Chapter 2: Rational. Functions. SHMth1: General Mathematics. Accountancy, Business and Management (ABM. Mr. Migo M. Mendoza

Transcription

1 Chapter 2: Rational Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza

2 Chapter 2: Rational Functions Lecture 6: Basic Concepts Lecture 7: Solving Rational Equations Lecture 8: Solving Rational Inequalities Lecture 9: Asymptotes of Rational Functions Lecture 10: Graphing Rational Functions

3 Family Activity 1: Constructing KWL Chart SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza

4 The Grading System Criteria Content Organization of Ideas Communication Skills Presentation and Aesthetic Consideration Behavior Percentage 18 points 5 points 5 points 3 points 4 points

5 What to do? Answer the question: What is a Rational Function? by constructing a KWL (Know, Want to Know, Learned) Chart. Afterwards, share your answer to the class. Please be guided that this is a time pressure family activity.

6 The KWL Chart Know Want to Learned Know

7 Lecture 6: Basic Concepts in Rational Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza

8 Rational Number (Q) It is a number that can be written as a fraction or ratio and whose numerators and denominators are integers provided that the denominator is not equal to 0.

9 The Definition of Rational Function If we let P(x) and Q(x) be two polynomials, then a function of the form: P( x) f ( x) Q( x) is called a RATIONAL FUNCTION.

10 The Domain of the Rational Function The domain of f(x) is the set of real numbers x except those for which Q(x) = 0.

11 Take Note: Since division by zero is impossible, a rational function has a DISCONTINUITY whenever its denominator is zero.

12 Did you know? The denominator of a rational function cannot be zero. Any value of the variable that would make the denominator zero is NOT PERMISSIBLE.

13 Take Note: The domain of a rational function is the set of all real numbers, except those that make the denominator zero.

14 Some Examples of Rational Function: f 3x 2 ( x) ; x x 2 2

15 Some Examples of Rational Function: f ( x) 2 4x x 3x ; x 3

16 Some Examples of Rational Function: f 1 ( x) ; x 3 x 3

17 A Short Review on Rational Functions: Find the domain of the rational function: x r( x) x( x 3)

18 Final Answer: The domain of r(x) is the set of all real numbers, except those that make the denominator zero. Thus, D { x x 0andx 3}

19 A Short Review on Rational Functions: Find the domain of the rational function: x 2 4 x 5 R( x) 2 x 2 x 8

20 Final Answer: The domain of r(x) is the set of all real numbers, except those that make the denominator zero. Thus, D { x x 4andx 2}

21 Rational Function in Real World SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza

22 In Pharmacology We use rational function to determine the medicine concentration after a period of time. Say: ( 5t C t) 0.01t 2 3.3

23 In Biology A biologist discovered a formula to determine how your blood brings oxygen to the rest of the body the HEMOGLOBIN. l l n n K d

24 In Investment Rational function is used to determine exact and ordinary interest which is use in Banker s Rule. I e Prt 365 I o Prt 360

25 In Consumer Loan Rational function is used to determine the borrower s equal payment at equal interval (annuity). R S n i n ( 1 i) 1

26 Lecture 7: Solving Rational Equations SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza

27 Rational Expression A rational expression is an expression that can be written as a ratio of two polynomials.

28 Family Activity 2: Am I Rational Expression? SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza

29 The Grading System Criteria Correctness Justification/ Reasoning Communication Skills Behavior Percentage 10 points 6 points 5 points 4 points

30 Am I Rational Expression? Using the definition of a rational expression, tell why the following is or not a rational expression. Have a sound justification.

31 Am I Rational Expression? ) ( 3 1 ) ( ) ( x x c x b x x x a ) ( 1 1 ) ( 3 x x e x x d

32 To sum it up In simplest manner, a rational expression can be described as a function where either the numerator, denominator, or both have a variable on it.

33 Lecture 7: Solving Rational Equations SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza

34 Something to think about What is your idea of a rational equation?

35 Example 46: Solve the rational equation: 5x

36 Something to think about How to solve rational equations?

37 Solving Rational Equations To solve a rational equation, we multiply each term of the equation by the least common denominator (LCD) of any fractions.

38 Solving Rational Equations The resulting equation should be equivalent to the original equation and be cleared of all fractions as long as we do not multiply by zero.

39 Steps in Solving an Equation Containing Rational Equations Step 1: Determine the LCD of all the denominators.

40 Steps in Solving an Equation Containing Rational Equations Step 2: Multiply each term of the equation by the LCD.

41 Steps in Solving an Equation Containing Rational Equations Step 3: Solve the resulting equation.

42 Steps in Solving an Equation Containing Rational Equations Step 4: Check your answer by substituting it into the original equation. Exclude from the solution any value that would make the LCD equal to zero. Such value is called EXTRANEOUS SOLUTION.

43 Final Answer: Hence, x 1 is the solution of the given equation.

44 Example 47: Solve the rational equation: x 4 1 x

45 Final Answer: Hence, x 6 is the root of the given equation.

46 Example 48: Solve the rational equation: x 2 x x 2

47 Final Answer: Hence, x 1 is the only root of the given equation.

48 Example 50: Solve the rational equation: x 12 x 2 3 x

49 Final Answer: The roots of the given rational equation are: x 6andx 3.

50 Example 51: Solve the rational equation: 1 3x 7 1 x 5 3x 2 19x 20 3x 4

51 Final Answer: The root of the given rational equation is: x 6

52 Performance Task 6: Please download, print and answer the Let s Practice 6. Kindly work independently.

53 Lecture 8: Solving Rational Inequalities SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza

54 Rational Inequalities An inequality that contains rational expressions is referred to as RATIONAL INEQUALITIES.

55 Rational Inequalities It is a rational equation that contains an inequality.

56 Example 50: Solve the rational inequality, then, graph its solution set: 2 x 1 x 1 4

57 Step 1: Solving Rational Inequalities Determine the critical numbers for f(x) by establishing the zeros of f(x) and excluded values for f(x). We can solve for the zeros of f(x) using the numerator of the rational function.

58 Step 2: Solving Rational Inequalities Plot the critical numbers in the number line into intervals and create a table for test of values of x.

59 Test of Values Intervals Test of Value x f(x) Sign of f(x)

60 Rational Inequality Theorem 1: If the rational inequality is of the form f(x) > 0 or f(x) 0, then all of the intervals with the positive sign are solutions. Also, the zeros of f(x) are part of the solution if f(x) 0.

61 Final Answer: The solution is the interval notation: x 9 x 4orx 1.

62 Example 51: Solve the rational inequality, then, graph its solution set: x 2 3x 5 1 x 2

63 Step 1: Solving Rational Inequalities Determine the critical numbers for f(x) by establishing the zeros of f(x) and excluded values for f(x). We can solve for the zeros of f(x) using the numerator of the rational function.

64 Step 2: Solving Rational Inequalities Plot the critical numbers in the number line into intervals and create a table for test of values of x.

65 Test of Values Intervals Test of Value x f(x) Sign of f(x)

66 Rational Inequality Theorem 2: If the rational inequality is of the form f(x) < 0 or f(x) 0, then all of the intervals with the negative sign are solutions. Also, the zeros of f(x) are part of the solution if f(x) 0.

67 Final Answer: Since the rational inequality is of the form f(x) < 0, then the solution is: x x 3or1 x 2.

68 Performance Task 7: Please download, print and answer the Let s Practice 7. Kindly work independently.

69 Lecture 9: Asymptotes of Rational Function SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza

70 Did you know? There are three (3) saddest love stories in Mathematics

71 The Painful Parallel You may encounter potential people, bump onto them, see them from afar, but will never actually get to know and meet them; even in the longest time.

72 The Painful Tangent Some people are only meant to meet one another at one point in their lives, but are forever parted.

73 The Painful Asymptote There are people who may get closer and closer to one another, but will never be together.

74 The Definition of the Asymptote It is a straight line associated with a curve such that as a point moves along infinite branch of the curve the distance from the point to the line approaches zero and the slope of the curve at the point approaches the slope of the line.

75 Understanding the Asymptote of a Rational Function

76 Types of Asymptote 1. Vertical Asymptote 2.Horizontal Asymptote 3.Oblique Asymptote

77 The Vertical Asymptote

78 The Horizontal Asymptote

79 The Oblique Asymptote

80 Understanding the Slope of a Line The Slope Song by Colin Dodds

81 Understanding the Slope of a Line The Slope Dude by Colin Dodds

82 The Behavior of the Graph of a Line 1. Increasing 2.Decreasing 3.Constant

83 Something to think about. What is the slope of a line?

84 Example 53: Find the vertical, horizontal, and oblique asymptotes (if any) for: 1 f ( x) x. ( 4)

85 Type 1: Vertical Asymptote Given a rational function: P( x) f ( x) ; Q( x) Q( x) If f(x) approaches infinity (or negative infinity) as x approaches a real number a from the right or left, then the line x = a is a VERTICAL ASYMPTOTE. 0.

86 Theorem 2.1: Vertical Asymptote If a is a real number such that Q(a) = 0 and P(a) 0, then the line x = a is a vertical asymptote of the graph f ) ( ) ( ) ( b b x x b x b a x a x a x a x Q x P x f m m m m n n n n

87 Type 2: Horizontal Asymptote Given a rational function: P( x) f ( x) ; Q( x) Q( x) If f(x) approaches infinity (or negative infinity) as f(x) approaches a real number b, then the line y = b is a HORIZONTAL ASYMPTOTE. 0.

88 Theorem 2.2.a: Horizontal Asymptote The horizontal asymptote of the graph f may be found by the following rules: If n < m, then y = 0 is a horizontal asymptote ) ( ) ( ) ( b b x x b x b a x a x a x a x Q x P x f m m m m n n n n

89 Theorem 2.2.b: Horizontal Asymptote The horizontal asymptote of the graph f may be found by the following rules: If n=m, then is a horizontal asymptote ) ( ) ( ) ( b b x x b x b a x a x a x a x Q x P x f m m m m n n n n b a y

90 Theorem 2.2.c: Horizontal Asymptote The horizontal asymptote of the graph f may be found by the following rules: If n>m, then there is no horizontal asymptote ) ( ) ( ) ( b b x x b x b a x a x a x a x Q x P x f m m m m n n n n

91 Type 3: Oblique Asymptote A rational function P( x) f ( x) ; Q( x) Q( x) 0. has an oblique asymptote if the degree of P(x) is greater than the degree of Q(x).

92 Final Answer: The vertical asymptote of the rational function is x = 4; The horizontal asymptote is y = 0; and The rational function does not contain any oblique asymptote.

93 Example 54: Find the vertical, horizontal, and oblique asymptotes (if any) for: 3x 2 2 x 4 f ( x) 2. x 4 x 3

94 Final Answer: The vertical asymptote of the rational function are x = 1 and x=3; The horizontal asymptote is y = 3; and The rational function does not contain any oblique asymptote.

95 Example 55: Find the vertical, horizontal, and f oblique asymptotes (if any) for: 2 x x ( x) x

96 Final Answer: The vertical asymptote of the rational function is x = -3; The horizontal asymptote is y = 1; and The rational function does not contain any oblique asymptote.

97 Example 56: Find the vertical, horizontal, and oblique asymptotes (if any) for: 4x 3 2x 2 7 f ( x) 2. x 2 x 3

98 Final Answer: The vertical asymptote of the rational function are x = -3 and x = 1; The graph has no horizontal asymptote; and The oblique asymptote is y = 4x - 6.

99 Performance Task 8: Please download, print and answer the Let s Practice 8. Kindly work independently.

100 Lecture 10: Graphing Rational Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza

101 Learning Expectations: This lecture focuses on the graphical solution of a rational function. It is necessary to determine the asymptotes which were already been discussed in the previous section. The additional important parts in setting the graph of a rational function are the intercepts, coordinates, domain and range. The steps provided on the next slides will guide us in establishing the graph of a rational function.

102 Example 57: Determine the domain, range, intercepts, and zeros of the rational function f ( x) 1 x 4 and sketch the graph.

103 Take Note: The technique in graphing rational functions includes finding the intercepts, zeroes and asymptotes of the rational function.

104 Steps in Graphing Rational Function Step 1: Determine the asymptotes of the graph.

105 The Asymptotes of the graph are: Vertical Asymptote: x 4 Horizontal Asymptote: y 0

106 Steps in Graphing Rational Function Step 2: Determine the x-intercepts and y-intercepts, if there are any.

107 Intercepts The intercepts of the graph of a rational function are the points of intersection of its graph and an axis.

108 The Y-Intercept The y-intecept of the graph of a rational function f(x), if it exists, occurs at f(0), provided that f(x) is defined at x = 0.

109 The X-Intercept The x-intercept of the graph of a rational function f(x), if it exists, occurs at zeroes of the numerator that are not zeroes of the denominators.

110 The x-intercept and y-intercept of the graph of f(x) are: x-intercept: There is no x-intercept. y-intercept: 1 0, 4

111 Steps in Graphing Rational Function Step 3: Consider the sign of f(x) in the intervals determined by zeros of P(x) and Q(x).

112 Steps in Graphing Rational Function Step 4: Identify the symmetry detected by the test.

113 Steps in Graphing Rational Function Step 5: Plot some points on either side of each vertical asymptote and check whether the graph crosses a horizontal asymptote.

114 The Table of Values x f(x) x f(x)

115 Steps in Graphing Rational Function Step 6: Sketch the graph, using the points plotted and using the asymptotes as guide. The graph is a smooth curve, except for breaks at the asymptotes.

116

117 Example 58: Determine the domain, range, intercepts, and zeros of the rational function f 2 x x ( x) x and sketch the graph.

118 Steps in Graphing Rational Function Step 1: Determine the asymptotes of the graph.

119 The Asymptotes of the graph are: Vertical Asymptote: x 3 x 3 Horizontal Asymptote: y 1

120 Steps in Graphing Rational Function Step 2: Determine the x-intercepts and y-intercepts, if there are any.

121 The x-intercept and y-intercept of the graph of f(x) are: x-intercept: 2, 0 y-intercept: 0, 2 3

122 Steps in Graphing Rational Function Step 3: Consider the sign of f(x) in the intervals determined by zeros of P(x) and Q(x).

123 Steps in Graphing Rational Function Step 4: Identify the symmetry detected by the test.

124 Steps in Graphing Rational Function Step 5: Plot some points on either side of each vertical asymptote and check whether the graph crosses a horizontal asymptote.

125 The Table of Values x f(x) x f(x)

126 Steps in Graphing Rational Function Step 6: Sketch the graph, using the points plotted and using the asymptotes as guide. The graph is a smooth curve, except for breaks at the asymptotes.

127

128 Example 59: Determine the domain, range, intercepts, and zeros of the rational function f ( x) 1 1 and sketch the graph. 2 x x

129 Steps in Graphing Rational Function Step 1: Determine the asymptotes of the graph.

130 The Asymptotes of the graph are: Vertical Asymptote: x 1 Horizontal Asymptote: y x 1

131 Steps in Graphing Rational Function Step 2: Determine the x-intercepts and y-intercepts, if there are any.

132 The x-intercept and y-intercept of the graph of f(x) are: x-intercept: There is no x-intercept. y-intercept: 0,1

133 Steps in Graphing Rational Function Step 3: Consider the sign of f(x) in the intervals determined by zeros of P(x) and Q(x).

134 Steps in Graphing Rational Function Step 4: Identify the symmetry detected by the test.

135 Steps in Graphing Rational Function Step 5: Plot some points on either side of each vertical asymptote and check whether the graph crosses a horizontal asymptote.

136 The Table of Values x f(x) x f(x)

137 Steps in Graphing Rational Function Step 6: Sketch the graph, using the points plotted and using the asymptotes as guide. The graph is a smooth curve, except for breaks at the asymptotes.

138

139 Performance Task 9: Please download, print and answer the Let s Practice 9. Kindly work independently.