Graph solutions to quadratic inequalities

Learning Objectives Identify the boundary parabola for a given quadratic inequality. Determine whether the boundary parabola should be a solid or dashed curve. Calculate the roots and vertex of the boundary parabola to accurately plot it. Use a test point to determine the correct region to shade on the coordinate plane. Graphically represent the complete solution set for a two-variable quadratic inequality. Interpret the shaded region as the set of all ordered pairs (x, y) that satisfy the inequality. A rocket's height is modeled by a quadratic function. How can we graph all the moments in time and heights where the rocket is *above* a certain altitude? 🚀 This tutorial builds on your knowledge of graphing parabolas and solving linear inequalities. You will learn how t...

TermDefinitionExample Quadratic InequalityAn inequality that involves a quadratic expression. In two variables, it takes the form y < ax² + bx + c, y > ax² + bx + c, y ≤ ax² + bx + c, or y ≥ ax² + bx + c.y ≥ x² - 2x - 3 is a quadratic inequality. The solution is the set of all points (x, y) that make the statement true. Boundary CurveThe parabola y = ax² + bx + c that separates the coordinate plane into two regions: one where the inequality is true and one where it is false.For the inequality y < 2x² + 1, the boundary curve is the parabola y = 2x² + 1. Solid vs. Dashed BoundaryA solid boundary curve is used for inequalities with ≤ or ≥, indicating that points on the parabola are included in the solution. A dashed boundary curve is used for < or >, indicating that points on...

Standard Form of a Quadratic Inequality y \ [inequality symbol] \ ax^2 + bx + c Where the inequality symbol can be >, <, ≥, or ≤. This form helps identify the coefficients a, b, and c needed to analyze and graph the boundary parabola. Quadratic Formula for Roots x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} Used to find the x-intercepts (roots) of the boundary parabola y = ax² + bx + c. These points are critical for accurately sketching the curve. Vertex Formula x_{vertex} = -\frac{b}{2a} Used to find the x-coordinate of the vertex of the parabola. Substitute this x-value back into the equation y = ax² + bx + c to find the y-coordinate, giving the vertex (h, k).