Hyperbolas. Derivating the Standard Equation.

You might be like many students that get nervous if you have to prove or derivate an equation or a formula. This lesson takes you on a proving tour with foci, a point in the hyperbola, and the distance formula. You will see that doing what you learned in the beginning of coordinate geometry enables you with some manipulation of perfect squares binomials to obtain the standard formula for hyperbolas. You will learn how to translate them into the 4 quadrants of the coordinate plane. Optionally, you may try to work the proof for yourself after some guidelines have been given. You may use your paper or the screen before you run the example and using the MARKER TOOLS menu. The definitions given at the bottom of the page may help you to better understand the lesson if you study them before starting the lesson itself.

Most students report struggling with hyperbolas...thing of the past. After you complete this lesson you will be perfectly able to find all the above answers. Have a good hyperbolic luck!

Lesson's Content

Lesson In PDF Format (no animations)


Lesson's Glossary

Asymptote: If the graph of a function gets close to a line but never intersects with this then line is an asymptote.

Axis of Symmetry: A line on which a graph is reflected onto itself. 

Center of the hyperbola: The point where the transversal and conjugate perpendicular axis intersect.

Completing the Square: Method that finds the constant term in an incomplete perfect square trinomial of a second degree equation to solve it.

Conjugate axis: The axis perpendicular to the transverse axis.

Conic section: A figure that is obtained slicing a double cone with a plane. (parabola, circle, hyperbola, and ellipse) 

Factoring: The process to brake a polynomial down into the product of several factors.

Factors: All whole numbers that are multiplied together to yield another number.

Factored Form: Any polynomial that is written as the product of polynomials of lower degree that may be obtained from the original polynomial.

Focus of an ellipse: Each one of the points in the major axis of the ellipse, from which sum of the distance from each to the set of points in the ellipse is constant.

Foci: Plural for focus.

Transverse axis: A segment in the hyperbola goes from vertex to vertex of the two branches, and it is contained in the line that goes through the foci.

Vertices of a hyperbola: Are identified as the endpoints in the line segment that is the transverse axis of the hyperbola.


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