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How do you graph a hyperbola?

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Quick Answer

To graph a hyperbola, find and mark the center, calculate the conjugate and transverse axes, and draw the rectangle that helps you give your hyperbola the correct shape before drawing in the curves. Once graphed, a hyperbola looks like a pair of parabolas with the vertices facing each other.

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Full Answer

  1. Draw a point at the center of the hyperbola

    Consider the equation [(y-5)^2/25] - [(x+4)^2/16] = 1. Take the x-coordinate for the center from the opposite of the constant in the numerator of the x-term, or -4 in this example. Take the y-coordinate for the center from the opposite of the constant in the numerator of the y-term, or 5 in this example. Draw a point on graph paper at (-4, 5).

  2. Locate the conjugate and transverse axes

    Take the square root of the constant in the denominator of the y-term; in the case of the example, the square root of 25 is 5. Move from the vertex in a direction parallel to the y-axis 5 points up and down from the center of the hyperbola, and draw two more points. For the purposes of this example, these points should be at (1,5) and (-9,5). Return to the vertex, and move to the right and left in the same number as the square root of the denominator of the x-term. In this example, the square root of 16 is 4. Make a slight mark at these points for reference; they should be at (-4, 9) and (-4, 1).

  3. Draw a rectangle to set up the hyperbola

    Draw a rectangle that goes through the four points that you found in Step 2; each of the four points should be a midpoint on its own segment. Draw dotted lines across the rectangle, making an X by sending the arrows through the opposing corners of the rectangle. Sketch the curves so that they fit within the dotted lines, making sure the arrows on the end of the hyperbolic curves appear to approach the dotted lines without ever crossing them.

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