How to find the asymptotes of a hyperbola?

Vertical asymptote:
Find the restriction on x. This is the easiest of the three asymptotes you will need to find (even if only two can show at a time). As a hyperbola is in the form: , you only need to find the restriction on the denominator, namely denominator can never be zero. Hence, let the denominator equal to zero to find the vertical asymptote.

Horizontal asymptote:
Find the restriction on y. To do this, you need to simplify the top and bottom to its lowest terms. If it simplifies to a form such as: , then the horizontal asymptote becomes y = a. You need to think to yourself, as x grows to infinite, and shrinks to negative infinite, what happens to the function? Does it slowly curve to a stop?

Oblique asymptote:
This is a pretty rare kind, but it still exists, so don't be naive to this sort of asymptote. This is a form of horizontal and vertical asymptote, only it's at an angle. That is, this asymptote is a set of x and y-coordinates that work in unison to produce a curvature or line.

Let's consider:

Now, in normal term, a horizontal asymptote would have a degree higher in the denominator than in the numerator. However, it's flipped in this case.

Now, you will need to long divide this set of polynomials to yield a straight y = x line, except it's been moved 11 units to the right to yield a y = x - 11 line.

Remember: these exist because the highest power is in the numerator and not the denominator.

If you need an equation, it would be y=2x+8, y would be how long it grew, x being weeks.if you need how long he was asleep, it would be 4 weeks if he started with no beard, the equation would be x=8÷2if he did have a beard before hand, the equation would be 2w+x=8