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Asymptotes and Rational Functions
Vertical Asymptotes
While working with rational functions, we learned that the restriction on the denominator could be the vertical asymptote. We determined a vertical asymptote equation by setting the denominator equal to 0 and solving for x.
The vertical asymptote is a vertical line, so it must be written as an equation x = value.
Horizontal Asymptotes
Rational functions will also offer us information about their horizontal asymptote.
Horizontal asymptotes are horizontal lines, so they must be written as y = value.
The little simple rational expression (below) will determine if a rational function has a horizontal asymptote or not, and will be determined by the power of x.
There are 3 cases we can use to determine the locations of the horizontal asymptote. Compare the largest power of the numerator to the largest power of the denominator.
Case 1: m = n
Assume that the value of m is 2 and n is also 2.
m and n are equal, so the horizontal asymptote will be the coefficient(s) of the largest power.
Looking at the function below, notice the coefficient of 2x2 is 2 and the coefficient of x2 is 1.
The coefficients of x will determine the equation of the horizontal asymptote. (H.A.)
The horizontal asymptote will always be a line.
Case 2: m < n
Assume that the value of m is 1 and n is 2.
m is less than n so the horizontal asymptote will be y = 0.
The coefficients will not be used, the horizontal asymptote will be the line y = 0 (x-axis).
Case 3: m > n
We will not have a horizontal asymptote; instead, it will be a slant asymptote.
If the numerator's power is greater than the denominator's power, divide the numerator by the denominator to find the result. It will be a line.
This line will have either a positive or a negative slope.
The power of the numerator is greater than the power of the denominator.
If we divide the numerator's expression by the denominator's expression, we would have the following equation: y = x - 3.
This would be a slant asymptote.
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