Hyperbolas

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Textbook Reference: Section 10.2 The Hyperbola

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Hyperbolas

Our next conic section we're going to look at is the hyperbola.

As a conic section, a hyperbola is what we get when we slice through our double-cone at a very steep angle (specifically, steeper than the edge of the cone).

What we end up with is a shape made of two disjoint branches, though we still consider both branches to be part of one hyperbola. (Think of it like how Michigan has two separate pieces, but it still counts as one state.)

The hyperbola has an "anatomy" of its own:

• Each branch of the hyperbola has a turning point called a vertex. (The word "vertex" comes from Latin vertere, meaning "to turn." The plural is vertices.)
• The segment connecting the vertices is called the transverse axis
• As you go further away from the center, the hyperbola straightens out toward two asymptotes that intersect at the center.
• The hyperbola also has two foci, that lie along the same line as the transverse axis.

A good intuition to have with the hyperbola is that it's kind of like an "anti-ellipse" — almost any property that you can think of for an ellipse, the hyperbola has the opposite property:

• An ellipse curves inward to form a closed curve.
  A hyperbola curves outward toward infinity.
• On an ellipse, the foci are closer to the center than the curve.
  On a hyperbola the foci are further from the center than the curve.
• If you pick any point on an ellipse, the sum of its distances to the foci is a fixed number.
  If you pick any point on a hyperbola, the difference of its distances to the foci is a fixed number.

These are just a few parts of the "anti-ellipse" analogy for hyperbolas. There are a number of others that you'll see as you further explore hyperbolas. Trust me, the analogy goes pretty deep.

You can see some real-world applications of hyperbolas here and here.

Preview Activity 20

Answer these questions and submit your answers as a document on Moodle. (Please submit as .docx or .pdf if possible.)

1. Using Desmos, graph the hyperbola with the following equation: \[\dfrac{(x+1)^2}{9}-\dfrac{(y-3)^2}{4}=1\]
   1. Where is the center?
   2. What is the length of the transverse axis?
   3. How are these pieces of information connected to the equation?
2. Now graph the hyperbola with the following equation: \[\dfrac{(y-3)^2}{4}-\dfrac{(x+1)^2}{9}=1\] This is called the conjugate of the first hyperbola.
   Compare and contrast the two graphs and their equations.
3. How do the previous questions continue the "anti-ellipse" analogy?
4. When you zoom out really far on the graph of a hyperbola, what do you notice?
5. Answer AT LEAST one of the following questions:
   1. What was something you found interesting about this activity?
   2. What was an "a-ha" moment you had while doing this activity?
   3. What was the muddiest point of this activity for you?
   4. What question(s) do you have about anything you've done?

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Posted by Bill Shillito