Regions between linear and quadratic boundaries
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| boundary/ˈbaʊndəri/ | biên |
Which points satisfy both boundaries?
- A point can lie above a parabola but still fail a second condition.
- This lesson studies boundary 边界: The line or curve separating points that satisfy an inequality from those that do not.
The method
- 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.
For y≥x²-2 AND y<x+4, why are the intersection points (-2,2) and (3,7) excluded?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A point must satisfy EVERY condition. The strict y<x+4 fails on the line itself, so both boundary meetings are excluded even though the solid parabola includes its own points.
A worked case
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.
Regions between linear and quadratic boundaries
Draw each equality boundary first
Match solution-set and boundary decisions to their conditions.
Find the left intersection x-coordinate for y=x²−2 and y=x+4.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Solve x²−x−6=0=(x+2)(x−3); the left root is −2.
Where it goes wrong
- 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.
Not true: Every intersection point is included if either boundary is solid. A point is included only if it satisfies every inequality, so a dashed boundary excludes it.
Find the right intersection x-coordinate.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The other root of x²−x−6=0 is 3.
Every intersection point is included if either boundary is solid.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A point is included only if it satisfies every inequality, so a dashed boundary excludes it.
Applying it
- 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.
At x=0, find the included lower y-bound of the combined region.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
At x=0 the parabola has y=−2; ≥ includes this lower boundary.
Match each boundary convention to its inequality.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The meeting x-values only give the horizontal extent of the region; membership of each point still needs both inequalities.
In your course
- AQA A-level Mathematics (7357), specification section B.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Boundary: The line or curve separating points that satisfy an inequality from those that do not.
- Draw each equality boundary first.
- Watch for: A dashed boundary is not part of the region, even where it meets a solid boundary.