3.7 Higher Order Rational Functions

Higher Order Rational Functions are functions in the form or

For now, in sketching these graphs, the most important features are the intercepts and the asymptotes.

Intercepts:

To find the y-intercept, set x to zero and calculate.

To find the x-intercepts, find the value(s) that make the numerator equal to zero.

Vertical Asymptotes:

To find the equations of any vertical asymptotes, find the value(s) that make the denominator equal to zero.

To find the behaviour of the graph around the vertical asymptotes pick a value close to the asymptote on one side and determine if the function is positive or negative. Positive means the graph goes up near that asymptote, negative means it goes down. Also, if the factor in the denominator that indicates a vertical asymptote has an odd degree, then the behaviour on each side of the asymptote will be opposite, for even degree, the behaviour will be the same.

Horizontal Asymptotes:

To find the equations of any horizontal asymptotes, determine if the function approaches a specific value with extremely large positive or large negative values of x.

A nice trick to find horizontal asymptotes is to look at the degree of the numerator and denominator. If the denominator has a higher degree, the horizontal asymptote will be y=0. If the numerator has a higher degree than the denominator, there will be no horizontal asymptote. If they have the same degree, the horizontal asymptote will be in the form , where a and b are the leading coefficients of the numerator and denominator respectively.

**A graph can cross a horizontal asymptote since it does not indicate an undefined value for the function and merely a trend with large values of x**

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