Explains the three rules for finding the horizontal asymptote of a rational function by comparing the degree of the numerator (n) to the degree of the denominator (m).
What does a horizontal asymptote represent on a graph?
A horizontal asymptote is a horizontal line, \(y=b\), that describes the end behavior of the function. The graph approaches this line as \(x\) approaches positive infinity (\(+\infty\)) or negative infinity (\(-\infty\)).
Consider \(f(x) = \frac{ax^n + ...}{bx^m + ...}\). What is the rule for the horizontal asymptote when \(n < m\)?
If the degree of the numerator (\(n\)) is less than the degree of the denominator (\(m\)), the horizontal asymptote is the line \(y=0\) (the x-axis).
Find the horizontal asymptote of \(f(x) = \frac{x+1}{x^2+3}\).
The degree of the numerator is \(1\) and the degree of the denominator is \(2\). Since \(1 < 2\), the horizontal asymptote is at \(y=0\).
Consider \(f(x) = \frac{ax^n + ...}{bx^m + ...}\). What is the rule for the horizontal asymptote when \(n = m\)?
If the degree of the numerator (\(n\)) is equal to the degree of the denominator (\(m\)), the horizontal asymptote is the line \(y = \frac{a}{b}\), where \(a\) and \(b\) are the leading coefficients of the numerator and denominator, respectively.
Find the horizontal asymptote of \(g(x) = \frac{8x^2 - 3x}{2x^2 + 1}\).
The degrees of the numerator and denominator are both \(2\). The leading coefficients are \(8\) and \(2\). The horizontal asymptote is at \(y = \frac{8}{2}\), which simplifies to \(y=4\).
Consider \(f(x) = \frac{ax^n + ...}{bx^m + ...}\). What is the rule regarding horizontal asymptotes when \(n > m\)?
If the degree of the numerator (\(n\)) is greater than the degree of the denominator (\(m\)), there is no horizontal asymptote. (If \(n=m+1\), there will be a slant asymptote instead).
Find the horizontal asymptote of \(h(x) = \frac{x^3-2x+1}{4x^2+5}\).
The degree of the numerator is \(3\) and the degree of the denominator is \(2\). Since \(3 > 2\), there is no horizontal asymptote.
Can the graph of a rational function ever cross its horizontal asymptote?
Yes. A horizontal asymptote describes the end behavior of the graph (as \(x \to \pm\infty\)). The graph can cross the asymptote at finite, or 'middle' values of \(x\).
Do you use the original or simplified function to find the horizontal asymptote?
You use the original function. Horizontal asymptotes are determined by the degrees of the polynomials, which represent the function's ultimate end behavior. Canceling factors (which create holes) does not change this end behavior.
