Regions between linear and quadratic boundaries

A point can lie above a parabola but still fail a second condition. Which points satisfy both boundaries?

• A point can lie above a parabola but still fail a second condition. Which points satisfy both boundaries?
• This lesson studies boundary 边界: The line or curve separating points that satisfy an inequality from those that do not.

Choose the mathematical structure

• Draw each equality boundary first. Use a solid line or curve for ≤ or ≥ and a dashed one for < or >. For y greater than a function, shade above its graph; for y less than it, shade below. For AND, keep only the overlap. Test a point away from every boundary to confirm the chosen side.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

$$x^2-2\le y

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Example

For y≥x²−2 AND y<x+4, draw the parabola solid and the line dashed. The boundaries meet when x²−2=x+4: (x−3)(x+2)=0, giving (−2,2) and (3,7). Between −2<x<3, the line is above the parabola. The allowed points satisfy x²−2≤y<x+4. At x=0, the vertical interval is −2≤y<4. The point (0,0) is included, (0,4) is excluded and (0,−3) is excluded. The intersection points are excluded because the line inequality is strict.

Test a tempting shortcut

• A dashed boundary is not part of the region, even where it meets a solid boundary. Solving for intersection x-values gives the horizontal extent, not all allowed coordinates. A region for OR is the union, which can be much larger than the overlap.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• For a different inequality such as y>x²−2, the permitted side is above the parabola and the curve itself is excluded. If a context also requires x≥0, clip the already combined region to that half-plane. Verify an interior point and a point outside each boundary separately.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• 7357 · A-level · B. Match the target tier and specification before assigning extensions.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.