Focuses on the specific case where a rational function has a slant (or oblique) asymptote and details how to find its equation using polynomial long division or synthetic division.
Under what specific condition does a rational function have a slant asymptote?
A rational function has a slant asymptote if and only if the degree of the numerator is exactly one greater than the degree of the denominator.
How do you find the equation of a slant asymptote for a rational function \(f(x) = \frac{P(x)}{Q(x)}\)?
Use polynomial long division to divide the numerator \(P(x)\) by the denominator \(Q(x)\). The equation of the slant asymptote is \(y = \text{quotient}\). The remainder is ignored.
Find the slant asymptote for the function \(f(x) = \frac{x^2 + 5x + 3}{x - 1}\).
Using polynomial long division to divide \(x^2 + 5x + 3\) by \(x - 1\), the quotient is \(x + 6\) and the remainder is \(9\). The equation of the slant asymptote is \(y = x + 6\).
When can you use synthetic division instead of long division to find the slant asymptote?
You can use synthetic division as a shortcut when the denominator is a linear factor of the form \((x-c)\).
Use synthetic division to find the slant asymptote for \(f(x) = \frac{2x^2 - 3x + 1}{x + 2}\).
Using \(-2\) for synthetic division with coefficients \(2, -3, 1\), the resulting quotient coefficients are \(2\) and \(-7\). This corresponds to the line \(y = 2x - 7\), which is the slant asymptote.
Can a rational function have both a slant asymptote and a horizontal asymptote?
No. The conditions for their existence are mutually exclusive. A horizontal asymptote exists if \(n \le m\), while a slant asymptote exists only if \(n = m+1\).
Does the function \(g(x) = \frac{x^4 - 1}{x^2 + 3}\) have a slant asymptote?
No. The degree of the numerator (\(4\)) is two greater than the degree of the denominator (\(2\)). Since the degree difference is not exactly one, there is no slant asymptote.
What part of the result from polynomial long division is discarded when finding the slant asymptote?
The remainder is discarded. As \(x \to \pm\infty\), the remainder term, \(\frac{\text{remainder}}{\text{divisor}}\), approaches \(0\), which is why the graph approaches the line defined by the quotient.
What is a slant (or oblique) asymptote?
A slant asymptote is a non-vertical, non-horizontal line that the graph of a function approaches as \(x\) approaches positive or negative infinity. Its equation is of the form \(y = mx+b\).
