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How to Graph Two Variable Quadratic Inequalities

Quadratic inequalities contain second-degree variables; two-variable inequalities include both x and y. Quadratic equations and inequalities produce parabolas when graphed. The generic form of the quadratic equation is y = ax^2 + bx + c. If a is negative, the parabola has the vertex on top and opens toward the bottom; if a is positive, it opens upward. The higher a is, the narrower the parabola will be. Finding the vertex and two other symmetrical points will help you graph the inequality.

Instructions

    • 1

      Find the vertex of the parabola using the equation. If the formula is y > 2x^2 - 5x + 3, and if the vertex is at the ordered pair (h,k), h = -b/2a. In the example, this would be 5/4.

      Then plug h in as the x-value to get the y-value, or k. This would be 2(5/4)^2 - 5(5/4) + 3, or 50/16 - 25/4 + 3, or 50/16 - 100/16 + 48/16, or -2/16, or -1/8.

      The vertex, then, is at (5/4, -1/8). Draw an empty circle at this point.

    • 2

      Pick two x-values that are equidistant from the vertex along the x-axis. The x-value of the vertex is 5/4, so choosing 0 and 5/2 would work. Plug both x-values in to get y-values. In the instance of 0, y > 2x^2 - 5x + 3 would give 3. In the instance of 5/2, 2(5/2)^2 - 5(5/2) + 3 = 50/4 - 50/4 + 3, or 3. Draw circles at (0,3) and (5/2, 3).

    • 3

      Draw a smooth curve through those three points, making a dotted line. Draw arrows on both ends to show that the parabola continues indefinitely. Shade in the area above the parabola.

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