3.7 Graphs of Rational Functions

A rational function is a quotient of two polynomial functions.

Parent function: has branches in 1st and 3rd quadrants Parent function: has branches in 1st and 3rd 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.

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.

Ex 2 Determine the asymptotes

Ex 3 find the asymptotes

Ex 4 Find asymptotes

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.

Ex 5 find asymptotes

Slant asymptote There is an oblique or slant asymptote if the degree of P(x) is EXACTLY one degree higher than Q(x).  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.

Ex 6 find the slant asymptote

Ex 7 Graph and find everything!!

Day 2

x2 – 3xy – 13x + 12y + 39 = 0