Ex 1: Find the Equation of Rational Function From a Graph — Transcript

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1. 0:00- IN THIS PROBLEM, WE WANT TO FIND AN EQUATION
2. 0:02FOR THE RATIONAL FUNCTION BASED UPON THE GRAPH.
3. 0:06TO DO THIS, WE'LL FIND THE FUNCTION IN FACTORED FORM
4. 0:09OR THIS FORM HERE, WHERE "A" IS A CONSTANT
5. 0:12AND WE'LL HAVE FACTORS IN THE NUMERATOR
6. 0:13AND FACTORS IN THE DENOMINATOR.
7. 0:15BEFORE WE DO THIS
8. 0:16LET'S IDENTIFY SOME OF THE KEY COMPONENTS FROM OUR GRAPH.
9. 0:19LET'S START WITH THE INTERCEPTS.
10. 0:21NOTICE HOW WE HAVE AN X INTERCEPT OF 3,
11. 0:24THIS WOULD BE THE POINT (3, 0)
12. 0:26AND WE HAVE A Y INTERCEPT OF -2.
13. 0:29THIS WOULD BE THE POINT (0, -2).
14. 0:33NOTICE HOW WE ALSO HAVE A VERTICAL ASYMPTOTE HERE
15. 0:38WHICH IS X = -1.
16. 0:42THIS IS THE INFORMATION WE NEED
17. 0:44TO DETERMINE OUR RATIONAL FUNCTION
18. 0:47BUT WE DO HAVE TO REMEMBER,
19. 0:48KNOWING THE VERTICAL ASYMPTOTE IS X = -1
20. 0:52GIVES US INFORMATION ABOUT A FACTOR IN THE DENOMINATOR
21. 0:56AND KNOWING THAT WE HAVE AN X INTERCEPT OF 3
22. 1:00GIVES US INFORMATION ABOUT A FACTOR IN THE NUMERATOR.
23. 1:04AND THEN WE CAN USE THE Y INTERCEPT OR THIS POINT HERE
24. 1:07TO DETERMINE THE VALUE OF OUR CONSTANT "A".
25. 1:11SO LET'S BEGIN TO SET THIS UP.
26. 1:12WE'RE GOING TO HAVE F OF X =
27. 1:15WE'LL FIND "A" IN THE SECOND STEP.
28. 1:17SO WE'LL HAVE OUR CONSTANT "A" HERE
29. 1:20AND NOW LET'S CONSIDER OUR VERTICAL ASYMPTOTE.
30. 1:22REMEMBER VERTICAL ASYMPTOTES ARE FOUND
31. 1:25BY DETERMINING WHICH X VALUES MAKE THE DENOMINATOR
32. 1:28EQUAL TO 0.
33. 1:29SO IF X = -1 MAKES OUR DENOMINATOR EQUAL TO 0
34. 1:33THAT MEANS WE MUST HAVE A FACTOR OF X + 1
35. 1:36IN THE DENOMINATOR.
36. 1:38NOTICE IF X IS -1 WE'D HAVE A ZERO IN THE DENOMINATOR
37. 1:41WHICH WOULD GIVE US THIS VERTICAL ASYMPTOTE.
38. 1:44NEXT, IF WE KNOW THAT THE X INTERCEPT IS (3, 0)
39. 1:47THAT MEANS F OF 3 MUST EQUAL 0,
40. 1:52MEANING WHEN X IS 3 THE FUNCTION VALUE IS 0.
41. 1:55WELL A RATIONAL FUNCTION IS EQUAL TO 0
42. 1:58WHEN THE NUMERATOR IS EQUAL TO 0.
43. 2:00THEREFORE, IF WE KNOW THE X INTERCEPT IS +3
44. 2:03OUR NUMERATOR MUST HAVE A FACTOR OF X - 3.
45. 2:09AND NOW WE ALMOST HAVE OUR FUNCTION
46. 2:11BUT WE STILL HAVE TO FIND THE VALUE OF "A".
47. 2:14AND WE CAN FIND "A" BY USING THE Y INTERCEPT
48. 2:16OR THIS POINT HERE.
49. 2:18IF THIS POINT ON THE FUNCTION HAS COORDINATES (0, -2)
50. 2:22THEN F OF 0 = -2.
51. 2:26SO IF WE SUBSTITUTE 0 FOR X
52. 2:28THIS FUNCTION VALUE MUST EQUAL -2.
53. 2:32SO LET'S GO AHEAD AND DO THAT.
54. 2:33F OF 0 WOULD BE EQUAL TO "A" x 0 - 3 IS -3, 0 + 1 = 1.
55. 2:44THIS MUST EQUAL -2.
56. 2:47SO WE HAVE WHAT, -3A = -2 DIVIDE BOTH SIDES BY -3
57. 2:55AND WE HAVE THE VALUE OF "A."
58. 2:56"A" IS EQUAL TO +2/3.
59. 3:01NOW LOOKING BACK AT THIS FUNCTION HERE,
60. 3:03THIS IS THE SAME AS "A" TIMES THIS FRACTION
61. 3:06AND SINCE "A" IS 2/3 WE CAN WRITE THE FUNCTION
62. 3:09AS F OF X = 2 x THE QUANTITY X - 3 DIVIDED BY 3
63. 3:17x THE QUANTITY X + 1.
64. 3:19NOTICE HOW WE HAVE 2/3 x THE FRACTION,
65. 3:23WHICH FITS THE FORM THAT WE HAVE HERE.
66. 3:25WE REALLY SHOULDN'T LEAVE THE FRACTION OF 2/3
67. 3:28IN THE NUMERATOR.
68. 3:29WE SHOULD WRITE IT THIS WAY.
69. 3:31THIS FUNCTION HERE HAS ALL THE CHARACTERISTICS
70. 3:35GIVEN BY THE GRAPH.
71. 3:37ALSO NOTICE THAT TO DOUBLE CHECK THIS
72. 3:40NOTICE HOW THE DEGREE OF THE NUMERATOR AND DENOMINATOR
73. 3:42ARE BOTH 1,
74. 3:44WHICH MEANS A HORIZONTAL ASYMPTOTE WOULD BE THE RATIO
75. 3:47OF THE LEADING COEFFICIENTS
76. 3:48WHICH IN THIS CASE WOULD BE Y = 2/3.
77. 3:52LOOKING AT OUR GRAPH, IT DOES APPEAR
78. 3:54THAT OUR HORIZONTAL ASYMPTOTE IS Y = 2/3.
79. 4:03OKAY. I HOPE THIS EXPLANATION WAS HELPFUL.