3.6 Graphs of Rational Functions

A rational function is a quotient of two polynomial functions.

Parent function: has branches in 1 st and 3 rd quadrants. No x or y-intercepts. Branches approach asymptotes.

Vertical asymptote – the line x = a is a VA for f(x) if f(x) approaches infinity or f(x) approaches negative infinity as x approaches a from either the left or the right. The VA is where the function is undefined or the value(s) that make the denominator = 0. Whenever the numerator and denominator have a common linear factor, a point discontinuity may appear. If, after dividing the common linear factors, the same factor remains in the denominator, a VA exists. Otherwise the graph will have point discontinuity. That means there is a hole in the graph at that point and not an asymptote Asymptote Hole

Ex 1 find any VA or holes

Horizontal Asymptote – the line y = b is a HA for f(x) if f(x) approaches b as x approaches infinity or as x approaches negative infinity. Can have 0 or 1 HA. May cross the HA but it levels off and approaches it as x approaches infinity.

Shortcut for HA’s If the degree of the denominator is > the degree of the numerator then there is a HA at y = 0. If the degree of the numerator is > the degree of the denominator then there is NO HA. If the degree of the numerator = the degree of the denominator then the HA is y = a/b where a is the leading coefficient of the numerator & b is the LC of the denominator.

Determine the asymptotes & the x and y-intercepts Degrees are equal  a/b

Find asymptotes, x-int, y-int Degree is bigger in the denominator  y=0

find asymptotes Degree is bigger in the denominator  y=0 Degrees are equal  y=a/b

Slant asymptote There is an oblique or slant asymptote if the degree of P(x) is EXACTLY one degree higher than Q(x). oblique or slant asymptote If this is the case the oblique asymptote is the quotient part of the division. Can have 0 or 1 slant asymptote. Can have a VA and slant, a HA and VA, but NOT a HA and slant. Either it has a slant or is has a HA

find the slant asymptote Degree is exactly one bigger in the numerator  divide

Graph and find everything!! Degree is bigger in the denominator  y=0 Test Points!!

Graph and find everything!!

Graph and find everything. x 2 – 3xy – 13x + 12y + 39 = 0

Homework pg 313 #5-9 odd, odd, odd, 59, 63