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For ,
(a) Identify the center.
(b) Identify the vertices.
© Identify the foci.
(d) Write equations for the asymptotes.
(e) Graph the hyperbola.
Give exact answers. Express numbers in simplest form using integers, fractions, or radicals.
Part 1 of 5
(a) The center is .
Part 2 of 5
(b) The vertices are and .
Part 3 of 5
© The foci are and .
Part:
Part 4 of 5
(d) The asymptotes are and .

Ask by Brewer Bowers. in the United States
Mar 25,2025

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Tutor-Verified Answer

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The asymptotes are and .

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The asymptotes for the hyperbola described by the equation can be derived using the formula for the slopes of the asymptotes, given by where is the center, is the distance from the center to the vertices, and relates to the distance from the center to the foci. Since we already have the center at , and and , the equations for the asymptotes become and .
So, the equations of the asymptotes are:
and .
Have you ever wondered how hyperbolas show up in real life? They pop up in everything from satellite dishes to the paths of spacecraft navigating through gravity wells. The shape reflects how the objects interact with gravitational forces, demonstrating that math isn’t just abstract—it’s alive in our everyday technology!
If you’re keen on exploring hyperbolas further, check out some interactive graphing tools online. Websites like Desmos allow you to manipulate the parameters of hyperbolas and visually see how changes affect their shapes. It’s a delightful way to deepen your understanding—who knew math could be so much fun?

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