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Amy has a master's degree in secondary education and has taught math at a public charter high school.
When it comes to algebra, we have equations and then we have standard form equations. Equations in standard form are those equations that are written in such a way so that we can see our useful information by just looking at the numbers. In this video lesson, we're going to be talking about the standard form equation of a hyperbola.
A hyperbola is a shape that looks like two arcs back to back with each other. You get this kind of shape when you stack two ice cream waffle cones on top of each other so that their tips are touching in the middle and then you cut it in a way so that you go through both cones. Just like with hyperbolas, you can stack your cones upright or you can lay them on the side. If your cones are upright, then your hyperbola will open up and down. If your cones are laid on a flat surface, then your hyperbola opens to the left and right. Because we have these two different directions that our hyperbola can be, we have two different forms of our standard form equation, one for the hyperbola that opens up and down and another for the hyperbola that opens sideways.
The standard form of a hyperbola that opens sideways is (x - h)^2 / a^2 - (y - k)^2 / b^2 = 1. For the hyperbola that opens up and down, it is (y - k)^2 / a^2 - (x - h)^2 / b^2 = 1. Notice that the x appears first for the hyperbola that opens sideways and the y appears first for the hyperbola that opens up and down. So you can link this with the axis that the cones are on. If it's the x-axis, the hyperbola opens sideways and the x appears first. If it's the y-axis, then the hyperbola opens up and down and the y appears first. Notice also that the h always appears with the x and the k with the y. And the a always comes before the b as it does in the alphabet. The letters h, k, a and b are there to provide us with useful information about our hyperbola, as we will see.
When you are given a problem asking you to write the standard form equation of a hyperbola, the problem will need to provide you with just a few bits of information. The information that you need to know is the location of the center of the hyperbola, the location of its vertices (the tips of the arcs of the hyperbola) and the location of the foci (the point in each arc where the ratio of the distance from any point on the arc to this focus point to the distance from this same point on the arc to a straight line is always the same).
When written in standard form, our hyperbola equation tells us that our center is located at (h, k) and that our vertices are a spaces away from the center in both directions. For hyperbolas that open up and down, the vertices are a spaces above and below the center. For hyperbolas that open sideways, the vertices are a spaces to the left and right of our center. The location of the foci is c spaces away from the center, where c^2 = a^2 + b^2. Just like for the vertices, if the hyperbola is upright, then the foci are located c spaces above and below the center, and if the hyperbola is sideways, then the foci are located c spaces to the left and right of the center.
Let's look at an example now. Our problem asks us to write the standard form equation of a hyperbola with a center located at (1, 2), with vertices located at (-2, 2) and (4, 2) and with foci located at (-4, 2) and (6, 2).
With this information, we can go ahead and write our complete standard form equation. The first thing we see is that our center is at (1, 2) so that means our h is 1 and our k is 2. So far, so good! We have two of our letters down. We just need a and b now.
The next step is to see how far away the vertices are from the center and in what direction. We can plot them on graph paper or just compare the location of the points to our center. We see that (-2, 2) is 3 spots to the left of our center, and the other point (4, 2) is 3 spots to the right of our center. That tells us that our hyperbola is sideways, so the x will appear first. Since the vertices are 3 spots away from the center in either direction, our a is 3.
To find our b, we will find how far away the foci are from the center. Looking at the points, we see that both are five spots away from our center. So that tells us that our c is 5. We can then use the formula c^2 = a^2 + b^2 to find our b. We know the value of c and a, which we can plug in. After plugging those in we can solve for b: 5^2 = 3^2 + b^2. To solve for b, I first evaluate all my squares to get 25 = 9 + b^2. Next, I subtract 9 from both sides to get 16 = b^2. Lastly, I take the square root of both sides to find that b = 4.
Now that I have all my information, I can now plug these into my standard form equation. I plug 1 in for h, 2 in for k, 3 in for a and 4 in for b. Since my hyperbola is sideways, I will use the standard form where the x appears first. (x - 1)^2 / 9 - (y - 2)^2 / 16 = 1.
Notice how I've squared my 3 to become 9 and my 4 to become 16. And guess what? I'm done! If my hyperbola were upright, then I would use the standard form equation where the y appears first.
So what have we learned? We've learned that a hyperbola looks like two arcs back to back with each other. Standard form equations are those equations that are written in such a way so that we can see our useful information by just looking at the numbers.
The standard form of a hyperbola that opens sideways is (x - h)^2 / a^2 - (y - k)^2 / b^2 = 1. For the hyperbola that opens up and down, it is (y - k)^2 / a^2 - (x - h)^2 / b^2 = 1. In both cases, the center of the hyperbola is given by (h, k).
The vertices are a spaces away from the center. In upright hyperbolas, it is a spaces above and below the center. For sideways hyperbolas, it is a spaces to the left and right of the center. The foci of the hyperbola are then located c spaces away from the center, where c^2 = a^2 + b^2. The foci are located in the same direction as the vertices.
To write our standard form equation, we just need to know the location of the center, the vertices and the foci. The center gives us the values of our h and k. The vertices give us the value of a, and they tell us the direction of our hyperbola. If the hyperbola is sideways, then the x appears first. If the hyperbola is upright, then the y appears first. The foci give us the value of c. To find our b value, we use the formula c^2 = a^2 + b^2 to solve for b.
By the end of the lesson you should be able to:
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