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DP IB Maths: AA HL

Revision Notes

Home / IB / Maths: AA HL / DP / Revision Notes / 2. Functions / 2.5 Reciprocal & Rational Functions / 2.5.1 Reciprocal & Rational Functions


2.5.1 Reciprocal & Rational Functions


Reciprocal Functions & Graphs

What is the reciprocal function?

  • The reciprocal function is defined by space f left parenthesis x right parenthesis equals 1 over x comma space x not equal to 0
  • Its domain is the set of all real values except 0
  • Its range is the set of all real values except 0
  • The reciprocal function has a self-inverse nature
    • space f to the power of negative 1 end exponent left parenthesis x right parenthesis equals f left parenthesis x right parenthesis
    • left parenthesis f ring operator f right parenthesis left parenthesis x right parenthesis equals x

What are the key features of the reciprocal graph?

  • The graph does not have a y-intercept
  • The graph does not have any roots
  • The graph has two asymptotes
    • A horizontal asymptote at the x-axis: space y equals 0
      • This is the limiting value when the absolute value of x gets very large
    • A vertical asymptote at the y-axis: space x equals 0
      • This is the value that causes the denominator to be zero
  • The graph has two axes of symmetry
    • y equals x
    • y equals negative x
  • The graph does not have any minimum or maximum points

Linear Rational Functions & Graphs

What is a rational function with linear terms?

  • A (linear) rational function is of the form space f left parenthesis x right parenthesis equals fraction numerator a x plus b over denominator c x plus d end fraction comma space x not equal to negative d over c
  • Its domain is the set of all real values except  negative d over c
  • Its range is the set of all real values except a over c
  • The reciprocal function is a special case of a rational function

What are the key features of linear rational graphs?

  • The graph has a y-intercept at stretchy left parenthesis 0 comma space b over d stretchy right parenthesis provided d not equal to 0
  • The graph has one root at stretchy left parenthesis negative b over a comma space 0 stretchy right parenthesis provided a not equal to 0
  • The graph has two asymptotes
    • A horizontal asymptote: space y equals a over c
      • This is the limiting value when the absolute value of x gets very large
    • A vertical asymptote: space x equals negative d over c
      • This is the value that causes the denominator to be zero
  • The graph does not have any minimum or maximum points
  • If you are asked to sketch or draw a rational graph:
    • Give the coordinates of any intercepts with the axes
    • Give the equations of the asymptotes

Exam Tip

  • If you draw a horizontal line anywhere it should only intersect this type of graph once at most
  • The only horizontal line that should not intersect the graph is the horizontal asymptote
    • This can be used to check your sketch in an exam

Worked Example

The function space f is defined by space f left parenthesis x right parenthesis equals fraction numerator 10 minus 5 x over denominator x plus 2 end fraction for x not equal to negative 2.

a)
Write down the equation of
(i)
the vertical asymptote of the graph of space f,
(ii)
the horizontal asymptote of the graph of space f.

2-4-1-ib-aa-sl-rational-func-a-we-solution

b)
Find the coordinates of the intercepts of the graph of space f with the axes.

2-4-1-ib-aa-sl-rational-func-b-we-solution

c)
Sketch the graph of space f.

2-4-1-ib-aa-sl-rational-func-c-we-solution

Quadratic Rational Functions & Graphs

How do I sketch the graph of a rational function where the terms are not linear?

  • A rational function can be written space f left parenthesis x right parenthesis equals fraction numerator g left parenthesis x right parenthesis over denominator h left parenthesis x right parenthesis end fraction
    • Where g and h are polynomials
  • To find the y-intercept evaluate fraction numerator g left parenthesis 0 right parenthesis over denominator h left parenthesis 0 right parenthesis end fraction
  • To find the x-intercept(s) solve space g left parenthesis x right parenthesis equals 0
  • To find the equations of the vertical asymptote(s) solve space h left parenthesis x right parenthesis equals 0
  • There will also be an asymptote determined by what f(x) tends to as x approaches infinity
    • In this course it will be either:
      • Horizontal
      • Oblique (a slanted line)
    • This can be found by writing space g left parenthesis x right parenthesis in the form space h left parenthesis x right parenthesis Q left parenthesis x right parenthesis plus r left parenthesis x right parenthesis
      • You can do this by polynomial division or comparing coefficients
    • The function then tends to the curve space y equals Q left parenthesis x right parenthesis

What are the key features of rational graphs: quadratic over linear?

  • For the rational function of the form f open parentheses x close parentheses equals fraction numerator a x squared plus b x plus c over denominator d x plus e end fraction
  • The graph has a y-intercept at stretchy left parenthesis 0 comma space c over e stretchy right parenthesis provided e not equal to 0
  • The graph can have 0, 1 or 2 roots
    • They are the solutions to a x squared plus b x plus c equals 0
  • The graph has one vertical asymptote x equals negative e over d
  • The graph has an oblique asymptote y equals p x plus q
    • Which can be found by writing a x squared plus b x plus c in the form left parenthesis d x plus e right parenthesis left parenthesis p x plus q right parenthesis plus r
      • Where p, q, r are constants
      • This can be done by polynomial division or comparing coefficients

2-5-1-ib-aa-hl-quad-rational-diagram-1

What are the key features of rational graphs: linear over quadratic?

  • For the rational function of the form f open parentheses x close parentheses equals fraction numerator a x plus b over denominator c x squared plus d x plus e end fraction
  • The graph has a y-intercept at stretchy left parenthesis 0 comma space b over e stretchy right parenthesis provided e not equal to 0
  • The graph has one root at x equals negative b over a
  • The graph has can have 0, 1 or 2 vertical asymptotes
    • They are the solutions to c x squared plus d x plus e equals 0
  • The graph has a horizontal asymptote 

2-5-1-ib-aa-hl-quad-rational-diagram-2

Exam Tip

  • If you draw a horizontal line anywhere it should only intersect this type of graph twice at most
    • This idea can be used to check your graph or help you sketch it

Worked Example

The function space f is defined by space f open parentheses x close parentheses equals fraction numerator 2 x squared plus 5 x minus 3 over denominator x plus 1 end fraction blank for x not equal to negative 1.

a)
(i)
Show that fraction numerator 2 x squared plus 5 x minus 3 over denominator x plus 1 end fraction equals p x plus q plus fraction numerator r over denominator x plus 1 end fraction for constants p comma space q and r which are to be found.
(ii)
Hence write down the equation of the oblique asymptote of the graph of space f.

2-5-1-ib-aa-hl-quad-rational-function-a-we-solution

b)
Find the coordinates of the intercepts of the graph of space f with the axes.

2-5-1-ib-aa-hl-quad-rational-function-b-we-solution

c)
Sketch the graph of space f.

2-5-1-ib-aa-hl-quad-rational-function-c-we-solution

 



  • 1. Number & Algebra
    • 1.1 Number & Algebra Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
          • 1.1.3 Partial Fractions
          • 1.2 Exponentials & Logs
            • 1.2.1 Introduction to Logarithms
              • 1.2.2 Laws of Logarithms
                • 1.2.3 Solving Exponential Equations
                • 1.3 Sequences & Series
                  • 1.3.1 Language of Sequences & Series
                    • 1.3.2 Arithmetic Sequences & Series
                      • 1.3.3 Geometric Sequences & Series
                        • 1.3.4 Applications of Sequences & Series
                          • 1.3.5 Compound Interest & Depreciation
                          • 1.4 Simple Proof & Reasoning
                            • 1.4.1 Proof
                            • 1.5 Further Proof & Reasoning
                              • 1.5.1 Proof by Induction
                                • 1.5.2 Proof by Contradiction
                                • 1.6 Binomial Theorem
                                  • 1.6.1 Binomial Theorem
                                    • 1.6.2 Extension of The Binomial Theorem
                                    • 1.7 Permutations & Combinations
                                      • 1.7.1 Counting Principles
                                        • 1.7.2 Permutations & Combinations
                                        • 1.8 Complex Numbers
                                          • 1.8.1 Intro to Complex Numbers
                                            • 1.8.2 Modulus & Argument
                                              • 1.8.3 Introduction to Argand Diagrams
                                              • 1.9 Further Complex Numbers
                                                • 1.9.1 Geometry of Complex Numbers
                                                  • 1.9.2 Forms of Complex Numbers
                                                    • 1.9.3 Complex Roots of Polynomials
                                                      • 1.9.4 De Moivre's Theorem
                                                        • 1.9.5 Roots of Complex Numbers
                                                        • 1.10 Systems of Linear Equations
                                                          • 1.10.1 Systems of Linear Equations
                                                            • 1.10.2 Algebraic Solutions
                                                          • 2. Functions
                                                            • 2.1 Linear Functions & Graphs
                                                              • 2.1.1 Equations of a Straight Line
                                                              • 2.2 Quadratic Functions & Graphs
                                                                • 2.2.1 Quadratic Functions
                                                                  • 2.2.2 Factorising & Completing the Square
                                                                    • 2.2.3 Solving Quadratics
                                                                      • 2.2.4 Quadratic Inequalities
                                                                        • 2.2.5 Discriminants
                                                                        • 2.3 Functions Toolkit
                                                                          • 2.3.1 Language of Functions
                                                                            • 2.3.2 Composite & Inverse Functions
                                                                              • 2.3.3 Symmetry of Functions
                                                                                • 2.3.4 Graphing Functions
                                                                                • 2.4 Other Functions & Graphs
                                                                                  • 2.4.1 Exponential & Logarithmic Functions
                                                                                    • 2.4.2 Solving Equations
                                                                                      • 2.4.3 Modelling with Functions
                                                                                      • 2.5 Reciprocal & Rational Functions
                                                                                        • 2.5.1 Reciprocal & Rational Functions
                                                                                        • 2.6 Transformations of Graphs
                                                                                          • 2.6.1 Translations of Graphs
                                                                                            • 2.6.2 Reflections of Graphs
                                                                                              • 2.6.3 Stretches Graphs
                                                                                                • 2.6.4 Composite Transformations of Graphs
                                                                                                • 2.7 Polynomial Functions
                                                                                                  • 2.7.1 Factor & Remainder Theorem
                                                                                                    • 2.7.2 Polynomial Division
                                                                                                      • 2.7.3 Polynomial Functions
                                                                                                        • 2.7.4 Roots of Polynomials
                                                                                                        • 2.8 Inequalities
                                                                                                          • 2.8.1 Solving Inequalities Graphically
                                                                                                            • 2.8.2 Polynomial Inequalities
                                                                                                            • 2.9 Further Functions & Graphs
                                                                                                              • 2.9.1 Modulus Functions
                                                                                                                • 2.9.2 Modulus Transformations
                                                                                                                  • 2.9.3 Modulus Equations & Inequalities
                                                                                                                    • 2.9.4 Reciprocal & Square Transformations
                                                                                                                  • 3. Geometry & Trigonometry
                                                                                                                    • 3.1 Geometry Toolkit
                                                                                                                      • 3.1.1 Coordinate Geometry
                                                                                                                        • 3.1.2 Radian Measure
                                                                                                                          • 3.1.3 Arcs & Sectors
                                                                                                                          • 3.2 Geometry of 3D Shapes
                                                                                                                            • 3.2.1 3D Coordinate Geometry
                                                                                                                              • 3.2.2 Volume & Surface Area
                                                                                                                              • 3.3 Trigonometry Toolkit
                                                                                                                                • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                                                  • 3.3.2 Non Right-Angled Trigonometry
                                                                                                                                    • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                                                    • 3.4 Trigonometry
                                                                                                                                      • 3.4.1 The Unit Circle
                                                                                                                                        • 3.4.2 Exact Values
                                                                                                                                        • 3.5 Trigonometric Functions & Graphs
                                                                                                                                          • 3.5.1 Graphs of Trigonometric Functions
                                                                                                                                            • 3.5.2 Transformations of Trigonometric Functions
                                                                                                                                              • 3.5.3 Modelling with Trigonometric Functions
                                                                                                                                              • 3.6 Trigonometric Equations & Identities
                                                                                                                                                • 3.6.1 Simple Identities
                                                                                                                                                  • 3.6.2 Compound Angle Formulae
                                                                                                                                                    • 3.6.3 Double Angle Formulae
                                                                                                                                                      • 3.6.4 Relationship Between Trigonometric Ratios
                                                                                                                                                        • 3.6.5 Linear Trigonometric Equations
                                                                                                                                                          • 3.6.6 Quadratic Trigonometric Equations
                                                                                                                                                          • 3.7 Inverse & Reciprocal Trig Functions
                                                                                                                                                            • 3.7.1 Reciprocal Trig Functions
                                                                                                                                                              • 3.7.2 Inverse Trig Functions
                                                                                                                                                              • 3.8 Further Trigonometry
                                                                                                                                                                • 3.8.1 Trigonometric Proof
                                                                                                                                                                  • 3.8.2 Strategy for Trigonometric Equations
                                                                                                                                                                  • 3.9 Vector Properties
                                                                                                                                                                    • 3.9.1 Introduction to Vectors
                                                                                                                                                                      • 3.9.2 Position & Displacement Vectors
                                                                                                                                                                        • 3.9.3 Magnitude of a Vector
                                                                                                                                                                          • 3.9.4 The Scalar Product
                                                                                                                                                                            • 3.9.5 Geometric Proof with Vectors
                                                                                                                                                                            • 3.10 Vector Equations of Lines
                                                                                                                                                                              • 3.10.1 Vector Equations of Lines
                                                                                                                                                                                • 3.10.2 Applications to Kinematics
                                                                                                                                                                                  • 3.10.3 Pairs of Lines in 3D
                                                                                                                                                                                    • 3.10.4 The Vector Product
                                                                                                                                                                                      • 3.10.5 Shortest Distances with Lines
                                                                                                                                                                                      • 3.11 Vector Planes
                                                                                                                                                                                        • 3.11.1 Vector Equations of Planes
                                                                                                                                                                                          • 3.11.2 Intersections of Lines & Planes
                                                                                                                                                                                            • 3.11.3 Angles Between Lines & Planes
                                                                                                                                                                                              • 3.11.4 Shortest Distances with Planes
                                                                                                                                                                                            • 4. Statistics & Probability
                                                                                                                                                                                              • 4.1 Statistics Toolkit
                                                                                                                                                                                                • 4.1.1 Sampling & Data Collection
                                                                                                                                                                                                  • 4.1.2 Statistical Measures
                                                                                                                                                                                                    • 4.1.3 Frequency Tables
                                                                                                                                                                                                      • 4.1.4 Linear Transformations of Data
                                                                                                                                                                                                        • 4.1.5 Outliers
                                                                                                                                                                                                          • 4.1.6 Univariate Data
                                                                                                                                                                                                            • 4.1.7 Interpreting Data
                                                                                                                                                                                                            • 4.2 Correlation & Regression
                                                                                                                                                                                                              • 4.2.1 Bivariate Data
                                                                                                                                                                                                                • 4.2.2 Correlation & Regression
                                                                                                                                                                                                                • 4.3 Probability
                                                                                                                                                                                                                  • 4.3.1 Probability & Types of Events
                                                                                                                                                                                                                    • 4.3.2 Conditional Probability
                                                                                                                                                                                                                      • 4.3.3 Bayes' Theorem
                                                                                                                                                                                                                        • 4.3.4 Sample Space Diagrams
                                                                                                                                                                                                                        • 4.4 Probability Distributions
                                                                                                                                                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                                                                                                                                                            • 4.4.2 Mean & Variance
                                                                                                                                                                                                                            • 4.5 Binomial Distribution
                                                                                                                                                                                                                              • 4.5.1 The Binomial Distribution
                                                                                                                                                                                                                                • 4.5.2 Calculating Binomial Probabilities
                                                                                                                                                                                                                                • 4.6 Normal Distribution
                                                                                                                                                                                                                                  • 4.6.1 The Normal Distribution
                                                                                                                                                                                                                                    • 4.6.2 Calculations with Normal Distribution
                                                                                                                                                                                                                                      • 4.6.3 Standardisation of Normal Variables
                                                                                                                                                                                                                                      • 4.7 Further Probability Distributions
                                                                                                                                                                                                                                        • 4.7.1 Probability Density Function
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                            • 5.2 Further Differentiation
                                                                                                                                                                                                                                              • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                  • 5.2.3 Higher Order Derivatives
                                                                                                                                                                                                                                                    • 5.2.4 Further Applications of Differentiation
                                                                                                                                                                                                                                                      • 5.2.5 Concavity & Points of Inflection
                                                                                                                                                                                                                                                        • 5.2.6 Derivatives & Graphs
                                                                                                                                                                                                                                                        • 5.3 Integration
                                                                                                                                                                                                                                                          • 5.3.1 Introduction to Integration
                                                                                                                                                                                                                                                            • 5.3.2 Applications of Integration
                                                                                                                                                                                                                                                            • 5.4 Further Integration
                                                                                                                                                                                                                                                              • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                  • 5.4.3 Definite Integrals
                                                                                                                                                                                                                                                                    • 5.4.4 Further Applications of Integration
                                                                                                                                                                                                                                                                    • 5.5 Optimisation
                                                                                                                                                                                                                                                                      • 5.5.1 Modelling with Differentiation
                                                                                                                                                                                                                                                                      • 5.6 Kinematics
                                                                                                                                                                                                                                                                        • 5.6.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                          • 5.6.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                          • 5.7 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.7.1 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.8 Advanced Differentiation
                                                                                                                                                                                                                                                                              • 5.8.1 First Principles Differentiation
                                                                                                                                                                                                                                                                                • 5.8.2 Applications of Chain Rule
                                                                                                                                                                                                                                                                                  • 5.8.3 Implicit Differentiation
                                                                                                                                                                                                                                                                                    • 5.8.4 Differentiating Further Functions
                                                                                                                                                                                                                                                                                    • 5.9 Advanced Integration
                                                                                                                                                                                                                                                                                      • 5.9.1 Integrating Further Functions
                                                                                                                                                                                                                                                                                        • 5.9.2 Further Techniques of Integration
                                                                                                                                                                                                                                                                                          • 5.9.3 Integrating with Partial Fractions
                                                                                                                                                                                                                                                                                            • 5.9.4 Advanced Applications of Integration
                                                                                                                                                                                                                                                                                              • 5.9.5 Modelling with Volumes of Revolution
                                                                                                                                                                                                                                                                                              • 5.10 Differential Equations
                                                                                                                                                                                                                                                                                                • 5.10.1 Numerical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                  • 5.10.2 Analytical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.10.3 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.11 MacLaurin Series
                                                                                                                                                                                                                                                                                                      • 5.11.1 Maclaurin Series
                                                                                                                                                                                                                                                                                                        • 5.11.2 Maclaurin Series from Differential Equations
                                                                                                                                                                                                                                                                                                        • 5.12 Further Limits (inc l'Hôpital's Rule)
                                                                                                                                                                                                                                                                                                          • 5.12.1 Further Limits
                                                                                                                                                                                                                                                                                                        Daniel Finlay

                                                                                                                                                                                                                                                                                                        Author: Daniel

                                                                                                                                                                                                                                                                                                        Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.


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