Graphing Functions (Edexcel IGCSE Further Pure Maths)

Revision Note

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

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 of space y equals straight f left parenthesis x right parenthesis if space straight f left parenthesis a right parenthesis equals b

  • The horizontal axis is used for the domain

    • these are the x-values

  • The vertical axis is used for the range

    • these are the corresponding y-values

  • Graphing a function can involve any or all of the following skills

    • creating a table of values

    • knowing the general shapes and characteristics of different types of graphs

    • finding axis intercepts

    • finding turning points (including maxima and minima)

    • identifying asymptotes

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

  • 'Sketch' questions will not provide a detailed grid to draw the graph on

    • A sketch is meant to show key features of a graph

      • It does not need to be perfectly to scale

      • It does not need to plot points precisely

    • When asked to sketch you should:

      • Show the correct general shape

      • Label any key points such as the intersections with the axes

      • Label the axes

      • Label the graph by giving its function or equation

  • 'Draw' questions will always provide a detailed grid to draw the graph on

    • A drawing of a graph is meant to be as accurate as possible

    • When asked to draw you should:

      • Use a ruler for any straight lines

      • Draw to scale

      • Plot any points precisely

      • Join points with a straight line or smooth curve

      • Label any key points such as the intersections with the axes

      • Label the axes

      • Label the graph by giving its function or equation

Can my calculator help me sketch/draw a graph?

  • Your calculator may be able to display graphs of functions

    • If so make sure you are familiar with how this feature works

  • This can be useful for checking your work

  • It can also help with identifying

    • the general shape of a graph

    • axis intercepts

    • turning points (including maxima and minima)

  • But be careful, especially on 'show that' questions

    • Make sure key parts of an answer are backed up by working

    • Marks may depend on this

Key Features of Graphs

What are the key features of graphs?

You should be familiar with the following key features and know how to identify them

  • Local minima and maxima

    • These are points where the graph has a minimum/maximum for a local region

      • They are the 'peaks' or 'valleys' of a graph

      • They are not necessarily the minimum or maximum for the whole graph

    • They are also called turning points

    • A graph can have multiple local minima or maxima

    • They occur where the gradient of the graph is equal to zero

      • So if you are graphing y equals straight f open parentheses x close parentheses

      • Turning points will occur where  straight f to the power of apostrophe open parentheses x close parentheses equals 0

  • Intercepts

    • y­­-intercepts are where the graph crosses the y-axis

      • At these points  x equals 0

    • x-intercepts are where the graph crosses the x-axis

      • At these points y = 0

      • These points are also called the roots of the function

  • Symmetry

    • Some graphs have lines of symmetry

      • E.g. a quadratic graph has a vertical line of symmetry through its minimum or maximum point

  • 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

      • Logarithmic graphs and graphs of rational functions have vertical asymptotes

Sketching Polynomials Notes Diagram 1

Examiner Tips and Tricks

  • Don't forget to label the graph, the axes and key features when drawing or sketching a graph

    • Marks may depend on this

  • Your calculator may be able to help you with drawing or sketching a graph

    • But don't forget to show working, especially in a 'show that' question

  • Sketching a graph can often be useful even if a question doesn't require it

Worked Example

A function is defined by  space straight f open parentheses x close parentheses equals x squared minus 4 x minus 5.

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

Remember, 'sketch' here means we only need to show the general shape and key features

This is a positive quadratic, so it is going to be a 'u-shaped' parabola

Find the y-intercept by substituting in x equals 0

straight f open parentheses 0 close parentheses equals open parentheses 0 close parentheses squared minus 4 open parentheses 0 close parentheses minus 5 equals negative 5

So the y-intercept is open parentheses 0 comma space minus 5 close parentheses

The x-intercepts will occur when  y equals straight f open parentheses x close parentheses equals 0

table row cell x squared minus 4 x minus 5 end cell equals 0 row cell open parentheses x minus 5 close parentheses open parentheses x plus 1 close parentheses end cell equals 0 end table

x equals 5 comma space space x equals negative 1

So the x-intercepts are open parentheses 5 comma space 0 close parentheses and open parentheses negative 1 comma space 0 close parentheses

The turning point will occur when  straight f to the power of apostrophe open parentheses x close parentheses equals 0

straight f to the power of apostrophe open parentheses x close parentheses equals 2 x minus 4

table row cell 2 x minus 4 end cell equals 0 row cell 2 x end cell equals 4 row x equals 2 end table

Substitute into straight f open parentheses x close parentheses to find the corresponding y-value

table row cell straight f open parentheses 2 close parentheses end cell equals cell open parentheses 2 close parentheses squared minus 4 open parentheses 2 close parentheses minus 5 end cell row blank equals cell 4 minus 8 minus 5 end cell row blank equals cell negative 9 end cell end table

So the turning point is open parentheses 2 comma space minus 9 close parentheses

Sketch a graph showing the correct general shape and incorporating all the key points identified above
Label the graph, the axes, and the key features

Graph of quadratic function

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Roger B

Author: Roger B

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at 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.