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# How is the equation for a hyperbola determined?

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Equations for hyperbolas are first found by using point-slope form to determine the asymptotes of lines, which form the center of the hyperbola. Hyperbolas exist in two basic shapes, which are opening horizontally from left and right, or vertically by opening up and down. The type of hyperbola graphed depends on the difference between forms, which is either negative or positive.

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The process of finding equations for hyperbolas begins with drawing a basic sketch of the hyperbolas. The value of the Y-term determines the shape of the hyperbola: if the term is negative and has a minus sign, the hyperbola will open to the left or right. If it is positive, however, the hyperbola opens on the vertical axis, and faces up or down. When finding the equation, it is best to begin by finding the center, as the rest of the graph builds around that value. Once the center is determined, the vertices can be determined, followed by the slopes of asymptotes. The asymptotes equate to the square root of the number listed as the Y-term. This number can then be divided by the square root of the term X. The numbers of the two terms always exist as a positive and negative, which form the asymptotes and allow for the graph to be sketched.

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