How to Know the Direction of Your Parabola

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A parabola's direction refers to the way it opens. Parabolic equations can have two forms: the standard form and the vertex form. Both forms can have either x or y as the input variable. As examples, the standard form with y as an input would be x = a y^2 + b y + c, and the vertex form with x as an input would be y = a * (x - h)^2 - k. In both forms, the direction of the parabola's opening is determined by the combination of the a-coefficient at the beginning of the equations and the selected input, with an x-input giving a vertical, or up or down, opening and a y-input giving a horizontal, or right or left, opening.

Vertical

  • Obtain a parabolic equation for an example. With this example, let the equation be y = 6x^2 - 4x + 10.

  • Find the a-coefficient of the equation. In this example, the a-coefficient is 6.

  • Compare the a-coefficient to zero. If the value is greater than zero, the parabola opens upward, and if the value is less than zero, it opens downward. In this example, the a-coefficient is 6 and greater than zero, so the parabola opens upward.

Horizontal

  • Obtain a parabolic equation for example purposes. With this example, let the equation be x = -2 * (y + 2)^2 - 3.

  • Find the a-coefficient in the equation. In this example, the a-coefficient is -2.

  • Compare the a-coefficient to zero. If its value is greater than zero, the parabola opens to the right, and if its value is less than zero, it opens to the left. In this example, the a-coefficient of -2 is less than zero, so the parabola opens to the left.

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