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

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 

  • e.g.

• The graph has a y-intercept at provided 

  • e.g. the y-intercept of is

  • e.g. does not have a y-intercept

• The graph can have 0, 1 or 2 roots

  • They are the solutions to 

    • e.g. has two roots and

    • e.g. has one root

    • e.g. has no roots

• The graph has one vertical asymptote 

  • e.g. the vertical asymptote of is

• 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

  • e.g. can be written

    • the oblique asymptote of is

Linear over quadratic

• For the rational function of the form 

  • e.g.

• The graph has a y-intercept at provided 

  • e.g. the y-intercept of is

  • e.g. does not have a y-intercept

• The graph has one root at 

  • e.g. the root of is

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

  • They are the solutions to 

    • e.g. has two vertical asymptotes and

    • e.g. has one vertical asymptote

    • e.g. has no vertical asymptotes

• The graph has a horizontal asymptote at