Final answer:
A rational function has a slant asymptote when the degree of the numerator is one greater than the degree of the denominator (m > n). Slant asymptotes occur because the graph of the function approaches a line as x goes to infinity, which may have either a positive or negative slope.
Step-by-step explanation:
A rational function in the form axm + ... bxn + ... has a slant (oblique) asymptote when m > n. This means the degree of the numerator is greater than the degree of the denominator by one. Asymptotes represent the behavior of the graph of a function at infinity or at certain specific points.
In the case of slant asymptotes, as the value of x goes to infinity, the graph of the function approaches a line that is not parallel to either axis. Unlike horizontal asymptotes, which occur when m equals n (m = n), slant asymptotes appear when the degree of the numerator is exactly one more than the degree of the denominator (m = n + 1), where m > n.
To determine the equation of the slant asymptote, you can perform polynomial long division or synthetic division on the function. The quotient, without the remainder, will give you the equation of the slant asymptote which may have either a positive or a negative slope.