Graphing Quadratic Inequalities Calculator

To graph a quadratic inequality of the form y > ax² + bx + c (or with <, ≥, ≤), you first treat the inequality as an equation to find the boundary parabola. The core calculation begins with determining the discriminant, denoted as Delta (Δ), using the formula Δ = b² - 4ac. Next, the calculator solves for the roots using the standard quadratic formula to find where the parabola crosses the x-axis. Depending on whether the inequality sign is strict (<, >) or inclusive (≤, ≥), the boundary is drawn with either a dashed or solid line. Finally, test points are evaluated to determine whether to shade the interior or exterior region of the parabola.

Let us walk through graphing the quadratic inequality y < 2x² - 4x - 6. Here, our coefficients are a = 2, b = -4, and c = -6. First, we compute the discriminant: Δ = (-4)² - 4(2)(-6) = 16 + 48 = 64. Because the discriminant is positive, the parabola has two distinct x-intercepts. Applying the quadratic formula, we find the roots: Root 1 = 3 and Root 2 = -1. The vertex x-coordinate is found using -b / (2a), which equals 4 / 4 = 1, placing the vertex at (1, -8). Because the inequality uses a strict 'less than' (<) symbol, we draw a dashed parabola opening upwards and shade the region beneath the curve.

Always double-check your inequality sign before sketching your boundary curve; strict inequalities require dashed lines, while inclusive ones require solid lines. When testing regions to determine where to shade, always pick a test point that does not lie directly on the parabola, with (0,0) being the easiest choice whenever possible. Lastly, pay close attention to the sign of coefficient 'a'—if it is negative, your parabola opens downward, which completely flips the interior and exterior shading rules.