The Hyperbola: Definition, Vertices, Foci & Graphing

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  • 0:02 The Hyperbola
  • 1:57 Two Vertices
  • 4:03 Two Foci
  • 5:44 Graphing A Hyperbola
  • 7:47 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this video lesson, we'll learn what a hyperbola is and how to graph one using the standard equation to find the center point, vertices and focus points.

The Hyperbola

What is a hyperbola? It can simply be described as two arcs back to back to each other. Why do you need to learn about hyperbolas? They are important to learn because they are one of the shapes that you get from slicing a cone. Picture two ice cream waffle cones, one on top of the other, with their tips touching in the middle. Take a knife and slice it in a way so that you cut through both cones. The shape that you are left with is what we call a hyperbola.

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As with many things in algebra, there is an equation to describe this shape. And because this is algebra, this equation actually helps us describe our hyperbola with accuracy. We can also extract a lot of useful information from it, and we can graph our hyperbola just by looking at the numbers and doing a little bit of basic math.

So, let's get started.

The standard equation of a hyperbola that we use is (x - h)^2/a^2 - (y - k)^2/b^2 = 1 for hyperbolas that open sideways. If our hyperbola opens up and down, then our standard equation is (y - k)^2/a^2 - (x - h)^2/b^2 = 1.

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How can you remember these? Notice that the h is always linked with the x and the k with the y. You can think h comes before k just like x comes before y, so h with x and k with y. The a^2 always comes before the b^2. If you are subtracting the y part, then your hyperbola will be sideways, but if you are subtracting the x part, then your hyperbola will be upright. As you know, our variables are x and y. The letters h, k, a, and b are there to help us describe our hyperbola, as we will come to see.

Two Vertices

Because our hyperbola is made up of two arcs, we will have two vertices, which are the tips of the arcs. We can find these vertices by looking at our h, k and a values. We begin by looking at our h and k numbers. By putting the letters together as in (h, k), we get the location of the center point between the two arcs, which is the center of our hyperbola. The a number tells us that the vertices are that many spaces away from the center in either direction.

If our x comes first in our equation, then we go a spaces to the left to find one tip and a spaces to the right to find the other tip. If the y comes first, then we go a spaces up to find one tip and a spaces down to find the other tip.

For example, if our equation is (x - 4)^2/4 - (y - 2)^2/9 = 1, we see that our h equals 4, our k equals 2, our a equals 2 because 2^2 equals 4 and our b is 3 since 3^2 equals 9. We found these values by comparing our equation to the standard form and seeing what numbers are in the same place as the letters we are concerned about. So since our h is 4 and our k is 2, our center is (4, 2). Because our a is 2 and our x appears first, our tips are 2 spaces to the left of the center and 2 spaces to the right. So our vertices are (2, 2) and (6, 2).

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Notice how we subtracted 2 from our x value in the center point to find our tip to the left, and we added 2 to our x value to find the tip to the right. If our y appeared first in our equation, then we would subtract our a value from the y value of our center point to find one tip and then add the a value to the y value to find the other tip.

Two Foci

In addition to our two vertices, we also have two focus points, or foci. There is one focus for each arc. The focus is the point for each arc where the ratio of the distance from any point on the arc to the focus to the distance from that point to a straight line is the same for all points on the arc.

To find this point, we use the formula c^2 = a^2 + b^2. We plug in the a and b terms from our standard equation for a hyperbola and then solve for c. If our x appears first in our standard equation, then we add and subtract this c value from the x value of our center point to find the foci. If our y value appears first in our standard equation, then we add and subtract our c value from the y value of our center point to find the foci.

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To continue our example, since our a is 2 and our b is 3, our c can be found by solving c^2 = 2^2 + 3^2. To solve for c, we first calculate the squares of our numbers. We get c^2 = 4 + 9 = 13. Then we take the square root to find that our c = 3.6056. We will add and subtract this c value to and from our x value of our center point to find our foci since our x appears first in our standard equation. Doing this, we get our foci of (0.3944, 2) and (7.6056, 2).

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