Back To CourseAlgebra II: High School
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Amy has a master's degree in secondary education and has taught math at a public charter high school.
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.
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.
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.
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).
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.
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.
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).
To graph our hyperbola, we will use our two tips that we found earlier, and we will now use our b value. We used our a value to find the tips of our arcs; now we will use the b value to help us graph our hyperbola. If the tips of our arcs are horizontal with each other, then we will use our b value to find two points above and below our center. If our arc tips are vertical with each other, then we will use our b value to find two points to the left and right of our center.
To do this, we add and subtract our b value to and from the y value of our center point to find the points above and below the center. To find the points to the left and right of our center, we add and subtract our b value from the x value of our center point.
Once we have done this, we will have four total points around the center. We use these points as the location of the sides of a rectangle bordering our center.
Once we have our rectangle, we can draw and extend the two diagonals. These diagonal lines tell us how far our arc curves. We can draw our arc so that it goes through the tip and then gets close to the diagonal lines we drew without crossing it.
In our example, our b equals 3. Since our arc tips are horizontal to each other, we will add and subtract our b value to and from the y value of our center point. Doing this, we find that our two other points are (4, -1) and (4, 5). Now we have four points. We can go ahead and draw a rectangle with sides that go through our four points. Next, we draw the diagonals and extend the lines. We then finish by drawing our hyperbola so that the arcs go through their tips and get close to the diagonals without crossing.
What have we learned? We learned that a hyperbola looks like two arcs back to back with each other. We also learned that the standard equation of a hyperbola 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. Our hyperbola has a center given by the point (h, k).
Our hyperbola also has two vertices, or tips. For hyperbolas that open sideways, the vertices are given by the points (h + a, k) and (h - a, k). For hyperbolas that open up and down, the vertices are given by the points (h, k + a) and (h, k - a).
Our hyperbola also has two focus points, or foci. For hyperbolas that open sideways, the foci are given by the points (h + c, k) and (h - c, k) where c^2 = a^2 + b^2. For hyperbolas that open up and down, the foci are given by the points (h, k + c) and (h, k - c).
To graph our hyperbola, we use the b value from our standard equation to plot two other points in addition to our vertices. For hyperbolas that open sideways, these two points are (h, k + b) and (h, k - b). For hyperbolas that open up and down, the points are (h + b, k) and (h - b, k). Once we have four points, we can draw a rectangle with sides that pass through each one of these points. Next, we draw two diagonals and extend them out. Then we can draw our hyperbolas so that they get close to our diagonal lines and pass through their tips.
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Back To CourseAlgebra II: High School
23 chapters | 203 lessons