Rational-Functions-GENERAL MATHEMATICS.pptx

RATIONAL FUNCTION IN GENERAL MATHEMATICS

1. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT Session 2 RATIONAL FUNCTIONS MSEUF, Lucena City April 18, 2017 Facilitator: Mr. WILLIAM M. VERZO

2. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT At the end of the session, the teacher-participants are expected to: represents real-life situations using rational functions . distinguishes rational function, rational equation, and rational inequality. solves rational equations and inequalities. represents a rational function through its: (a) table of values, (b) graph, and (c) equation. finds the domain and range of a rational function. determines the: (a) intercepts; (b) zeroes; and asymptotes of rational functions. graphs rational functions. solves problems involving rational functions, equations, and inequalities.

3. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT PRIMING ACTIVITY 3x 2 -8x+4 11x 2 -99 16x 3 +128 Factor the following completely: 4. x 3 +2x 2 -4x-8 2x 2 -x-15 6. 10x 3 -80 (3x-2)(x-2) 11(x+3)(x-3) 16(x+2)(x 2 -2x+4) (x-2)(x+2) 2 (2x+5)(x-3) 10(x-2)(x 2 +2x+4)

4. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT PRIMING ACTIVITY Solve the following rational equation.

5. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT PRIMING ACTIVITY

6. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT ACTIVITY 1 (Group Work) (15 min) Your task is to complete the table to show that the time it takes to reach the top of wall depends on the climber’s speed. Compare your results and describe their properties. The team leaders of the groups report their conclusions to the whole class. Climbing the Wall

7. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT ANALYSIS What did you find difficult about this task? What task did you find most difficult to do? Why? What information can you get from the equation of a rational graph? What have you learned about the key features of the rational function? What are some common errors which students may commit? How can we prevent such error? Feel free to offer a suggestion.

8. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT ABSTRACTION LEARNING AREA STANDARD : At the end of the course, the students must know how to solve problems involving rational, exponential and logarithmic functions; to solve business-related problems; and to apply logic to real-life situations. CONTENT STANDARD : The learner demonstrates understanding of key concepts of rational functions.

9. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT ABSTRACTION PERFORMANCE STANDARD: The learner is able to accurately formulate and solve real-life problems involving rational functions.

10. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT ABSTRACTION LEARNING COMPETENCIES: The learners ... represents real-life situations using rational functions. distinguishes rational function, rational equation, and rational inequality. solves rational equations and inequalities. represents a rational function through its: (a) table of values, (b) graph, and (c) equation. finds the domain and range of a rational function.

11. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT ABSTRACTION LEARNING COMPETENCIES: The learners ... determines the: (a) intercepts; (b) zeroes; and asymptotes of rational functions. graphs rational functions. solves problems involving rational functions, equations, and inequalities.

12. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT Many real-world problems can be modeled by rational functions. REPRESENTING REAL LIFE SITUATIONS USING RATIONAL FUNCTIONS

13. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT REPRESENTING REAL LIFE SITUATIONS USING RATIONAL FUNCTIONS

14. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT RATIONAL FUNCTIONS Rational Expression It is the quotient of two polynomials. A rational function is any ratio of two polynomials, where denominator cannot be ZERO! Examples: Not Rational:

15. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT RATIONAL FUNCTIONS Asymptotes Asymptotes are the boundary lines that a rational function approaches, but never crosses. We draw these as Dashed Lines on our graphs. There are three types of asymptotes: Vertical Horizontal (Graph can cross these) Slant

16. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT RATIONAL FUNCTIONS Vertical Asymptotes Vertical Asymptotes exist where the denominator would be zero. They are graphed as Vertical Dashed Lines There can be more than one! To find them, set the denominator equal to zero and solve for “x”

17. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT RATIONAL FUNCTIONS Vertical Asymptotes Set the denominator equal to zero x – 1 = 0, so x = 1 This graph has a vertical asymptote at x = 1 Find the vertical asymptotes for the following function:

18. 1 2 6 3 4 5 7 8 9 10 4 3 2 7 5 6 8 9 x-axis y-axis 1 -2 -6 -3 -4 -5 -7 -8 -9 10 -4 -3 -2 -1 -7 -5 -6 -8 -9 -1 Vertical Asymptote at X = 1

19. Other Examples: Find the vertical asymptotes for the following functions:

20. Horizontal Asymptotes Horizontal Asymptotes are also Dashed Lines drawn horizontally to represent another boundary. To find the horizontal asymptote you compare the degree of the numerator with the degree of the denominator

21. Horizontal Asymptote (HA) Given Rational Function: Compare DEGREE of Numerator to Denominator If N < D , then y = 0 is the HA If N > D, then the graph has NO HA If N = D, then the HA is

22. Example #1 Find the horizontal asymptote for the following function: Since the degree of numerator is equal to degree of denominator (m = n) Then HA: y = 1/1 = 1 This graph has a horizontal asymptote at y = 1

23. 1 2 6 3 4 5 7 8 9 10 4 3 2 7 5 6 8 9 x-axis y-axis 1 -2 -6 -3 -4 -5 -7 -8 -9 10 -4 -3 -2 -1 -7 -5 -6 -8 -9 -1 Horizontal Asymptote at y = 1

24. Other Examples: Find the horizontal asymptote for the following functions:

25. Slant Asymptotes (SA) Slant asymptotes exist when the degree of the numerator is one larger than the denominator. Cannot have both a HA and SA To find the SA, divide the Numerator by the Denominator. The results is a line y = mx + b that is the SA.

26. Example of SA 27 -2

27. 28 Holes A hole exists when the same factor exists in both the numerator and denominator of the rational expression and the factor is eliminated when you reduce!

28. Example of Hole Discontinuity 29

29. Domain: ( –  , 0)  (0,  ) Range: ( –  , 0)  (0,  ) Find domain and graph. x y – 2 – ½ – 1 – 1 – ½ – 2 undefined ½ 2 1 1 2 ½ It is discontinuous at x = 0.

30. Domain: ( –  , 0)  (0,  ) Range: ( –  , 0)  (0,  ) Find domain and graph. x y – 2 – ½ – 1 – 1 – ½ – 2 undefined ½ 2 1 1 2 ½ decreases on the intervals ( – ,0) and (0, ).

31. Domain: ( –  , 0)  (0,  ) Range: ( –  , 0)  (0,  ) Find domain and graph. x y – 2 – ½ – 1 – 1 – ½ – 2 undefined ½ 2 1 1 2 ½ The y-axis is a vertical asymptote, and the x-axis is a horizontal asymptote.

32. Domain: ( –  , 0)  (0,  ) Range: ( –  , 0)  (0,  ) Find domain and graph. x y – 2 – ½ – 1 – 1 – ½ – 2 undefined ½ 2 1 1 2 ½ It is an odd function and its graph is symmetric with respect to the origin.

33. Domain: ( –  , 0)  (0,  ) Range: (0,  ) Find domain and graph. x y  3  2 ¼  1 1  ½ 4  ¼ 16 undefined increases on the interval ( – ,0) and decreases on the interval (0, ).

34. Domain: ( –  , 0)  (0,  ) Range: (0,  ) Find domain and graph. x y  3  2 ¼  1 1  ½ 4  ¼ 16 undefined It is discontinuous at x = 0.

35. Domain: ( –  , 0)  (0,  ) Range: (0,  ) Find domain and graph. x y  3  2 ¼  1 1  ½ 4  ¼ 16 undefined The y -axis is a vertical asymptote, and the x -axis is a horizontal asymptote.

36. Domain: ( –  , 0)  (0,  ) Range: (0,  ) Find domain and graph. x y  3  2 ¼  1 1  ½ 4  ¼ 16 undefined It is an even function, and Its graph is symmetric with respect to the y-axis.

37. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT RATIONAL EQUATIONS

38. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT RATIONAL INEQUALITIES

39. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT RATIONAL FUNCTIONS Lesson 4: Representations of Rational Functions Lesson 4:

40. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT GRAPHING RATIONAL FUNCTIONS General Steps to Graph a Rational Function 1) Factor the numerator and the denominator 2) State the domain and the location of any holes in the graph 3) Simplify the function to lowest terms 4) Find the y-intercept (x = 0) and the x-intercept(s) (y = 0) 5) Identify any existing asymptotes (vertical, horizontal, or oblique

41. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT GRAPHING RATIONAL FUNCTIONS General Steps to Graph a Rational Function 6) Identify any points intersecting a horizontal or oblique asymptote. 7) Use test points between the zeros and vertical asymptotes to locate the graph above or below the x-axis 8) Analyze the behavior of the graph on each side of an asymptote 9) Sketch the graph

42. The Graph of a Rational Function General Steps to Graph a Rational Function 4) Find the y-intercept (x = 0) and the x-intercept(s) (y = 0) y-intercept (x = 0) x-intercept(s) (y = 0)       Use numerator factors            

43. The Graph of a Rational Function General Steps to Graph a Rational Function 5) Identify any existing asymptotes (vertical, horizontal, or oblique Horiz . Or Oblique Asymptotes Vertical Asymptotes     Use denominator factors             Examine the largest exponents Same Horiz . - use coefficients  

44. The Graph of a Rational Function

45. The Graph of a Rational Function General Steps to Graph a Rational Function 6) Identify any points intersecting a horizontal or oblique asymptote.            

46. The Graph of a Rational Function General Steps to Graph a Rational Function   7) Use test points between the zeros and vertical asymptotes to locate the graph above or below the x-axis -4 -2 2 3                    

47. The Graph of a Rational Function General Steps to Graph a Rational Function   7) Use test points between the zeros and vertical asymptotes to locate the graph above or below the x-axis -4 -2 2 3                  

48. The Graph of a Rational Function General Steps to Graph a Rational Function       -4 -2 2 3         8) Analyze the behavior of the graph on each side of an asymptote          

49. The Graph of a Rational Function General Steps to Graph a Rational Function     8) Analyze the behavior of the graph on each side of an asymptote             -4 -2 2 3        

50. The Graph of a Rational Function 9) Sketch the graph

51. The Graph of a Rational Function Example 1) Factor the numerator and the denominator 2) State the domain and the location of any holes in the graph 3) Simplify the function to lowest terms       Domain:   Hole in the graph at  

52. The Graph of a Rational Function General Steps to Graph a Rational Function 4) Find the y-intercept (x = 0) and the x-intercept(s) (y = 0) y-intercept (x = 0) x-intercept(s) (y = 0)       Use numerator factors      

53. The Graph of a Rational Function General Steps to Graph a Rational Function 5) Identify any existing asymptotes (vertical, horizontal, or oblique Horiz . Or Oblique Asymptotes Vertical Asymptotes     Use denominator factors         Examine the largest exponents Same Horiz . - use coefficients  

54. The Graph of a Rational Function General Steps to Graph a Rational Function 6) Identify any points intersecting a horizontal or oblique asymptote.            

55. The Graph of a Rational Function General Steps to Graph a Rational Function   7) Use test points between the zeros and vertical asymptotes to locate the graph above or below the x-axis                     -3 2

56. The Graph of a Rational Function General Steps to Graph a Rational Function     8) Analyze the behavior of the graph on each side of an asymptote           -3 2

57. The Graph of a Rational Function 9) Sketch the graph

58. The Graph of a Rational Function Example 1) Factor the numerator and the denominator 2) State the domain and the location of any holes in the graph 3) Simplify the function to lowest terms       Domain:   No holes

59. The Graph of a Rational Function General Steps to Graph a Rational Function 4) Find the y-intercept (x = 0) and the x-intercept(s) (y = 0) y-intercept (x = 0) x-intercept(s) (y = 0)       Use numerator factors            

60. The Graph of a Rational Function General Steps to Graph a Rational Function 5) Identify any existing asymptotes (vertical, horizontal, or oblique Horiz . or Oblique Asymptotes Vertical Asymptotes     Use denominator factors         Examine the largest exponents Oblique: Use long division                  

61. The Graph of a Rational Function General Steps to Graph a Rational Function 6) Identify any points intersecting a horizontal or oblique asymptote.            

62. The Graph of a Rational Function General Steps to Graph a Rational Function   7) Use test points between the zeros and vertical asymptotes to locate the graph above or below the x-axis                     -2 1 -1    

63. The Graph of a Rational Function General Steps to Graph a Rational Function     8) Analyze the behavior of the graph on each side of an asymptote           1

64. The Graph of a Rational Function 9) Sketch the graph

65. Review: STEPS for GRAPHING HOLES ___________________________________________ ___________________________________________ EX _________________________________________ EX _________________________________________ Discontinuous part of the graph where the line jumps over. Represented by a little open circle. Hole @ x = 3 Hole @ x = 2 No hole at x = 0

66. VERTICAL ASYMPTOTES ___________________________________________ ___________________________________________ EX _________________________________________ EX _________________________________________ Discontinuous part of the graph where the line cannot cross over. Represented by a dotted line called an asymptote . VA @ x = 2 Hole @ x =0 VA @ x = 2, -5 Review: STEPS for GRAPHING

67. HORIZONTAL ASYMPTOTES n = degree of numerator d = degree of denominator _______________________________________________ _______________________________________________ _______________________________________________ Case 1 n > d No HA Case 2 n < d HA @ y = 0 Case 1 n = d HA is the ratio of coefficients HA @ y = 4 / 5 Review: STEPS for GRAPHING

68. Finding holes and asymptotes VA: x=-1, -5 HA: y=0 (power of the denominator is greater than the numerator) Holes: none VA: none (graph is the same as y=x-1 once the (x-2)s cancel HA: none (degree of the numerator is greater than the denominator) Hole: x=2

69. Let’s try some VA: x=3 HA: none (power of the numerator is greater than the denominator) Holes: x=2 VA: x=-5,0 ( cancel the (x-3)s HA: y=0 (degree of the denominator is greater than the numerator) Hole: x=3 Find the vertical, horizontal asymptotes and any holes

70. GRAPHING y = x / (x – 3) 1) HOLES? no holes since nothing cancels VERTICAL ASYMPTOTES? Yes ! VA @ x =3 4) T-CHART X Y = x/(x – 3) 4 Y = 4 2 Y = -2 3) HORIZONTAL ASYMPTOTES? Yes ! HA @ y =1 5 Y = 0 Y = 5/2

71. GRAPHING 1) HOLES? VERTICAL ASYMPTOTES? 3) HORIZONTAL ASYMPTOTES? 4) The graph - What cancels? Graph the function y=x with a hole at x=-1 hole @ x = -1 None! None!

72. GRAPHING 1) HOLES? VERTICAL ASYMPTOTES? 4) T-CHART X 6 Y = 1/2 -3 Y = -5/8 3) HORIZONTAL ASYMPTOTES? 1 2 Y = 1/12 Y = 0 3 Y = -1 / 10 WAIT – What about the Horizontal Asymptote here? hole @ x = 0 Yes ! VA @ x =-2 , 5 Yes ! HA @ y =0 (Power of the denominator is greater than the numerator)

73. Remember, Horizontal Asymptotes only describe the ends of the function (left and right). What happens in the middle is ‘fair game’. T-CHART X -1 Y = 1/2 4 Y = -1/3 2 Y = 0 To find out what the graph looks like between the vertical asymptotes, go to a T Chart and plug in values close to the asymptotes. Left Right Middle

74. Let’s try one: Sketch the Graph 1) HOLES? VERTICAL ASYMPTOTES? 4) T-CHART X Y = 0 -1 Y = 1/4 3) HORIZONTAL ASYMPTOTES? -2 2 Y = .22 Y=-2 3 Y = -3/4 none Yes ! VA @ x = 1 Yes ! HA @ y =0 (Power of the denominator is greater than the numerator)

75. Problems Find the vertical asymptotes, horizontal asymptotes, slant asymptotes and holes for each of the following functions. Vertical: x = -2 Horizontal : y = 1 Slant: none Hole: at x = - 5 Vertical: x = 3 Horizontal : none Slant: y = 2 x +11 Hole: none

76. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT APPLICATION On a manila paper, paste/draw some small pictures of objects such that they are positioned at different coordinates. Then, draw circles that contain these pictures. Using the pictures and the circles drawn on the grid, formulate problems involving the equation of the circle and then solve them.

77. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT CLOSURE “The ability to simplify means to eliminate the unnecessary so that the necessary may speak.” Hans Hofmann – early 20th century teacher and painter

78. MANUEL S. ENVERGA UNIVERSITY FOUNDATION An Autonomous University BASIC EDUCATION DEPARTMENT