Hyperbolas. Graphing Them From The Equation. Finding The Equation From The Graph.

You hear hyperbola, and almost jump out of your seat! ... How should I determine if the hyperbola is vertical or horizontal? How will I differentiate the equation of an ellipse, from the equation of a hyperbola? What is the transverse axis? What is the conjugate axis? How do I graph the asymptotes? What numbers and how should I look to complete the square to find the standard form of the hyperbola equation? Is there an easy way of graphing it?

Most students report struggling with hyperbolas...not anymore. After you complete this lesson you will be perfectly able to find all the above answers. This lesson takes you into a journey that is visually rich and enables you to follow sequence and color clues to decipher the secrets inside the hyperbola, its graphing and its equation!

Lesson's Content

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.

Interactive Algebraic Applets

A hyperbola is defined, as the space for which the absolute value of the difference

of the distances from a point in this space; to two points called foci, remains constant.

You may interact with this applet, by dragging the point in the hyperbola in each

one of the branches. You may jump it from one branch to the other by dragging it.

The absolute value of the difference above referred will remain constant; as

the distances are updated.

Hyperbolas are sometimes challenging because the student has to

do a lot of math for each graph to be displayed in paper. This

interactive applet saves that time; by allowing the student to

manipulate parameters a, b, h, and k in the standard formula.

All that it is required is to drag the point at the sliders associated

to these values, and thus view the graph and the corresponding

equations updated.

The general equation for conics has parameters A, B, C, D, E, and F.

The most important ones are A, and C; which are the coefficients for the

quadratic x, and y terms. Depending on whether they are equal, or different

in value, and sign: You get a different conic. For example, if these values

are different in sign and value, but different from cero; you have a

hyperbola. If they are equal in value an sign, you have a circle, etc.

You will orient yourself with a table in the applet, in how to change

these parameters to display the different conics.

Vocabulary Puzzle

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