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DP IB Maths: AI HL

Revision Notes

Home / IB / Maths: AI HL / DP / Revision Notes / 2. Functions / 2.2 Further Functions & Graphs / 2.2.2 Graphing Functions


2.2.2 Graphing Functions


Graphing Functions

How do I graph the function y = f(x)?

  • A point space left parenthesis a comma space b right parenthesis lies on the graph space y equals f left parenthesis x right parenthesis if space f left parenthesis a right parenthesis equals b
  • The horizontal axis is used for the domain
  • The vertical axis is used for the range
  • You will be able to graph some functions by hand
  • For some functions you will need to use your GDC
  • You might be asked to graph the sum or difference of two functions
    • Use your GDC to graph space y equals f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis or space y equals f left parenthesis x right parenthesis minus g left parenthesis x right parenthesis
    • Just type the functions into the graphing mode

What is the difference between “draw” and “sketch”?

  • If asked to sketch you should:
    • Show the general shape
    • Label any key points such as the intersections with the axes
    • Label the axes
  • If asked to draw you should:
    • Use a pencil and ruler
    • Draw to scale
    • Plot any points accurately
    • Join points with a straight line or smooth curve
    • Label any key points such as the intersections with the axes
    • Label the axes

How can my GDC help me sketch/draw a graph?

  • You use your GDC to plot the graph
    • Check the scales on the graph to make sure you see the full shape
  • Use your GDC to find any key points
  • Use your GDC to check specific points to help you plot the graph

Key Features of Graphs

What are the key features of graphs?

  • You should be familiar with the following key features and know how to use your GDC to find them
  • Local minimums/maximums
    • These are points where the graph has a minimum/maximum for a small region
    • They are also called turning points
      • This is where the graph changes its direction between upwards and downwards directions
    • A graph can have multiple local minimums/maximums
    • A local minimum/maximum is not necessarily the minimum/maximum of the whole graph
      • This would be called the global minimum/maximum
    • For quadratic graphs the minimum/maximum is called the vertex
  • Intercepts
    • y­­ – intercepts are where the graph crosses the y-axis
      • At these points x = 0
    • x – intercepts are where the graph crosses the x-axis
      • At these points y = 0
      • These points are also called the zeros of the function or roots of the equation
  • Symmetry
    • Some graphs have lines of symmetry
      • A quadratic will have a vertical line of symmetry
  • Asymptotes
    • These are lines which the graph will get closer to but not cross
    • These can be horizontal or vertical
      • Exponential graphs have horizontal asymptotes
      • Graphs of variables which vary inversely can have vertical and horizontal asymptotes

Sketching Polynomials Notes Diagram 1

Exam Tip

  • Most GDC makes/models will not plot/show asymptotes just from inputting a function
    • Add the asymptotes as additional graphs for your GDC to plot
    • You can then check the equations of your asymptotes visually
    • You may have to zoom in or change the viewing window options to confirm an asymptote
  • Even if using your GDC to plot graphs and solve problems sketching them as part of your working is good exam technique
    • Label the key features of the graph and anything else relevant to the question on your sketch

Worked Example

Two functions are defined by

space f open parentheses x close parentheses equals x squared minus 4 x minus 5 and space g open parentheses x close parentheses equals 2 plus fraction numerator 1 over denominator x plus 1 end fraction.

a)
Draw the graph space y equals f left parenthesis x right parenthesis.

2-2-2-ib-ai-key-features-of-graphs-a-we-solution

b)
Sketch the graph space y equals g left parenthesis x right parenthesis.

2-2-2-ib-ai-key-features-of-graphs-b-we-solution

Intersecting Graphs

How do I find where two graphs intersect?

  • Plot both graphs on your GDC
  • Use the intersect function to find the intersections
  • Check if there is more than one point of intersection

Solving Equations Graphically Notes Diagram 1

How can I use graphs to solve equations?

  • One method to solve equations is to use graphs
  • To solve space f left parenthesis x right parenthesis equals a
    • Plot the two graphs space y equals f left parenthesis x right parenthesis and space y equals a on your GDC
    • Find the points of intersections
    • The x-coordinates are the solutions of the equation
  • To solve space f left parenthesis x right parenthesis equals g left parenthesis x right parenthesis
    • Plot the two graphs space y equals f left parenthesis x right parenthesis and space space y equals g left parenthesis x right parenthesis on your GDC
    • Find the points of intersections
    • The x-coordinates are the solutions of the equation
  • Using graphs makes it easier to see how many solutions an equation will have

Exam Tip

  • You can use graphs to solve equations
    • Questions will not necessarily ask for a drawing/sketch or make reference to graphs
    • Use your GDC to plot the equations and find the intersections between the graphs

Worked Example

Two functions are defined by

space f left parenthesis x right parenthesis equals x cubed minus x and space g left parenthesis x right parenthesis equals 4 over x.

a)
Sketch the graph space y equals f left parenthesis x right parenthesis.

2-2-2-ib-ai-sl-intersecting-graphs-a-we-solution

b)
Write down the number of real solutions to the equation space x cubed minus x equals 2.

2-2-2-ib-ai-sl-intersecting-graphs-b-we-solution

c)
Find the coordinates of the points where space y equals f left parenthesis x right parenthesis and space y equals g left parenthesis x right parenthesis intersect.

2-2-2-ib-ai-sl-intersecting-graphs-c-we-solution

d)
Write down the solutions to the equation space x cubed minus x equals 4 over x.

2-2-2-ib-ai-sl-intersecting-graphs-d-we-solution



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Approximation & Estimation
          • 1.1.3 GDC: Solving Equations
          • 1.2 Exponentials & Logs
            • 1.2.1 Exponents
              • 1.2.2 Logarithms
              • 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
                      • 1.4 Financial Applications
                        • 1.4.1 Compound Interest & Depreciation
                          • 1.4.2 Amortisation & Annuities
                          • 1.5 Complex Numbers
                            • 1.5.1 Intro to Complex Numbers
                              • 1.5.2 Modulus & Argument
                                • 1.5.3 Introduction to Argand Diagrams
                                • 1.6 Further Complex Numbers
                                  • 1.6.1 Geometry of Complex Numbers
                                    • 1.6.2 Forms of Complex Numbers
                                      • 1.6.3 Applications of Complex Numbers
                                      • 1.7 Matrices
                                        • 1.7.1 Introduction to Matrices
                                          • 1.7.2 Operations with Matrices
                                            • 1.7.3 Determinants & Inverses
                                              • 1.7.4 Solving Systems of Linear Equations with Matrices
                                              • 1.8 Eigenvalues & Eigenvectors
                                                • 1.8.1 Eigenvalues & Eigenvectors
                                                  • 1.8.2 Applications of Matrices
                                                • 2. Functions
                                                  • 2.1 Linear Functions & Graphs
                                                    • 2.1.1 Equations of a Straight Line
                                                    • 2.2 Further Functions & Graphs
                                                      • 2.2.1 Functions
                                                        • 2.2.2 Graphing Functions
                                                          • 2.2.3 Properties of Graphs
                                                          • 2.3 Modelling with Functions
                                                            • 2.3.1 Linear Models
                                                              • 2.3.2 Quadratic & Cubic Models
                                                                • 2.3.3 Exponential Models
                                                                  • 2.3.4 Direct & Inverse Variation
                                                                    • 2.3.5 Sinusoidal Models
                                                                      • 2.3.6 Strategy for Modelling Functions
                                                                      • 2.4 Functions Toolkit
                                                                        • 2.4.1 Composite & Inverse Functions
                                                                        • 2.5 Transformations of Graphs
                                                                          • 2.5.1 Translations of Graphs
                                                                            • 2.5.2 Reflections of Graphs
                                                                              • 2.5.3 Stretches of Graphs
                                                                                • 2.5.4 Composite Transformations of Graphs
                                                                                • 2.6 Further Modelling with Functions
                                                                                  • 2.6.1 Properties of Further Graphs
                                                                                    • 2.6.2 Natural Logarithmic Models
                                                                                      • 2.6.3 Logistic Models
                                                                                        • 2.6.4 Piecewise Models
                                                                                      • 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
                                                                                                    • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                      • 3.3.2 Non Right-Angled Trigonometry
                                                                                                        • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                        • 3.4 Further Trigonometry
                                                                                                          • 3.4.1 The Unit Circle
                                                                                                            • 3.4.2 Simple Identities
                                                                                                              • 3.4.3 Solving Trigonometric Equations
                                                                                                              • 3.5 Voronoi Diagrams
                                                                                                                • 3.5.1 Voronoi Diagrams
                                                                                                                  • 3.5.2 Toxic Waste Dump Problem
                                                                                                                  • 3.6 Matrix Transformations
                                                                                                                    • 3.6.1 Matrix Transformations
                                                                                                                      • 3.6.2 Determinant of a Transformation Matrix
                                                                                                                      • 3.7 Vector Properties
                                                                                                                        • 3.7.1 Introduction to Vectors
                                                                                                                          • 3.7.2 Position & Displacement Vectors
                                                                                                                            • 3.7.3 Magnitude of a Vector
                                                                                                                              • 3.7.4 The Scalar Product
                                                                                                                                • 3.7.5 The Vector Product
                                                                                                                                  • 3.7.6 Components of Vectors
                                                                                                                                    • 3.7.7 Geometric Proof with Vectors
                                                                                                                                    • 3.8 Vector Equations of Lines
                                                                                                                                      • 3.8.1 Vector Equations of Lines
                                                                                                                                        • 3.8.2 Shortest Distances with Lines
                                                                                                                                        • 3.9 Modelling with Vectors
                                                                                                                                          • 3.9.1 Kinematics with Vectors
                                                                                                                                            • 3.9.2 Constant & Variable Velocity
                                                                                                                                            • 3.10 Graph Theory
                                                                                                                                              • 3.10.1 Introduction to Graph Theory
                                                                                                                                                • 3.10.2 Walks & Adjacency Matrices
                                                                                                                                                  • 3.10.3 Minimum Spanning Trees
                                                                                                                                                    • 3.10.4 Chinese Postman Problem
                                                                                                                                                      • 3.10.5 Travelling Salesman Problem
                                                                                                                                                        • 3.10.6 Bounds for Travelling Salesman Problem
                                                                                                                                                      • 4. Statistics & Probability
                                                                                                                                                        • 4.1 Statistics Toolkit
                                                                                                                                                          • 4.1.1 Sampling
                                                                                                                                                            • 4.1.2 Data Collection
                                                                                                                                                              • 4.1.3 Statistical Measures
                                                                                                                                                                • 4.1.4 Frequency Tables
                                                                                                                                                                  • 4.1.5 Linear Transformations of Data
                                                                                                                                                                    • 4.1.6 Outliers
                                                                                                                                                                      • 4.1.7 Univariate Data
                                                                                                                                                                        • 4.1.8 Interpreting Data
                                                                                                                                                                        • 4.2 Correlation & Regression
                                                                                                                                                                          • 4.2.1 Bivariate Data
                                                                                                                                                                            • 4.2.2 Correlation Coefficients
                                                                                                                                                                              • 4.2.3 Linear Regression
                                                                                                                                                                              • 4.3 Further Correlation & Regression
                                                                                                                                                                                • 4.3.1 Non-linear Regression
                                                                                                                                                                                  • 4.3.2 Logarithmic Scales
                                                                                                                                                                                    • 4.3.3 Linearising using Logarithms
                                                                                                                                                                                    • 4.4 Probability
                                                                                                                                                                                      • 4.4.1 Probability & Types of Events
                                                                                                                                                                                        • 4.4.2 Conditional Probability
                                                                                                                                                                                          • 4.4.3 Sample Space Diagrams
                                                                                                                                                                                          • 4.5 Probability Distributions
                                                                                                                                                                                            • 4.5.1 Discrete Probability Distributions
                                                                                                                                                                                              • 4.5.2 Expected Values
                                                                                                                                                                                              • 4.6 Random Variables
                                                                                                                                                                                                • 4.6.1 Linear Combinations of Random Variables
                                                                                                                                                                                                  • 4.6.2 Unbiased Estimates
                                                                                                                                                                                                  • 4.7 Binomial Distribution
                                                                                                                                                                                                    • 4.7.1 The Binomial Distribution
                                                                                                                                                                                                      • 4.7.2 Calculating Binomial Probabilities
                                                                                                                                                                                                      • 4.8 Normal Distribution
                                                                                                                                                                                                        • 4.8.1 The Normal Distribution
                                                                                                                                                                                                          • 4.8.2 Calculations with Normal Distribution
                                                                                                                                                                                                          • 4.9 Further Normal Distribution (inc Central Limit Theorem)
                                                                                                                                                                                                            • 4.9.1 Sample Mean Distribution
                                                                                                                                                                                                              • 4.9.2 Confidence Interval for the Mean
                                                                                                                                                                                                              • 4.10 Poisson Distribution
                                                                                                                                                                                                                • 4.10.1 Poisson Distribution
                                                                                                                                                                                                                  • 4.10.2 Calculating Poisson Probabilities
                                                                                                                                                                                                                  • 4.11 Hypothesis Testing
                                                                                                                                                                                                                    • 4.11.1 Hypothesis Testing
                                                                                                                                                                                                                      • 4.11.2 Chi-squared Test for Independence
                                                                                                                                                                                                                        • 4.11.3 Goodness of Fit Test
                                                                                                                                                                                                                        • 4.12 Further Hypothesis Testing
                                                                                                                                                                                                                          • 4.12.1 Hypothesis Testing for Mean (One Sample)
                                                                                                                                                                                                                            • 4.12.2 Hypothesis Testing for Mean (Two Sample)
                                                                                                                                                                                                                              • 4.12.3 Binomial Hypothesis Testing
                                                                                                                                                                                                                                • 4.12.4 Poisson Hypothesis Testing
                                                                                                                                                                                                                                  • 4.12.5 Hypothesis Testing for Correlation
                                                                                                                                                                                                                                    • 4.12.6 Type I & Type II Errors
                                                                                                                                                                                                                                    • 4.13 Transition Matrices & Markov Chains
                                                                                                                                                                                                                                      • 4.13.1 Markov Chains
                                                                                                                                                                                                                                        • 4.13.2 Transition Matrices
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                              • 5.1.3 Modelling with Differentiation
                                                                                                                                                                                                                                              • 5.2 Further Differentiation
                                                                                                                                                                                                                                                • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                  • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                    • 5.2.3 Related Rates of Change
                                                                                                                                                                                                                                                      • 5.2.4 Second Order Derivatives
                                                                                                                                                                                                                                                        • 5.2.5 Further Applications of Differentiation
                                                                                                                                                                                                                                                          • 5.2.6 Concavity & Points of Inflection
                                                                                                                                                                                                                                                          • 5.3 Integration
                                                                                                                                                                                                                                                            • 5.3.1 Trapezoid Rule: Numerical Integration
                                                                                                                                                                                                                                                              • 5.3.2 Introduction to Integration
                                                                                                                                                                                                                                                                • 5.3.3 Applications of Integration
                                                                                                                                                                                                                                                                • 5.4 Further Integration
                                                                                                                                                                                                                                                                  • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                    • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                      • 5.4.3 Further Applications of Integration
                                                                                                                                                                                                                                                                        • 5.4.4 Volumes of Revolution
                                                                                                                                                                                                                                                                        • 5.5 Kinematics
                                                                                                                                                                                                                                                                          • 5.5.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                            • 5.5.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                            • 5.6 Differential Equations
                                                                                                                                                                                                                                                                              • 5.6.1 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                • 5.6.2 Separation of Variables
                                                                                                                                                                                                                                                                                  • 5.6.3 Slope Fields
                                                                                                                                                                                                                                                                                    • 5.6.4 Approximate Solutions to Differential Equations
                                                                                                                                                                                                                                                                                    • 5.7 Further Differential Equations
                                                                                                                                                                                                                                                                                      • 5.7.1 Coupled Differential Equations
                                                                                                                                                                                                                                                                                        • 5.7.2 Second Order Differential Equations
                                                                                                                                                                                                                                                                                      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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