Graphing More Complex Rational Functions Dr. Shildneck Fall, 2014

To graph Find all vertical asymptotes Find the horizontal asymptotes Find any slant asymptotes Draw the asymptotes. Plot points Sketch Curves

Example 1 Graph

Finding Vertical Asymptotes To find the vertical asymptotes of a rational function, find all values of x such that the denominator is zero.

To find the horizontal asymptote There is at most one horizontal asymptote: If the degree of p(x) < degree of q(x), (top power < bottom power) the horizontal asymptote is y = 0. If the degree of p(x) = degree of q(x), (top power = bottom power) the horizontal asymptote is If the degree of p(x) > degree of q(x), (top power > bottom power) there is NO horizontal asymptote. Leading coefficient of p(x) Leading coefficient of q(x) y =

Example: Horizontal Asymptote degree of top = 2 degree of bottom = 1 2 > 1 So, there is No Horizontal Asymptote.

To find slant asymptotes There are only slant asymptotes if the degree of p(x) = 1 + degree of q(x). Divide p(x) by q(x). The result (excluding the remainder) is the equation of the slant asymptote.

Example: Slant Asymptotes Degree of Top = 1 + Degree of Bottom, so Divide: Slant Asymptote – ( ) – ( ) Doesn’t matter.

Example: Draw Asymptotes Vertical Horizontal none Slant

Example: Plotting Points Note: There are only two sections created by vertical asymptotes. Plot points on each side of the vertical asymptote. 2 1

Example: Plotting Points -4 -3 -2 -1 Y -6.5 -6 A S M -1.5

Example: Plotting Points

Example: Sketch Branches

Example 2 Graph:

Finding Vertical Asymptotes To find the vertical asymptotes of a rational function, find all values of x such that the denominator is zero. or

Example: Horizontal Asymptote degree of top = 2 degree of bottom = 2 2 =2, so, the Horizontal Asymptote is

Example: Slant Asymptotes Degree of Top = Degree of Bottom Thus, there is no slant asymptote.

Example: Draw Asymptotes Vertical Horizontal Slant none

Example: Plotting Points There are three sections created by vertical asymptotes. Plot points in each section created by the vertical asymptotes. In middle sections plot 3-5 points. 2 3 1

Example: Plotting Points -5 -4 -3 -2 2 3 4 5 Y -75 16 ≈ -4.7 -48 7 -6.8 A S 12 2.4

Example: Plotting Points

Example: Sketch Branches

Assignment Worksheet #4