**Hyperbola:**

The standard equation for hyperbolas is:

where b^{2} = c^{2} - a^{2}

vertices (Â± a,0) (0, Â± a)foci (Â± c,0) (0, Â± 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 c^{2} = a^{2} + b^{2}

ex.

Graph 9x^{2} - 16y^{2}= 144

a^{2} = 16 ; b^{2} = 9

major axis: x-axis

vertices: (Â± 4,0)

c^{2} = a+ b

c^{2} = 16 + 9

c^{2} = 25

foci: (Â± 5,0)

Graph 36x^{2 }- 4y^{2 }+ 144 = 0

36x^{2} - 4y^{2} = -144 factor -1 out

4y^{2 }- 36x^{2} = 144

a^{2 }= 36 ; b^{2 }= 4

major axis: y-axis

vertices: (0,Â± 6)

c^{2}= a^{2 }+ b^{2}

c^{2}= 36 + 4

c^{2} = 40

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