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Understanding Hyperbolas: Definitions, Graphing, and Equations

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This section explores the geometric definition and properties of hyperbolas, including the standard forms centered at (0,0) and (h,k). Learn to graph hyperbolas, find their asymptotes, and determine their eccentricity. The essential properties of hyperbolas, such as the distance from two fixed points (foci) being constant, are discussed alongside how to convert and manipulate equations into standard forms. Visual and practical methods are provided for plotting hyperbolas and their key features.

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Understanding Hyperbolas: Definitions, Graphing, and Equations

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  1. Section 8-3 The Hyperbola

  2. Section 8-3 • the geometric definition of a hyperbola • standard form of a hyperbola with a center at (0 , 0) • translating a hyperbola – center at (h , k) • graphing a hyperbola • finding the equations of the asymptotes • finding the equation of a hyperbola • eccentricity and orbits • reflective properties of hyperbola

  3. Geometry of a Hyperbola • hyperbola – the set of all points whose distance from two fixed points (the foci) have a constant difference • all the points are coplanar • the line through the foci is called the focal axis • the midway point between the foci is called the center

  4. Geometry of a Hyperbola F1 F2 V1 center V2 F1 and F2 are the foci V1 and V2 are the vertices (chord between called the transverse axis)

  5. Geometry of a Hyperbola F1 F2 V1 center V2 d2 d1 F1 and F2 are the foci d1 - d2 = constant V1 and V2 are the vertices (chord between called the transverse axis)

  6. Standard Form: Center (0 , 0) • 2a = length of the transverse axis (endpoints are the vertices) • 2b = length of the conjugate axis • c = focal radius (distance from the center to each foci) • c2 = a2 + b2 (use to find c)

  7. Standard Form: Center (h , k)

  8. Graphing a Hyperbola • convert the equation into standard form, if necessary (complete the square) • find and plot the center • use “a” to plot the vertices (same direction as the variable a2 is underneath) • use “b” to plot two other points • draw a rectangle using these four points • draw the diagonals of the rectangle (dashed), these are the asymptotes • draw in the hyperbola (use vertices) • plot the foci using “c” (c is the distance from the center to each focus)

  9. Equations of the Asymptotes • the equations of the asymptotes can be found by replacing the 1 on the right-side of the equation with a 0 and then solving for y

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