Find properties of hyperbolas (Tutorial)

Learning Objectives Identify the center, vertices, and foci of a hyperbola from its standard equation. Determine the orientation (horizontal or vertical transverse axis) of a hyperbola. Derive the equations of the asymptotes for a given hyperbola. Calculate the eccentricity and explain its significance for a hyperbola. Convert the general form of a hyperbola's equation to its standard form by completing the square. Sketch the graph of a hyperbola, accurately plotting its key features and asymptotes. Ever wondered about the shape of a sonic boom's shockwave or the design of a nuclear cooling tower? ✈️ These are real-world examples of the hyperbola we're about to explore! This tutorial will guide you through the key properties of hyperbolas, the final conic sec...

TermDefinitionExample HyperbolaA set of all points (x, y) in a plane, the difference of whose distances from two distinct fixed points, called foci, is a positive constant.The graph of the equation x²/9 - y²/16 = 1 is a hyperbola. Transverse AxisThe line segment that connects the two vertices of a hyperbola. Its length is 2a.For the hyperbola x²/9 - y²/16 = 1, the transverse axis is horizontal and has a length of 2 * 3 = 6. Conjugate AxisThe line segment perpendicular to the transverse axis, passing through the center. Its length is 2b.For the hyperbola x²/9 - y²/16 = 1, the conjugate axis is vertical and has a length of 2 * 4 = 8. Foci (singular: Focus)The two fixed points used to define the hyperbola. They lie on the same axis as the vertices. The distance from the center to each focus...

Standard Equation of a Horizontal Hyperbola \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 Use this form when the transverse axis is horizontal. The center is (h, k). The term with 'a²' is always under the positive variable, and 'a' is the distance from the center to a vertex. Standard Equation of a Vertical Hyperbola \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 Use this form when the transverse axis is vertical. The center is (h, k). The positive term is the y-term, indicating the hyperbola opens up and down. Foci and Asymptotes Relationship Foci: c^2 = a^2 + b^2 \\ Asymptotes (Horizontal): y - k = \pm \frac{b}{a}(x - h) \\ Asymptotes (Vertical): y - k = \pm \frac{a}{b}(x - h) The relationship c² = a² + b² is fundamental for finding the foci. The...