Key Concepts & Glossary

Key Equations

Hyperbola, center at origin, transverse axis on x-axis	$\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$
Hyperbola, center at origin, transverse axis on y-axis	$\frac{{y}^{2}}{{a}^{2}}-\frac{{x}^{2}}{{b}^{2}}=1$
Hyperbola, center at $\left(h,k\right)$ , transverse axis parallel to x-axis	$\frac{{\left(x-h\right)}^{2}}{{a}^{2}}-\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1$
Hyperbola, center at $\left(h,k\right)$ , transverse axis parallel to y-axis	$\frac{{\left(y-k\right)}^{2}}{{a}^{2}}-\frac{{\left(x-h\right)}^{2}}{{b}^{2}}=1$

A hyperbola is the set of all points $\left(x,y\right)$ in a plane such that the difference of the distances between $\left(x,y\right)$ and the foci is a positive constant.
The standard form of a hyperbola can be used to locate its vertices and foci.
When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form.
When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola.
Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the dimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides.

center of a hyperbola: the midpoint of both the transverse and conjugate axes of a hyperbola

conjugate axis: the axis of a hyperbola that is perpendicular to the transverse axis and has the co-vertices as its endpoints

hyperbola: the set of all points $\left(x,y\right)$ in a plane such that the difference of the distances between $\left(x,y\right)$ and the foci is a positive constant

transverse axis: the axis of a hyperbola that includes the foci and has the vertices as its endpoints