Hyperbola:

The standard equation for hyperbolas is:

where b² = c² - a²

vertices (± a,0) (0, ± a)foci (± c,0) (0, ± c)transverseaxis on x-axis, on y-axis,length 2a length 2aconjugateaxis on y-axis, on x-axis,length 2b length 2b

a is always larger than b; and a,b, and c are related by c² = a² + b²

ex.
Graph 9x² - 16y²= 144

a² = 16 ; b² = 9
major axis: x-axis
vertices: (± 4,0)

c² = a+ b
c² = 16 + 9
c² = 25

foci: (± 5,0)

Graph 36x² - 4y² + 144 = 0
36x² - 4y² = -144 factor -1 out
4y² - 36x² = 144

a² = 36 ; b² = 4
major axis: y-axis
vertices: (0,± 6)

c²= a² + b²
c²= 36 + 4
c² = 40

(to find the asymptotes, let the x term equal the y term and solve for y)