How to Graph Quadratic Inequalities
Graphing a quadratic inequality involves transforming an algebraic relation of the form $y \lessgtr ax^2 + bx + c$ into a clear two-dimensional planar visualization consisting of a parabolic boundary curve and an infinite shaded solution set.
1. Identify the Parabolic Boundary Line
The first step is replacing the inequality operator with an equality:
y = ax^2 + bx + c
- If the inequality uses strict comparisons ($<$ or $>$), draw a dashed parabola to signal that points directly on the boundary are excluded from the solution set.
- If the inequality includes equality ($\le$ or $\ge$), draw a solid parabola indicating boundary inclusion.
2. Locate Key Geometric Coordinates
- Vertex $(h, k)$: The extremum of the parabola occurs at $h = -\frac{b}{2a}$, with $k = f(h)$.
- Axis of Symmetry: The vertical mirror line $x = h$.
- Roots / X-intercepts: Solve $ax^2 + bx + c = 0$ via the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
- Y-intercept: Evaluate at $x = 0$, giving coordinate $(0, c)$.
3. Determine Solution Shading via Test Points
Choose any point not located directly on the boundary curve. The origin $(0, 0)$ is the standard benchmark unless the curve passes through it:
- Substitute $(0, 0)$ into the original inequality: $0 > c$ (or whichever relation applies).
- If the inequality holds true, shade the entire half-plane containing $(0, 0)$.
- If false, shade the opposing region.
Quick Reference: Quadratic Boundary & Shading Rules
| Inequality Type | Boundary Curve Style | Vertex Direction ($a > 0$) | Shaded Region |
|---|---|---|---|
| $y > ax^2 + bx + c$ | Dashed | Opens Upward | Interior / Above curve |
| $y \ge ax^2 + bx + c$ | Solid | Opens Upward | Interior & On curve |
| $y < ax^2 + bx + c$ | Dashed | Opens Upward | Exterior / Below curve |
| $y \le ax^2 + bx + c$ | Solid | Opens Upward | Exterior & On curve |