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

Reciprocal Functions & Graphs

What is the reciprocal function?

• The reciprocal function is defined by

• Its domain is the set of all real values except 0

• Its range is the set of all real values except 0

• The reciprocal function has a self-inverse nature

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What are the key features of the reciprocal graph?

• The graph does not have a y-intercept

• The graph does not have any roots

• The graph has two asymptotes

  • A horizontal asymptote at the x-axis: 

    • This is the limiting value when the absolute value of x gets very large

  • A vertical asymptote at the y-axis: 

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

• The graph has two axes of symmetry

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• The graph does not have any minimum or maximum points

Linear Rational Functions & Graphs

What is a rational function with linear terms?

• A (linear) rational function is of the form 

• Its domain is the set of all real values except 

• Its range is the set of all real values except 

• The reciprocal function is a special case of a rational function

What are the key features of linear rational graphs?

• The graph has a y-intercept at provided 

• The graph has one root at provided 

• The graph has two asymptotes

  • A horizontal asymptote: 

    • This is the limiting value when the absolute value of x gets very large

  • A vertical asymptote: 

    • 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

Quadratic Rational Functions & Graphs

How do I sketch the graph of a rational function where the terms are not linear?

• A rational function can be written 

  • Where g and h are polynomials

• To find the y-intercept evaluate 

• To find the x-intercept(s) solve 

• To find the equations of the vertical asymptote(s) solve 

• There will also be an asymptote determined by what f(x) tends to as x approaches infinity

  • In this course it will be either:

    • Horizontal

    • Oblique (a slanted line)

  • This can be found by writing in the form 

    • You can do this by polynomial division or comparing coefficients

  • The function then tends to the curve 

What are the key features of rational graphs: quadratic over linear?

• For the rational function of the form 

• The graph has a y-intercept at provided 

• The graph can have 0, 1 or 2 roots

  • They are the solutions to 

• The graph has one vertical asymptote 

• The graph has an oblique asymptote 

  • Which can be found by writing in the form 

    • Where p, q, r are constants

    • This can be done by polynomial division or comparing coefficients

What are the key features of rational graphs: linear over quadratic?

• For the rational function of the form 

• The graph has a y-intercept at provided 

• The graph has one root at 

• The graph has can have 0, 1 or 2 vertical asymptotes

  • They are the solutions to 

• The graph has a horizontal asymptote