Chapter 2: Rational Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza
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
Family Activity 1: Constructing KWL Chart SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza
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
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.
The KWL Chart Know Want to Learned Know 1. 2. 3. 1. 2. 3.
Lecture 6: Basic Concepts in Rational Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza
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.
The Definition of Rational Function If we let P() and Q() be two polynomials, then a function of the form: P( ) f ( ) Q( ) is called a RATIONAL FUNCTION.
The Domain of the Rational Function The domain of f() is the set of real numbers ecept those for which Q() = 0.
Take Note: Since division by zero is impossible, a rational function has a DISCONTINUITY whenever its denominator is zero.
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.
Take Note: The domain of a rational function is the set of all real numbers, ecept those that make the denominator zero.
Some Eamples of Rational Function: f 3 2 ( ) ; 2 2
Some Eamples of Rational Function: f ( ) 2 4 3 2 1 2 ; 3
Some Eamples of Rational Function: f 1 ( ) ; 3 3
A Short Review on Rational Functions: Find the domain of the rational function: r( ) ( 3)
Final Answer: The domain of r() is the set of all real numbers, ecept those that make the denominator zero. Thus, D { 0and 3}
A Short Review on Rational Functions: Find the domain of the rational function: 2 4 5 R( ) 2 2 8
Final Answer: The domain of r() is the set of all real numbers, ecept those that make the denominator zero. Thus, D { 4and 2}
Rational Function in Real World SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza
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
In Biology A biologist discovered a formula to determine how your blood brings oygen to the rest of the body the HEMOGLOBIN. l l n n K d
In Investment Rational function is used to determine eact and ordinary interest which is use in Banker s Rule. I e Prt 365 I o Prt 360
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
Lecture 7: Solving Rational Equations SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza
Rational Epression A rational epression is an epression that can be written as a ratio of two polynomials.
Family Activity 2: Am I Rational Epression? SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza
The Grading System Criteria Correctness Justification/ Reasoning Communication Skills Behavior Percentage 10 points 6 points 5 points 4 points
Am I Rational Epression? Using the definition of a rational epression, tell why the following is or not a rational epression. Have a sound justification.
Am I Rational Epression? 2 3 4 ) ( 3 1 ) ( 3 3 3 3 ) ( 2 2 2 c b a 2 2 1 ) ( 1 1 ) ( 3 e d
To sum it up In simplest manner, a rational epression can be described as a function where either the numerator, denominator, or both have a variable on it.
Lecture 7: Solving Rational Equations SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza
Something to think about What is your idea of a rational equation?
Eample 46: Solve the rational equation: 5 6 1 2 1 3
Something to think about How to solve rational equations?
Solving Rational Equations To solve a rational equation, we multiply each term of the equation by the least common denominator (LCD) of any fractions.
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.
Steps in Solving an Equation Containing Rational Equations Step 1: Determine the LCD of all the denominators.
Steps in Solving an Equation Containing Rational Equations Step 2: Multiply each term of the equation by the LCD.
Steps in Solving an Equation Containing Rational Equations Step 3: Solve the resulting equation.
Steps in Solving an Equation Containing Rational Equations Step 4: Check your answer by substituting it into the original equation. Eclude from the solution any value that would make the LCD equal to zero. Such value is called EXTRANEOUS SOLUTION.
Final Answer: Hence, 1 is the solution of the given equation.
Eample 47: Solve the rational equation: 4 1 3 1 2 3
Final Answer: Hence, 6 is the root of the given equation.
Eample 48: Solve the rational equation: 3 2 1 2 2
Final Answer: Hence, 1 is the only root of the given equation.
Eample 50: Solve the rational equation: 12 2 3
Final Answer: The roots of the given rational equation are: 6and 3.
Eample 51: Solve the rational equation: 1 3 7 1 5 3 2 19 20 3 4
Final Answer: The root of the given rational equation is: 6
Performance Task 6: Please download, print and answer the Let s Practice 6. Kindly work independently.
Lecture 8: Solving Rational Inequalities SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza
Rational Inequalities An inequality that contains rational epressions is referred to as RATIONAL INEQUALITIES.
Rational Inequalities It is a rational equation that contains an inequality.
Eample 50: Solve the rational inequality, then, graph its solution set: 2 1 1 4
Step 1: Solving Rational Inequalities Determine the critical numbers for f() by establishing the zeros of f() and ecluded values for f(). We can solve for the zeros of f() using the numerator of the rational function.
Step 2: Solving Rational Inequalities Plot the critical numbers in the number line into intervals and create a table for test of values of.
Test of Values Intervals Test of Value f() Sign of f()
Rational Inequality Theorem 1: If the rational inequality is of the form f() > 0 or f() 0, then all of the intervals with the positive sign are solutions. Also, the zeros of f() are part of the solution if f() 0.
Final Answer: The solution is the interval notation: 9 4or 1.
Eample 51: Solve the rational inequality, then, graph its solution set: 2 3 5 1 2
Step 1: Solving Rational Inequalities Determine the critical numbers for f() by establishing the zeros of f() and ecluded values for f(). We can solve for the zeros of f() using the numerator of the rational function.
Step 2: Solving Rational Inequalities Plot the critical numbers in the number line into intervals and create a table for test of values of.
Test of Values Intervals Test of Value f() Sign of f()
Rational Inequality Theorem 2: If the rational inequality is of the form f() < 0 or f() 0, then all of the intervals with the negative sign are solutions. Also, the zeros of f() are part of the solution if f() 0.
Final Answer: Since the rational inequality is of the form f() < 0, then the solution is: 3or1 2.
Performance Task 7: Please download, print and answer the Let s Practice 7. Kindly work independently.
Lecture 9: Asymptotes of Rational Function SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza
Did you know? There are three (3) saddest love stories in Mathematics
The Painful Asymptote There are people who may get closer and closer to one another, but will never be together.
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.
The Painful Tangent Some people are only meant to meet one another at one point in their lives, but are forever parted.
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.
Types of Asymptote 1. Vertical Asymptote 2.Horizontal Asymptote 3.Oblique Asymptote
Eample 53: Find the vertical, horizontal, and oblique asymptotes (if any) for: 1 f ( ). ( 4)
Type 1: Vertical Asymptote Given a rational function: P( ) f ( ) ; Q( ) Q( ) If f() approaches infinity (or negative infinity) as approaches a real number a from the right or left, then the line = a is a VERTICAL ASYMPTOTE. 0.
Theorem 2.1: Vertical Asymptote If a is a real number such that Q(a) = 0 and P(a) 0, then the line = a is a vertical asymptote of the graph f. 0 1 1 1 0 1 1 1...... ) ( ) ( ) ( b b b b a a a a Q P f m m m m n n n n
Type 2: Horizontal Asymptote Given a rational function: P( ) f ( ) ; Q( ) Q( ) If f() approaches infinity (or negative infinity) as f() approaches a real number b, then the line y = b is a HORIZONTAL ASYMPTOTE. 0.
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. 0 1 1 1 0 1 1 1...... ) ( ) ( ) ( b b b b a a a a Q P f m m m m n n n n
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. 0 1 1 1 0 1 1 1...... ) ( ) ( ) ( b b b b a a a a Q P f m m m m n n n n
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. 0 1 1 1 0 1 1 1...... ) ( ) ( ) ( b b b b a a a a Q P f m m m m n n n n
Type 3: Oblique Asymptote A rational function P( ) f ( ) ; Q( ) Q( ) 0. has an oblique asymptote if the degree of P() is greater than the degree of Q().
Final Answer: The vertical asymptote of the rational function is = 4; The horizontal asymptote is y = 0; and The rational function does not contain any oblique asymptote.
Eample 54: Find the vertical, horizontal, and oblique asymptotes (if any) for: 3 2 2 4 f ( ) 2. 4 3
Final Answer: The vertical asymptote of the rational function are = 1 and =3; The horizontal asymptote is y = 3; and The rational function does not contain any oblique asymptote.
Eample 55: Find the vertical, horizontal, and f oblique asymptotes (if any) for: 2 ( ) 2 6. 9
Final Answer: The vertical asymptote of the rational function is = -3; The horizontal asymptote is y = 1; and The rational function does not contain any oblique asymptote.
Eample 56: Find the vertical, horizontal, and oblique asymptotes (if any) for: 4 3 2 2 7 f ( ) 2. 2 3
Final Answer: The vertical asymptote of the rational function are = -3 and = 1; The graph has no horizontal asymptote; and The oblique asymptote is y = 4-6.
Performance Task 8: Please download, print and answer the Let s Practice 8. Kindly work independently.
Lecture 10: Graphing Rational Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza
Eample 57: Determine the domain, range, intercepts, and zeros of the rational function f ( ) 1 4 and sketch the graph.
Take Note: The technique in graphing rational functions includes finding the intercepts, zeroes and asymptotes of the rational function.
Steps in Graphing Rational Function Step 1: Determine the asymptotes of the graph.
Steps in Graphing Rational Function Step 2: Determine the -intercepts and y-intercepts, if there are any.
Intercepts The intercepts of the graph of a rational function are the points of intersection of its graph and an ais.
The Y-Intercept The y-intecept of the graph of a rational function r(), if it eists, occurs at r(0), provided that r() is defined at = 0.
The X-Intercept The -intercept of the graph of a rational function r(), if it eists, occurs at zeroes of the numerator that are not zeroes of the denominators.
Steps in Graphing Rational Function Step 3: Consider the sign of f() in the intervals determined by zeros of P() and Q().
Steps in Graphing Rational Function Step 4: Identify the symmetry detected by the test.
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.
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, ecept for breaks at the asymptotes.
Eample 58: Determine the domain, range, intercepts, and zeros of the rational function f 2 ( ) 2 6 9 and sketch the graph.
Steps in Graphing Rational Function Step 1: Determine the asymptotes of the graph.
Steps in Graphing Rational Function Step 2: Determine the -intercepts and y-intercepts, if there are any.
Steps in Graphing Rational Function Step 3: Consider the sign of f() in the intervals determined by zeros of P() and Q().
Steps in Graphing Rational Function Step 4: Identify the symmetry detected by the test.
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.
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, ecept for breaks at the asymptotes.
Eample 59: Determine the domain, range, intercepts, and zeros of the rational function f ( ) 1 1 and sketch the graph. 2
Steps in Graphing Rational Function Step 1: Determine the asymptotes of the graph.
Steps in Graphing Rational Function Step 2: Determine the -intercepts and y-intercepts, if there are any.
Steps in Graphing Rational Function Step 3: Consider the sign of f() in the intervals determined by zeros of P() and Q().
Steps in Graphing Rational Function Step 4: Identify the symmetry detected by the test.
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.
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, ecept for breaks at the asymptotes.
Performance Task 9: Please download, print and answer the Let s Practice 9. Kindly work independently.