hyperbolas
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The document discusses how to graph hyperbolas when the center is not at the origin. It provides an example of a hyperbola with center (2, -1), which shifts the original graph 2 units right and 1 unit down. It then discusses key features of the hyperbola like vertices, foci, asymptotes, and their locations relative to the shifted center point.
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hyperbolas
- 1. Hyperbolas translated away from the center
- 2. Example : Suppose the center is not at the origin (0, 0) but is at some other point such as (2, -1). To graph this hyperbola requires us to remember how graphs are moved horizontally and vertically by a change in the equation. Using Example #1 above, we have This will move the graph in our previous example 2 units right and 1 unit down. Both graphs are shown below.
- 3. Note that a = 3, b = 4, In this graph the transverse axis is horizontal. Thus each focus is a distance of 5 horizontally from the center. One focus is at (7, -1) and one is at (-3, -1). Note that the graphing calculator does not do a good job of showing the top and bottom halves of the branches of the hyperbola joining at the vertices which are located at (-3, -1) and (5, -1). Using this as a model, other equations describing hyperbolas with centers at (2, -1) can be written. and the slope of the asymptotes is If a = 3 and b = 2, and the transverse axis is horizontal, the equation is The slope of the asymptotes is and the vertices are located at (-1, -1) and (5, -1).
- 4. The foci are located at If a = 2 and b = 4, and the transverse axis is vertical, the equation is