Standard Forms

Algebra 2 · Axiom Academy

LESSON Standard Forms of a Hyperbola Two equations, one decision: the positive term tells you which way the branches open and where the vertices sit. Center a hyperbola at the origin and it is one of these two equations. The variable in the positive (leading) term is the direction the branches open, and the axis the vertices sit on. Flip which term is positive and the whole curve swings a quarter turn. x^2 positive opens left–right, vertices y^2 positive opens up–down, vertices 2. Reading a, b, and the Vertices Once the orientation is set, read a and b straight from the denominators. Here is the catch coming from ellipses: a^2 is the denominator under the positive term, not the larger denominator. For a hyperbola, a can be smaller than b . x^2 is positive, so it opens left–right and a^2 = 4 . Even though b = 3 is larger than a = 2 , the vertices land on the x-axis, the axis of the positive term. 3. Vertices, Foci, and the a–b–c Triangle The foci sit on the same axis as the vertices, a distance c from the center and always beyond the vertices. For a hyperbola the relationship is a sum : c^2 = a^2 + b^2 (an ellipse subtracts; a hyperbola adds). y^2 is positive, so it opens up–down with a = 3 and b = 4 (again ). The right triangle with legs a and b has hypotenuse c , so c = 5 . Two equations, but only one decision — which term is positive — sets the orientation, the vertices, and where the foci go.

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