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Graphs of Rational Functions (Edexcel IGCSE Further Pure Maths)

Revision Note

Dan Finlay

Written by: Dan Finlay

Reviewed by: Lucy Kirkham

Linear Rational Functions & Graphs

What is a linear rational function?

  • A (linear) rational function is of the form  straight f left parenthesis x right parenthesis equals fraction numerator a x plus b over denominator c x plus d end fraction comma space x not equal to negative d over c

  • Its domain is the set of all real values except  negative d over c

    • Because  x equals negative d over c  would make the denominator zero

  • Its range is the set of all real values except a over c

    • Because there's no value of x for which  fraction numerator a x plus b over denominator c x plus d end fraction equals a over c

  • The reciprocal function  straight f open parentheses x close parentheses equals 1 over x  is a special case of a rational function

What are the key features of linear rational graphs?

  • The graph has a y-intercept at stretchy left parenthesis 0 comma space b over d stretchy right parenthesis provided d not equal to 0

    • The is found by substituting x equals 0

  • The graph has one x-intercept at stretchy left parenthesis negative b over a comma space 0 stretchy right parenthesis provided a not equal to 0

    • This is found by setting the numerator equal to zero and solving the equation

  • The graph has two asymptotes

    • A horizontal asymptote: space y equals a over c

      • This is the limiting value when the value of x gets very large in the positive or negative direction

    • A vertical asymptote: space x equals negative d over c

      • This is the value that causes the denominator to be zero

  • The graph does not have any minimum or maximum points

  • If you are asked to sketch or draw a rational graph:

    • Give the coordinates of any intercepts with the axes

    • Give the equations of the asymptotes

Examiner Tips and Tricks

  • A horizontal line should only intersect this type of graph once at most

  • The only horizontal line that should not intersect the graph is the horizontal asymptote

    • This can be used to check your sketch in an exam

Worked Example

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

(a) Write down the equations of the horizontal and vertical asymptotes of the graph of y equals straight f open parentheses x close parentheses.

The horizontal asymptote is the limiting value as x becomes very large in the positive or negative direction

When that happens, the '10' in the numerator and '2' in the denominator no longer affect the value of the function much

For large values of x

straight f open parentheses x close parentheses almost equal to fraction numerator negative 5 x over denominator x end fraction equals negative 5

That means the horizontal asymptote will be at  y equals negative 5

Horizontal asymptote:  bold italic y bold equals bold minus bold 5

The vertical asymptote occurs where the denominator becomes zero

table row cell x plus 2 end cell equals 0 row x equals cell negative 2 end cell end table

Vertical asymptote:  bold italic x bold equals bold minus bold 2

(b) Find the coordinates of the intercepts of the graph of space y equals straight f open parentheses x close parentheses with the coordinate axes.

The y-intercept occurs when x equals 0

straight f open parentheses 0 close parentheses equals fraction numerator 10 minus 5 open parentheses 0 close parentheses over denominator open parentheses 0 close parentheses plus 2 end fraction equals 10 over 2 equals 5

y-intercept:  stretchy left parenthesis 0 comma space 5 stretchy right parenthesis

The x-intercept occurs when y equals straight f open parentheses x close parentheses equals 0

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 10 minus 5 x over denominator x plus 2 end fraction end cell equals 0 row cell 10 minus 5 x end cell equals 0 row cell 5 x end cell equals 10 row x equals 2 end table

x-intercept:  stretchy left parenthesis 2 comma space 0 stretchy right parenthesis

(c) Sketch the graph of  y equals straight f open parentheses x close parentheses.

Don't forget to include (and label) the asymptotes, and to label the axis intercepts

graph of rational function

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    Author
    Reviewer
    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.

    Lucy Kirkham

    Author: Lucy Kirkham

    Expertise: Head of STEM

    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.