Asymptotes

Definition

Asymptotes are lines that a curve approaches but never touches or intersects. They provide insight into the behavior of the graph at extreme values.

5 Must Know Facts For Your Next Test

1. Asymptotes can be vertical, horizontal, or oblique.
2. For hyperbolas, asymptotes intersect at the center and define the slope of the branches.
3. The equations of asymptotes for a hyperbola centered at $(h, k)$ with horizontal transverse axis are $y = k \pm \frac{b}{a}(x - h)$. For a vertical transverse axis, they are $y = k \pm \frac{a}{b}(x - h)$.
4. Asymptotes help to sketch hyperbolas accurately by providing guidelines for their branches.
5. Asymptotic behavior describes how functions behave as inputs approach infinity or specific values.

Review Questions

Related terms

Hyperbola:

A type of conic section formed by intersecting a double cone with a plane, consisting of two separate curves.

Transverse Axis:

The line segment that passes through the foci of a hyperbola and whose endpoints lie on the hyperbola.

Conjugate Axis:

The line segment perpendicular to the transverse axis at its midpoint in a hyperbola; it determines the distance between vertices along this axis.