Reflections, rotations and enlargements · Higher
導入済み| English | 意味 |
|---|---|
| centre of enlargement/ˈsentə ɒv enˈlɑːdʒmənt/ | 拡大・縮小の中心 |
Where is the enlargement centre?
- A logo is enlarged around a point away from the origin. Multiplying its coordinates alone puts it in the wrong place.
- This lesson studies centre of enlargement 位似中心: The point from which each point's displacement is multiplied by a scale factor.
Choose the mathematical structure
- A translation adds a vector. A reflection reverses signed perpendicular distance from a mirror line. A rotation needs a centre, angle and direction. For enlargement from C, use new P=C+k(P-C).
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
For an enlargement with centre C=(1,1) and factor 2, why is the image of P=(3,2) found from P-C first?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Enlargement scales the vector from the centre to each point. P-C=(2,1) doubles to (4,2), and adding back the centre gives (5,3). Scaling P directly would wrongly assume the origin as centre.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For C=(1,1),P=(3,2),k=2, the image is C+2(P-C)=(5,3). With k=1/2 it is (2,1.5); with k=-2 it is (-3,-1), on the opposite side of the centre. Reflecting (3,2) in the y-axis gives (-3,2), and a 90° anticlockwise rotation about the origin gives (-2,3). Identify centre, factor, line, angle and direction as applicable.
Reflections, rotations and enlargements
A translation adds a vector
Explain why scaling P directly would give a different enlargement.
For C=(1,1),P=(3,2),k=2, find the image x-coordinate.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Image=C+2(P-C). Its x-coordinate is 1+2(3-1)=5.
Test a tempting shortcut
- A rotation needs its centre and direction, not just an angle. A negative enlargement factor places the image on the opposite side of the centre. A translation does not change orientation or size.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every enlargement is centred at the origin unless its scale factor is negative. This claim is false. Explain which definition or assumption it violates.
For that enlargement, find the image y-coordinate.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Its y-coordinate is 1+2(2-1)=3.
Every enlargement is centred at the origin unless its scale factor is negative.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
A rotation needs its centre and direction, not just an angle. A negative enlargement factor places the image on the opposite side of the centre. A translation does not change orientation or size.
Interpret a new situation
- Describe a transformation completely before constructing the image. Check corresponding distances and angles. For combined transformations, apply them in the stated order; they usually do not commute.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Reflect (3,2) in the y-axis. Find the new x-coordinate.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Reflection in the vertical axis negates x: 3 becomes -3.
Match each transformation to its defining data.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Each transformation is fully specified only by its own data; an angle without a centre or a factor without a centre is ambiguous.
Use this in your course
- 8300 · Higher · 3.4. 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.
The point from which each point's displacement is multiplied by a scale factor. Choose the relationship, show the method, check its assumptions and interpret the result.