To shade quadratic inequalities, you first need to graph the quadratic function (parabola) and then determine which region satisfies the inequality. The type of line (solid or dashed) and the shading indicate the solution set.
Steps to Shade Quadratic Inequalities:
- Replace the inequality sign with an equals sign and graph the quadratic equation. This gives you the parabola, which serves as the boundary line.
- Determine whether the parabola should be solid or dashed:
- Solid Line: If the inequality is ≤ or ≥, the parabola is part of the solution set, so draw a solid line.
- Dashed Line: If the inequality is < or >, the parabola is not part of the solution set, so draw a dashed line.
- Choose a test point: Select a point that is not on the parabola. A common choice is (0,0), provided it doesn't lie on the parabola itself.
- Substitute the test point into the original inequality:
- If the inequality is true: Shade the region that contains the test point. This means all the points in that region satisfy the inequality.
- If the inequality is false: Shade the region that does not contain the test point. This means all the points in the other region satisfy the inequality.
Example:
Let's say we want to graph and shade the inequality y > x^2 - 2x -3.
- Graph the parabola
y = x^2 - 2x - 3. You can find the vertex, x-intercepts, and y-intercept to help you sketch it. The x-intercepts are at (-1,0) and (3,0), and the y-intercept is at (0,-3). The vertex is at (1,-4). - Determine the type of line: Since the inequality is
>, we use a dashed line. - Choose a test point: Let's use (0,0).
- Substitute:
0 > (0)^2 - 2(0) - 3which simplifies to0 > -3. - Evaluate: This is true, so we shade the region containing (0,0), which is the area above the parabola.
Therefore, the solution is the area above the dashed parabola.
