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When does a curve cross an asymptote?

Hey guys, just revising for Monday's FP1.
When sketching curves, how do you know if a curve crosses an asymptote or not? Is there a general rule?

Thanks!

Reply 1

generalebriety

Asymptotes are where the graph has no value. A curve will never cross its own asymptote, because at that asymptote, the curve is not defined. Think of the graph of y = 1/x. There are asymptotes at y=0 and at x=0. The curve never crosses them because at y=0 there's no possible x value (0 = 1/x?) and at x=0 there's no possible y value (y = 1/0?).

Reply 2

Shickles

Ahem, you can have curves crossing an asymptote, actually! I think they only occur in rational functions. I can't think of an example right now but it does occur, trust me! Usually the curves will only cross the horizontal asymptote, not the vertical ones, mind.

Reply 3

JeremyC

A curve can cross it's horizontal asymptote; take the curve

\large y = \frac{(x-3)(2x-5)}{(x+1)(x+2)}

for instance, which has a horizontal asymptote of y = 2 - the curve intersects the asymptote when

\large x = \frac{11}{17}

. Indeed, this is just a particular example of something more general - according to Further Pure Mathematics (by Brian and Mark Gaulter) I have next to me: "When the denominator is a quadratic expression ... the curve will normally cross the horizontal asymptote" - note that here they refer to a quadratic expression with two real roots.

Reply 4

insparato

Hmm apparantly they can! Interesting stuff.

Reply 5

JeremyC

Indeed.

A vertical asymptote occurs at a particular (finite) value of x if the curve tends to

\pm \infty

for that value of x - for instance, if the denominator is zero for that value of x (in fact, I can't think of any other causes of a vertical asymptote). Thus, there is no way that the curve can cross the vertical asymptote, since there is only one y value for each x value.

A horizontal asymptote is what the curve tends towards when x tends to

\pm \infty

, and it is thus potentially possible for the curve to cross the horizontal asymptote - but this is by no means guaranteed. As Shickles and I mentioned earlier, this can occur in rational functions with a quadratic denominator.

Reply 6

generalebriety

I stand corrected. :smile:

Reply 7

Shickles

Very interesting! Who knew that curves could cut asymptotes, eh? I didn't know this until a week ago :redface:

Reply 8

Profesh

Shickles

Hey guys, just revising for Monday's FP1.
When sketching curves, how do you know if a curve crosses an asymptote or not? Is there a general rule?

Thanks!

Ideally; when the Moon is in the Seventh House, and Jupiter aligns with Mars.

Reply 9

generalebriety

Profesh

Ideally; when the Moon is in the Seventh House, and Jupiter aligns with Mars.

Wow, a rare visit from Profesh to the maths forum...

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