The Standard Form of the Equation of a Hyperbola with Center at (H, K) Is

The standard form of the equation of a hyperbola with center at (h, k) is

vertical transverse axis—branches open up & down

·  The equation of a hyperbola is ALWAYS = 1. The difference remains constant.

·  a2 is always first.

·  a2 + b2 = c2 c2 is always the largest value.

Note the relationship between a2 and b2 is OPPOSITE of what is remaining constant. i.e. A difference remains constant but c2 represents the sum of a2 and b2.

·  The transverse axis runs through the foci and the vertices.

·  If a is under the x, then the transverse axis runs parallel to the x axis.

·  If a is under the y, then the transverse axis runs parallel to the y axis.

·  The center is at (h,k).

·  A tells you the distance from center to the vertices (along the trAnsverse axis).

·  B is counted from the vertices along the conjugate axis.

·  C tells you where the focus points are. We call these foCi. Think “C”—foCi.

·  The foci are always inside the hyperbola and on the transverse axis. Think “focus in”.

The asymptotes are “boundary” lines that are diagonal lines and intersect at the center. They do not touch the branches of the hyperbola. The branches of the hyperbola will approach the asympototes. The slope can be found using rise/run. One slope will be positive and one slope will be negative. To determine if it is b/a or a/b, think rise over run. The numerator will always be the square root of what is under y. The denominator will always be the square root of what is under x.

When graphing asymptotes, do NOT reduce the slope. This will allow you to form a box that has the vertices on two of it’s sides.

When writing the equation of the asymptotes, remember to use point slope form. The point will be the center and the slope will come from a and b. The slope should be reduced in the final equation of the asymptotes.