Graphing Functions (DP IB Analysis & Approaches (AA)) : Revision Note

Did this video help you?

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

Did this video help you?

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

Examiner Tips and Tricks

  • 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

Did this video help you?

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

Examiner Tips and Tricks

  • 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
👀 You've read 1 of your 5 free revision notes this week
An illustration of students holding their exam resultsUnlock more revision notes. It's free!

By signing up you agree to our Terms and Privacy Policy.

Already have an account? Log in

Did this page help you?

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

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.

Download notes on Graphing Functions