Can the graph of a rational function cross a horizantal asymptote ?

Answer 1

Yes! Take the example of the function . We know that this function must be 0 when x = 1, so the graph must pass through the point (1, 0). If we look at the behavior of the graph as x gets extremeley positive or extremely negative, we can also observe that the x² in the denominator grows much quicker than the numerator, and the value of y tends closer and closer towards 0, making the line y = 0 a horizontal asymptote. By adding a constant term a to end of the function, we can find a whole family of rational functions of the form

In general, if we want a rational function whose graph crosses its horizontal asymptote, we need to make sure that

1. The denominator has a higher degree than the numerator (a necessary condition to make sure we have a horizonal asymptote)
2. The zeroes of the function aren't undefined (e.g. the zeroes of a/x² are undefined when a ≠ 0, since the equation 0 = a/x² has no solutions)

Answer 2

Yes. Graph y = 6/(x-2) + 4/(x+7) = (10x+50)/(x^2 + 9x + 14)

I don't know what any rule says, but this type of rational function does cross the horizontal asymptote