Wetzel

Math 110

3.5 – Graphs of Rational Functions

Rational Functions -  

            Undefined where the denominator is equal to zero. 

            To find these values, set the denominator equal to zero and solve.

Asymptotes –

            Vertical –

·        x = a number

·        set the denominator equal to zero and solve.

            Horizontal –

·        y = a number

·        write the num. and den. in standard form and note their degree

·        If the den. has a larger degree than the num., there is a horizontal asymptote at y = 0.

·        If the den. and num. have the same degree, there is a horizontal asymptote at where a is the leading coefficient of the num. and b is the leading coefficient of the den.

            Oblique -  (Slant)

·        y = mx + b

·        If the numerator’s degree is exactly one degree higher than the denominator’s, there is an oblique asymptote.

·        To find any oblique asymptote, divide the num. by the den.

                                    (disregard any remainders)

            Ex.:      Find any and all asymptotes

            1.)                      2.)                    3.) 

Graphing Rational Functions –

            Steps:

            1.)  Find any vertical asymptotes

            2.)  Find any horizontal or oblique asymptotes

            3.)  Find the y-intercept

            4.)  Find the zeros (x-intercepts) and their multiplicity

            5.)  Determine if the graph will cross its non-vertical asymptotes

                        horizontal:  y = b

                                    f(x) = b

                        oblique:  y = mx + b

                                    f(x) = mx + b

            6.)  Plot the selected points.  Choose additional test points in each interval between the vertical asymptotes and x-intercepts.

            7.)  Sketch the graph.

            Ex.:

            4.) 

HW:  6^th Ed.  p.348  #1-55odd, 59, 61, 63

         5^th Ed.  p. 341 #1-41odd, 45, 47, 49