# Understanding Hyperbolas

Let's look at some of the key parts of a hyperbola. In this example, we have a vertical hyperbola.

Notice that (h, k) is the center of the entire hyperbola but does not actually fall on the hyperbola itself. Each hyperbola has a vertex and two asymptotes guide how wide or how narrow the curve is.

There are two patterns for hyperbolas.

By examining the equation, we can determine the following:

1.

2.

3.

**If it is vertical or horizontal.**If the x term is positive, the parabola is horizontal (the curves open left and right). If the y term is positive, the parabola is vertical (the curves open up and down). Unlike an ellipse, it does not matter which denominator is larger.2.

**The center point.**As with all conic sections, the center point is (h, k). Notice that the h is always with the x and the k is always with the y. There is also a negative in front of each, so you must take the opposite.3.

**The a and b values.**These will be needed to graph the parabola. Notice that a is always under the positive term and b is always under the negative term.Let's identify some key information from these hyperbolas:

First of all, we know it is a horizontal hyperbola since the x term is positive. That means the curves open left and right.

Next, we can find the center point by identifying h and k. h is with the x, and we must take the opposite, so h = 3. k is with the y and we must take the opposite so k = -2. The center point is (3, -2).

We can find a and b by taking the square root of the denominators. The square root of 9 is 3 so a = 3. The square root of 25 is 5, so b = 5. Notice that a is not always the largest number.

Summary: This is a horizontal hyperbola. The center is at (3, -2). a = 3, b = 5

Next, we can find the center point by identifying h and k. h is with the x, and we must take the opposite, so h = 3. k is with the y and we must take the opposite so k = -2. The center point is (3, -2).

We can find a and b by taking the square root of the denominators. The square root of 9 is 3 so a = 3. The square root of 25 is 5, so b = 5. Notice that a is not always the largest number.

Summary: This is a horizontal hyperbola. The center is at (3, -2). a = 3, b = 5

First of all, we know it is a vertical hyperbola since the y term is positive. That means the curves open up and down.

Next, we can find the center point by identifying h and k. Remember h is with the x and k is with the y. We must take the opposite of each. So our center point is (-3, -1)

We can find a and b by taking the square root of the denominators. a = 7 and b = 2 (Remember a is always with the positive term.)

Summary: This is a vertical hyperbola. The center is at (-3, -1). a = 7, b = 2

Next, we can find the center point by identifying h and k. Remember h is with the x and k is with the y. We must take the opposite of each. So our center point is (-3, -1)

We can find a and b by taking the square root of the denominators. a = 7 and b = 2 (Remember a is always with the positive term.)

Summary: This is a vertical hyperbola. The center is at (-3, -1). a = 7, b = 2

**Practice:**Tell whether each parabola is vertical or horizontal. Then find the center point, a, and b.

**Answers:**1) vertical, center: (4, -5), a = 1, b = 4 2) horizontal, center: (7, -5), a = 3, b = 5 3) horizontal, center: (-2, -5), a = 7, b = 2 4) vertical, center: (1, 0) a = 10, b = 8 5) horizontal center: (3, -1), a = 9, b = 5

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