slope is equal to the change in x over the change in y (rise over run), and 2 is equal to 2/1, we can “rise” 2 units and “run” 1 unit, which gives us the point (1,4). Now that we have two points, we can draw a straight line through them, giving us our final graph.

29. Think It Over What have you learned so far? This presentation has addressed the following: Slope Parallel lines Reciprocals Perpendicular lines Equation forms Graphing

30. Evaluation It is important that you apply the information that you have learned to more complex problems. The quiz that follows will be not be taken for a grade. However, I want you to work out every problem on looseleaf paper and turn your work in to the sub at the end of class.

31. MAKE SURE THAT YOU HAVE PENCIL AND PAPER. PUT YOUR NAME AND DATE ON THE SHEET YOU TURN IN. YOU MAY NOT TALK TO YOUR NEIGHBORS. QUIZ BEGIN QUIZ

32. QUIZ QUESTIONS: 9 9 10 11 12 6 6 5 5 4 4 3 3 2 2 1 1 8 8 7 7 FINISHED

33. WHAT IS THE NEGATIVE RECIPROCAL OF 3/2? QUIZ QUESTION 1 2/3 -2/3 3 3 -3/2 3/2

34. TO FIND THE RECIPROCAL, DIVIDE 1 BY THE ORIGINAL NUMBER. IN THIS PROBLEM, MAKE SURE YOU’RE FINDING THE NEGATIVE RECIPROCAL. INCORRECT. TRY THIS PROBLEM AGAIN TRY THIS PROBLEM AGAIN

35. GREAT JOB! YOU CORRECTLY IDENTIFIED THE NEGATIVE RECIPROCAL OF THE ORIGINAL NUMBER! CORRECT! BACK TO QUIZ

36. WHAT IS THE SLOPE OF THE LINE THAT RUNS THROUGH THE POINTS (-1,6) AND (3,3)? QUIZ QUESTION 2 -4/3 2/9 -1/4 -3/4 9/2

37. REMEMBER THAT SLOPE IS RISE OVER RUN. INCORRECT. TRY THIS PROBLEM AGAIN. TRY THIS PROBLEM AGAIN.

38. GREAT JOB! YOU KNOW HOW TO CALCULATE SLOPE: (Y 2 -Y 1 )/(X 2 -X 1 ) CORRECT! BACK TO QUIZ

39. IDENTIFY TWO LINES THAT ARE PARALLEL: QUIZ QUESTION 3 4x+y+7=0 y=4x-2 4x+y+7=0 y=4x-2 y=3x+7 (1/6)x+(1/2)y=0 y=3x+7 (1/6)x+(1/2)y=0 5x+y-3=0 y=(-1/5)x+4 5x+y-3=0 y=(-1/5)x+4 y=-3x+1 6x+2y-1=0 y=-3x+1 6x+2y-1=0

40. REMEMBER THAT PARALLEL LINES HAVE EQUAL SLOPES, AND PERPENDICULAR LINES HAVE NEGATIVE RECIPROCAL SLOPES. (YOU MOST LIKELY MISSED THIS QUESTION BECAUSE OF ARITHMETIC ERRORS, SO DOUBLE CHECK YOUR WORK.) INCORRECT. TRY THIS PROBLEM AGAIN. TRY THIS PROBLEM AGAIN.

41. GREAT JOB! YOU KNOW THE QUALITIES OF SLOPES OF PARALLEL LINES AND CORRECTLY IDENTIFIED THE PAIR OF PARALLEL LINES! CORRECT! BACK TO QUIZ

42. IDENTIFY THE EQUATION IN STANDARD EQUATION FORM OF A LINE WITH SLOPE -2 THAT PASSES THROUGH THE POINT (-1,4). QUIZ QUESTION 4 2x+y-2=0 2x+y+6=0 -2x+y+2=0 y=-2x+2

43. MAKE SURE THAT THE LINE IS IN STANDARD EQUATION FORM. IT IS MOST EFFECTIVE TO START WITH POINT-SLOPE FORM AND THEN MANIPULATE. INCORRECT. TRY THIS PROBLEM AGAIN. TRY THIS PROBLEM AGAIN. HINT

44. STANDARD EQUATION FORM: AX+BY+C=0 POINT-SLOPE FORM: Y-Y 1 =M(X-X 1 ) HINT TRY THIS PROBLEM AGAIN. TRY THIS PROBLEM AGAIN.

45. GREAT JOB! YOU USED THE CORRECT EQUATION FORM AND PROPERLY MANIPULATED THE INFORMATION! CORRECT! BACK TO QUIZ

46. IDENTIFY THE TWO SLOPES WHICH WOULD CREATE PERPENDICULAR LINES WHEN GRAPHED. QUIZ QUESTION 5 6 and 1/3 4 and -1/4 2 and 2 -1 and 1 1/7 and 7

47. RECALL THAT PERPENDICULAR LINES HAVE SLOPES WHICH ARE NEGATIVE RECIPROCALS OF ONE ANOTHER. INCORRECT. TRY THIS PROBLEM AGAIN. TRY THIS PROBLEM AGAIN.

48. GREAT JOB! YOU KNOW WHAT A RECIPROCAL IS AND APPLIED YOUR KNOWLEDGE TO IDENTIFY THE PERPENDICULAR SLOPES! CORRECT! BACK TO QUIZ

49. IDENTIFY THE SLOPE OF THE LINE. QUIZ QUESTION 6 0 0 1/2 1/4 2 2 4 4

50. REMEMBER THAT SLOPE IS RISE OVER RUN. INCORRECT. TRY THIS PROBLEM AGAIN. TRY THIS PROBLEM AGAIN.

51. GREAT JOB! YOU KNOW HOW TO CALCULATE SLOPE: (Y 2 -Y 1 )/(X 2 -X 1 ) CORRECT! BACK TO QUIZ

52. IDENTIFY THE EQUATION OF THIS LINE. QUIZ QUESTION 7 y=(-7/9)x+(1/3) y=(-7/9)x+(39/9) y=(-9/7)x-(13/7) y=(-7/9)x-(1/3)

53. FIRST, USE THE TWO POINTS TO FIND THE SLOPE. NEXT, USE THE SLOPE AND ONE OF THE POINTS TO PLUG INTO POINT-SLOPE FORM. FINALLY, MANIPULATE THE EQUATION INTO ITS FINAL FORM. POINT-SLOPE FORM: Y-Y 1 =M(X-X 1 ) INCORRECT. TRY THIS PROBLEM AGAIN. TRY THIS PROBLEM AGAIN.

54. GREAT JOB! YOU CORRECTLY FOUND THE SLOPE BETWEEN TWO POINTS ON THE LINE, THEN USED POINT-SLOPE FORM TO FIND THE CORRECT EQUATION OF THE LINE! CORRECT! BACK TO QUIZ

55. IDENTIFY THE PAIR OF POINTS THAT WOULD PRODUCE THE LINE Y=3X+1 QUIZ QUESTION 8 (0,1) and (-1,4) (0,-1) and (1,2) (0,1) and (2,4) (1,4) and (-1,-2)

56. ATTEMPT THIS PROBLEM BY TESTING THE PAIRS OF POINTS. USE THE POINTS TO FIND THE SLOPE, THEN PLUG THE SLOPE AND ONE OF THE POINTS INTO POINT SLOPE FORM. FINALLY, MANIPULATE TO THE FINAL EQUATION, Y=3X+1 INCORRECT. TRY THIS PROBLEM AGAIN. TRY THIS PROBLEM AGAIN.

57. GREAT JOB! YOU CORRECTLY IDENTIFIED THE PAIR OF POINTS WHICH PRODUCE THE LINE Y=3X+1 CORRECT! BACK TO QUIZ

58. IDENTIFY THE LINE THAT PASSES THROUGH THE POINT (O,-2) AND IS PERPENDICULAR TO THE LINE Y=(-1/6)X+3 QUIZ QUESTION 9 y=6x+2 y=(-1/6)x-2 y=6x-2 y=6x+12

59. IT IS IMPORTANT TO USE POINT-SLOPE FORM Y-Y 1 =M(X-X 1 ) TO COMPLETE THIS PROBLEM. RECALL THAT PERPENDICULAR LINES HAVE SLOPES WHICH ARE NEGATIVE RECIPROCALS. INCORRECT. TRY THIS PROBLEM AGAIN. TRY THIS PROBLEM AGAIN.

60. GREAT JOB! YOU CORRECTLY IDENTIFIED THE LINE THAT RUNS THROUGH THE POINT (0,-2) AND HAS SLOPE 6. CORRECT! BACK TO QUIZ

61. IDENTIFY THE PAIR OF LINES THAT ARE PERPENDICULAR. QUIZ QUESTION 10 y=-3x+1 3x+6y-1=0 y=-3x+1 3x+6y-1=0 y=0.5x-2 -x+2y+4=0 y=0.5x-2 -x+2y+4=0 y=(-1/3)x+12 6x+2y-12=0 y=(-1/3)x+12 6x+2y-12=0 y=-2x+3 x-2y+3=0 y=-2x+3 x-2y+3=0

62. THE BEST WAY TO APPROACH THIS PROBLEM IS TO MANIPULATE FROM ONE FORM OF THE EQUATION TO ANOTHER. WATCH OUT FOR ANY SIMPLE MISTAKES IN ARITHMETIC! RECALL THAT PERPENDICULAR LINES HAVE SLOPES WHICH ARE NEGATIVE RECIPROCALS! INCORRECT. TRY THIS PROBLEM AGAIN. TRY THIS PROBLEM AGAIN.

63. GREAT JOB! YOU CORRECTLY IDENTIFIED THE TWO EQUATIONS THAT HAVE SLOPES WHICH ARE NEGATIVE RECIPROCALS OF ONE ANOTHER. CORRECT! BACK TO QUIZ

64. IDENTIFY THE Y-INTERCEPT OF THE LINE THAT PASSES THROUGH THE POINTS (7,-2) AND (4,19). QUIZ QUESTION 11 (0,49) (0,-1) (O,-7) (0,47)

65. THE GOAL OF THIS PROBLEM IS TO FIND THE Y-INTERCEPT. FIRST, USE THE POINTS TO FIND SLOPE. THEN, PLUG THE SLOPE AND ONE OF THE POINTS INTO POINT-SLOPE FORM AND MANIPULATE INTO SLOPE-INTERCEPT FORM. THIS WILL GIVE YOU THE INTERCEPT. INCORRECT. TRY THIS PROBLEM AGAIN. TRY THIS PROBLEM AGAIN.

66. GREAT JOB! YOU CORRECTLY USED THE TWO POINTS TO FIND THE Y-INTERCEPT OF THE LINE BETWEEN THE POINTS! CORRECT! BACK TO QUIZ

67. FIND THE EQUATION OF A LINE PERPENDICULAR TO THIS LINE THAT RUNS THROUGH THE POINT (2,2). QUIZ QUESTION 12 y=(-3/4)x+(7/2) y=(3/4)x+(1/2) y=(3/4)x-(1/2) y=(-4/3)x-(2/3)

68. THE EASIEST WAY TO DO THIS PROBLEM IS TO FIRST USE THE TWO POINTS ON THE GRAPH TO FIND THE SLOPE OF THE ORIGINAL LINE. THEN, TO FIND THE PERPENDICULAR LINE, USE POINT-SLOPE FORM. USE THE POINT (2,2) AND THE NEGATIVE RECIPROCAL OF THE ORIGINAL SLOPE FOR THE SLOPE OF THE PERPENDICULAR LINE. INCORRECT. TRY THIS PROBLEM AGAIN. TRY THIS PROBLEM AGAIN.

69. GREAT JOB! YOU CORRECTLY USED THE TWO POINTS TO FIND THE SLOPE OF THE ORIGINAL LINE, THEN USED WHAT YOU KNOW ABOUT PERPENDICULAR LINES TO FIND THE EQUATION OF THE PERPENDICULAR LINE. CORRECT! BACK TO QUIZ

70. YOU ARE DONE WITH THE QUIZ! YOU SHOULD NOW BE ABLE TO (OBJECTIVE) WITHOUT TOO MUCH TROUBLE! PLEASE MAKE SURE THAT YOU HAVE SHOWN ALL OF YOUR WORK ON THE PAPER THAT YOU WILL TURN IN TO THE SUB. IF YOU NEED TO REVIEW QUIZ PROBLEMS TO SHOW WORK, USE THE BUTTON BELOW. CONGRATS BACK TO QUIZ!

71. What have you learned? You also learned that parallel lines have identical slopes, and that perpendicular lines have negative reciprocal slopes.

72. What have you learned? You also learned how to graph lines by plotting the y- intercept, and using slope to find other points on the line.

73. What have you learned? Standard Equation Form Ax+By=C Point-Slope Form y-y 1 =m(x-x 1 ) Slope-Intercept Form y=mx+b You now know how to identify and graph parallel and perpendicular lines!

74. PLEASE USE THE REST OF THE CLASS PERIOD REVIEWING CONCEPTS OR STAYING QUIET OTHERWISE. All Done! Have a great day! HIT THE ESC BUTTON TO END PRESENTATION