Functions and Models Reference
Hyperbolas
A hyperbola has two branches and two asymptotes. In standard form, the positive term tells us whether the branches open left-right or up-down.
Fact Table
| Form | Opens | Asymptotes |
|---|---|---|
| $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$ | Left and right. | $y-k=\pm\frac ba(x-h)$ |
| $\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$ | Up and down. | $y-k=\pm\frac ab(x-h)$ |
| Foci | Along the transverse axis. | $c^2=a^2+b^2$ |
Content Formulas
Horizontal Hyperbola
$$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$$
Vertical Hyperbola
$$\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$$
Focal Relationship
$$c^2=a^2+b^2$$
Center
$$(h,k)$$
Unlike ellipses, hyperbolas use $c^2=a^2+b^2$. The foci sit outside the vertices.
Classic Example
Read a Hyperbola
Find the center, vertices, asymptotes, and foci of $\frac{(x-1)^2}{9}-\frac{(y+2)^2}{4}=1$.
$$h=1,\quad k=-2,\quad a=3,\quad b=2$$
$$\text{center: }(1,-2)$$
$$\text{vertices: }(1\pm3,-2)=(-2,-2),(4,-2)$$
$$y+2=\pm\frac23(x-1)$$
$$c^2=9+4=13,\quad c=\sqrt{13}$$
$$\text{foci: }(1\pm\sqrt{13},-2)$$