Can a graph cross an oblique asymptote?
I tried to Google the answer to my question, but I can't find a site that gives me a detailed explanation. help. thank you.
No, if you are dealing with polynomial fractions, to get a slant asymptote, you have to have the numerator just one degree greater than the denominator, so that the asymptote is a straight line. If it is greater than one degree, the asymptote is a "curve".
Yes, a graph can cross an oblique asymptote. An oblique asymptote is a slanted line that represents the behavior of a function as it approaches positive or negative infinity. It occurs when the degree of the numerator is one more than the degree of the denominator in a rational function.
To understand why a graph can cross an oblique asymptote, let's look at the equation of an oblique asymptote. Suppose we have a rational function f(x) = (ax^2 + bx + c) / (dx + e), where the degree of the numerator is one more than the degree of the denominator.
To find the equation of the oblique asymptote, we perform long division or synthetic division between the numerator and the denominator. This gives us a quotient Q(x) and a remainder R(x). The quotient represents the equation of the oblique asymptote.
However, it's important to note that the remainder R(x) does not affect the equation of the oblique asymptote. The remainder represents the discrepancy between the actual function f(x) and the asymptote. If the remainder R(x) approaches zero as x approaches infinity or negative infinity, the graph will have an oblique asymptote. But if the remainder does not approach zero, the graph may cross the oblique asymptote.
So, in conclusion, a graph can cross an oblique asymptote if the remainder of the division does not approach zero. Keep in mind that it is also possible for a graph to have multiple oblique asymptotes, or no oblique asymptotes at all.