• IB
  • IB Docs (2) Team
    Logout
  • Maths
  • Biology
  • Chemistry
  • Physics
  • Combined Science
  • English Language
  • Geography
  • Other Subjects
GCSE Maths
Edexcel Topic QuestionsRevision NotesPast PapersPast Papers Questions
AQA Topic QuestionsRevision NotesPast Papers
OCR Topic QuestionsRevision NotesPast Papers
GCSE Biology
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsRevision NotesPast Papers
OCR Gateway Topic QuestionsRevision NotesPast Papers
GCSE Chemistry
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsRevision NotesPast Papers
OCR Gateway Topic QuestionsRevision NotesPast Papers
GCSE Physics
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsRevision NotesPast Papers
OCR Gateway Topic QuestionsRevision NotesPast Papers
GCSE Combined Science
Edexcel Combined: Biology Topic QuestionsRevision NotesPast Papers
Edexcel Combined: Chemistry Topic QuestionsRevision NotesPast Papers
Edexcel Combined: Physics Revision NotesPast Papers
AQA Combined: Biology Topic QuestionsRevision NotesPast Papers
AQA Combined: Chemistry Topic QuestionsRevision NotesPast Papers
AQA Combined: Physics Topic QuestionsRevision NotesPast Papers
OCR Gateway Combined: Biology Topic QuestionsRevision Notes
OCR Gateway Combined: Chemistry Revision Notes
OCR Gateway Combined: Physics Revision Notes
GCSE English Language
AQA Revision NotesPractice PapersPast Papers
Edexcel Past Papers
OCR Past Papers
GCSE Geography
AQA Topic QuestionsRevision Notes
Edexcel Topic Questions
GCSE Other Subjects
AQA English LiteratureBusinessComputer ScienceEconomicsFurther MathsGeographyHistoryPsychologySociologyStatistics
Edexcel English LiteratureBusinessComputer ScienceGeographyHistoryPsychologyStatistics
OCR English LiteratureBusinessComputer ScienceEconomicsPsychology
OCR Gateway GeographyHistory
  • Maths
  • Biology
  • Chemistry
  • Physics
  • Double Science
  • Economics
  • English Language
  • Geography
  • Other Subjects
IGCSE Maths
Edexcel Topic QuestionsRevision NotesPast PapersBronze-Silver-Gold Questions
CIE (Extended) Topic QuestionsRevision NotesPast Papers
CIE (Core) Topic QuestionsPast Papers
IGCSE Biology
Edexcel Topic QuestionsRevision NotesPast Papers
CIE 2020-2022 Topic QuestionsRevision NotesPast Papers
CIE 2023-2025 Topic QuestionsRevision NotesPast Papers
IGCSE Chemistry
Edexcel Topic QuestionsRevision NotesPast Papers
CIE 2020-2022 Topic QuestionsRevision NotesPast Papers
CIE 2023-2025 Topic QuestionsRevision NotesPast Papers
IGCSE Physics
Edexcel Topic QuestionsRevision NotesPast Papers
CIE 2020-2022 Topic QuestionsRevision NotesPast Papers
CIE 2023-2025 Topic QuestionsRevision NotesPast Papers
IGCSE Double Science
Edexcel Double: Biology Topic QuestionsRevision NotesPast Papers
Edexcel Double: Chemistry Topic QuestionsRevision NotesPast Papers
Edexcel Double: Physics Topic QuestionsRevision NotesPast Papers
IGCSE Economics
CIE Topic QuestionsRevision NotesPast Papers
IGCSE English Language
CIE Revision NotesPractice PapersPast Papers
Edexcel Past Papers
IGCSE Geography
CIE Revision NotesTopic QuestionsPast Papers
Edexcel Topic QuestionsRevision NotesPast Papers
IGCSE Other Subjects
CIE Additional MathsEnglish LiteratureBusinessComputer ScienceHistorySociology
Edexcel English LiteratureBusinessComputer ScienceHistoryFurther Maths
  • Maths
  • Biology
  • Chemistry
  • Physics
  • English Language
  • Other Subjects
AS Maths
Edexcel Pure MathsMechanicsStatistics
AQA Pure MathsMechanicsStatistics
OCR Pure MathsMechanicsStatistics
CIE Pure 1Pure 2MechanicsProbability & Statistics 1
Edexcel IAS Pure 1Pure 2MechanicsStatistics
AS Biology
AQA Topic QuestionsRevision NotesPast Papers
OCR Topic QuestionsRevision NotesPast Papers
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Revision Notes
AS Chemistry
Edexcel Revision Notes
AQA Topic QuestionsRevision NotesPast Papers
OCR Revision Notes
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Revision Notes
AS Physics
Edexcel Revision Notes
AQA Topic QuestionsRevision NotesPast Papers
OCR Revision NotesPast Papers
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Revision Notes
AS English Language
AQA Past Papers
Edexcel Past Papers
OCR Past Papers
AS Other Subjects
AQA BusinessComputer ScienceEconomicsEnglish LiteratureFurther MathsGeographyHistoryPsychologySociology
Edexcel BusinessEconomicsEnglish LiteratureFurther MathsGeographyHistoryPsychology
OCR BusinessComputer ScienceEconomicsEnglish LiteratureFurther Maths AGeographyHistoryPsychologySociology
CIE Further Maths
  • Maths
  • Biology
  • Chemistry
  • Physics
  • English Language
  • Economics
  • Further Maths
  • Psychology
  • Other Subjects
A Level Maths
Edexcel Pure MathsMechanicsStatistics
AQA Pure MathsMechanicsStatistics
OCR Pure MathsMechanicsStatistics
CIE Pure 1Pure 3MechanicsProbability & Statistics 1Probability & Statistics 2
Edexcel IAL Pure 1Pure 2Pure 3Pure 4Mechanics 1Mechanics 2Statistics 1Statistics 2Decision 1
A Level Biology
Edexcel Topic QuestionsPast Papers
Edexcel A (SNAB) Revision Notes
AQA Topic QuestionsRevision NotesPast Papers
OCR Topic QuestionsRevision NotesPast PapersGold Questions
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Topic QuestionsRevision NotesPast Papers
A Level Chemistry
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsRevision NotesPast Papers
OCR Topic QuestionsRevision NotesPast PapersGold Questions
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Topic QuestionsRevision NotesPast Papers
A Level Physics
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsRevision NotesPast Papers
OCR Topic QuestionsRevision NotesPast Papers
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Topic QuestionsRevision NotesPast Papers
A Level English Language
AQA Past Papers
CIE Past Papers
Edexcel Past Papers
OCR Past Papers
Edexcel IAL Past Papers
A Level Economics
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsPast Papers
OCR Past Papers
CIE Past Papers
A Level Further Maths
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Past Papers
OCR Past Papers
CIE Past Papers
Edexcel IAL Past Papers
A Level Psychology
AQA Topic QuestionsRevision NotesPast Papers
CIE Past Papers
Edexcel Past Papers
OCR Past Papers
Edexcel IAL Past Papers
A Level Other Subjects
AQA BusinessComputer ScienceEconomicsEnglish LiteratureGeographyHistorySociology
CIE BusinessComputer ScienceEconomicsEnglish LiteratureGeographySociology
Edexcel BusinessEconomics AEnglish LiteratureGeographyHistory
OCR BusinessComputer ScienceEconomicsEnglish LiteratureGeographyHistorySociology
Edexcel IAL English LiteratureGeography
CIE IAL History
  • Biology
  • Chemistry
  • Physics
  • Other Subjects
O Level Biology
CIE Topic QuestionsPast Papers
O Level Chemistry
CIE Topic QuestionsPast Papers
O Level Physics
CIE Topic QuestionsPast Papers
O Level Other Subjects
CIE Additional MathsMaths D
  • Maths
  • Biology
  • Chemistry
  • Physics
Pre U Maths
CIE Topic QuestionsPast Papers
Pre U Biology
CIE Topic QuestionsPast Papers
Pre U Chemistry
CIE Topic QuestionsPast Papers
Pre U Physics
CIE Topic QuestionsPast Papers
  • Maths
  • Biology
  • Chemistry
  • Physics
  • Economics
IB Maths
Maths: AA HL Topic QuestionsRevision NotesPractice Papers
Maths: AI HL Topic QuestionsRevision NotesPractice Papers
Maths: AA SL Topic QuestionsRevision NotesPractice Papers
Maths: AI SL Topic QuestionsRevision NotesPractice Papers
IB Biology
Biology: SL Topic QuestionsRevision NotesPractice Papers
Biology: HL Topic QuestionsRevision NotesPractice Papers
IB Chemistry
Chemistry: SL Topic QuestionsRevision NotesPractice Papers
Chemistry: HL Topic QuestionsRevision NotesPractice Papers
IB Physics
Physics: SL Topic QuestionsRevision NotesPractice Papers
Physics: HL Topic QuestionsRevision NotesPractice Papers
IB Economics
Economics: SL Revision Notes

DP IB Maths: AA HL

Revision Notes

Home / IB / Maths: AA HL / DP / Revision Notes / 2. Functions / 2.7 Polynomial Functions / 2.7.3 Polynomial Functions


2.7.3 Polynomial Functions


Sketching Polynomial Graphs

In exams you’ll commonly be asked to sketch the graphs of different polynomial functions with and without the use of your GDC.

What’s the relationship between a polynomial’s degree and its zeros?

  • If a real polynomial P(x) has degree n, it will have n zeros which can be written in the form a + bi, where a, b ∈ ℝ
    • For example:
      • A quadratic will have 2 zeros
      • A cubic function will have 3 zeros
      • A quartic will have 4 zeros
    • Some of the zeros may be repeated
  • Every real polynomial of odd degree has at least one real zero

How do I sketch the graph of a polynomial function without a GDC?

  • Suppose P open parentheses x close parentheses equals a subscript n x to the power of n plus a subscript n minus 1 end subscript x to the power of n minus 1 end exponent plus blank horizontal ellipsis plus a subscript 1 x plus a subscript 0 is a real polynomial with degree n
  • To sketch the graph of a polynomial you need to know three things:
    • The y-intercept
      • Find this by substituting x = 0 to get y = a0
    • The roots
      • You can find these by factorising or solving y = 0
    • The shape
      • This is determined by the degree (n) and the sign of the leading coefficient (an)

How does the multiplicity of a real root affect the graph of the polynomial?

  • The multiplicity of a root is the number of times it is repeated when the polynomial is factorised
    • If x equals k is a root with multiplicity m then space left parenthesis x minus k right parenthesis to the power of m is a factor of the polynomial
  • The graph either crosses the x-axis or touches the x-axis at a root x = k where k is a real number
    • If x = k has multiplicity 1 then the graph crosses the x-axis at (k, 0)
    • If x = k has multiplicity 2 then the graph has a turning point at (k, 0) so touches the x-axis 
      • If x = k has odd multiplicity m ≥ 3 then the graph has a stationary point of inflection at (k, 0) so crosses the x-axis
      • If x = k has even multiplicity m ≥ 4 then the graph has a turning point at (k, 0) so touches the x-axis

2-7-3-ib-aa-hl-polynomial-roots

How do I determine the shape of the graph of the polynomial?

  • Consider what happens as x tends to ± ∞
    • If an is positive and n is even then the graph approaches from the top left and tends to the top right
      • limit as x rightwards arrow negative infinity of f left parenthesis x right parenthesis equals limit as x rightwards arrow plus infinity of f left parenthesis x right parenthesis equals plus infinity
    • If an is negative and n is even then the graph approaches from the bottom left and tends to the bottom right
      • limit as x rightwards arrow negative infinity of f left parenthesis x right parenthesis equals limit as x rightwards arrow plus infinity of f left parenthesis x right parenthesis equals plus infinity
    • If an is positive and n is odd then the graph approaches from the bottom left and tends to the top right
      • limit as x rightwards arrow negative infinity of f left parenthesis x right parenthesis equals negative infinity and limit as x rightwards arrow plus infinity of f left parenthesis x right parenthesis equals plus infinity
    • If an is negative and n is odd then the graph approaches from the top left and tends to the bottom right
      • limit as x rightwards arrow negative infinity of f left parenthesis x right parenthesis equals plus infinity and limit as x rightwards arrow plus infinity of f left parenthesis x right parenthesis equals negative infinity
  • Once you know the shape, the real roots and the y-intercept then you simply connect the points using a smooth curve
  • There will be at least one turning point in-between each pair of roots
    • If the degree is n then there is at most n – 1 stationary points (some will be turning points)
      • Every real polynomial of even degree has at least one turning point
      • Every real polynomial of odd degree bigger than 1 has at least one point of inflection
    • If it is a calculator paper then you can use your GDC to find the coordinates of the turning points
    • You won’t need to find their location without a GDC unless the question asks you to

2-7-3-ib-aa-hl-polynomial-shape

Exam Tip

  • If it is a calculator paper then you can use your GDC to find the coordinates of any turning points
  • If it is the non-calculator paper then you will not be required to find the turning points when sketching unless specifically asked to

Worked Example

a)
The function space f is defined by space f left parenthesis x right parenthesis equals left parenthesis x plus 1 right parenthesis left parenthesis 2 x minus 1 right parenthesis left parenthesis x minus 2 right parenthesis squared. Sketch the graph of space y equals f left parenthesis x right parenthesis.

2-7-3-ib-aa-hl-sketching-polynomial-a-we-solution

b)
The graph below shows a polynomial function. Find a possible equation of the polynomial.
2-7-3-we-image

2-7-3-ib-aa-hl-sketching-polynomial-b-we-solution

Solving Polynomial Equations

What is “The Fundamental Theorem of Algebra”?

  • Every real polynomial with degree n can be factorised into n complex linear factors
    • Some of which may be repeated
    • This means the polynomial will have n zeros (some may be repeats)
  • Every real polynomial can be expressed as a product of real linear factors and real irreducible quadratic factors
    • An irreducible quadratic is where it does not have real roots
      • The discriminant will be negative: b2 – 4ac < 0
  • If a + bi (b ≠ 0) is a zero of a real polynomial then its complex conjugate a – bi is also a zero
  • Every real polynomial of odd degree will have at least one real zero

How do I solve polynomial equations?

  • Suppose you have an equation P(x) = 0 where P(x) is a real polynomial of degree n
    • P open parentheses x close parentheses equals a subscript n x to the power of n plus a subscript n minus 1 end subscript x to the power of n minus 1 end exponent plus blank horizontal ellipsis plus a subscript 1 x plus a subscript 0
  • You may be given one zero or you might have to find a zero x = k by substituting values into P(x) until it equals 0
  • If you know a root then you know a factor
    • If you know x = k is a root then (x – k) is a factor
    • If you know x = a + bi is a root then you know a quadratic factor (x – (a + bi))( x – (a – bi))
      • Which can be written as ((x – a) - bi)((x – a) + bi) and expanded quickly using difference of two squares
  • You can then divide P(x) by this factor to get another factor
    • For example: dividing a cubic by a linear factor will give you a quadratic factor
  • You then may be able to factorise this new factor

Exam Tip

  • If a polynomial has three or less terms check whether a substitution can turn it into a quadratic
    • For example: x to the power of 6 plus 3 x cubed plus 2 can be written as open parentheses x cubed close parentheses squared plus 3 open parentheses x cubed close parentheses plus 2

Worked Example

Given that space x equals 1 half is a zero of the polynomial defined by space f left parenthesis x right parenthesis equals 2 x cubed minus 3 x squared plus 5 x minus 2, find all three zeros of space f.

2-7-3-ib-aa-hl-solving-polynomials-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
                                                                                                                                                                                                                                                                                                        Lucy Kirkham

                                                                                                                                                                                                                                                                                                        Author: Lucy

                                                                                                                                                                                                                                                                                                        Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels. Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all.


                                                                                                                                                                                                                                                                                                        Save My Exams Logo
                                                                                                                                                                                                                                                                                                        Resources
                                                                                                                                                                                                                                                                                                        Home Join Support

                                                                                                                                                                                                                                                                                                        Members
                                                                                                                                                                                                                                                                                                        Members Home Account Logout

                                                                                                                                                                                                                                                                                                        Company
                                                                                                                                                                                                                                                                                                        About Us Contact Us Jobs Terms Privacy Facebook Twitter

                                                                                                                                                                                                                                                                                                        Quick Links
                                                                                                                                                                                                                                                                                                        GCSE Revision Notes IGCSE Revision Notes A Level Revision Notes Biology Chemistry Physics Maths 2022 Advance Information

                                                                                                                                                                                                                                                                                                         
                                                                                                                                                                                                                                                                                                        © IB Documents (2) Team & u/aimlesskr
                                                                                                                                                                                                                                                                                                        IBO was not involved in the production of, and does not endorse, the resources created by Save My Exams.