Reflections, rotations and enlargements
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| centre of enlargement/ˈsentə ɒv enˈlɑːdʒmənt/ | tâm co dãn |
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.
The method
- 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).
For an enlargement with centre C=(1,1) and factor 2, why is the image of P=(3,2) found from P-C first?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
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.
A worked case
For C=(1,1),P=(3,2),k=2, P-C=(2,1), so new P=(1,1)+2(2,1)=(5,3). Reflection of (3,2) in the y-axis gives (-3,2). A 90° anticlockwise rotation about the origin gives (-2,3).
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.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Image=C+2(P-C). Its x-coordinate is 1+2(3-1)=5.
Where it goes wrong
- 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.
Not true: Every enlargement is centred at the origin unless its scale factor is negative. An enlargement can have any centre; a negative scale factor puts the image on the other side of the centre.
For that enlargement, find the image y-coordinate.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Its y-coordinate is 1+2(2-1)=3.
Every enlargement is centred at the origin unless its scale factor is negative.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
An enlargement can have any centre; a negative scale factor puts the image on the other side of the centre.
Applying it
- 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.
Reflect (3,2) in the y-axis. Find the new x-coordinate.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Reflection in the vertical axis negates x: 3 becomes -3.
Match each transformation to its defining data.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Each transformation is fully specified only by its own data; an angle without a centre or a factor without a centre is ambiguous.
In your course
- AQA A-level Mathematics (7357), specification section J.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Centre of enlargement: The point from which each point's displacement is multiplied by a scale factor.
- A translation adds a vector.
- Watch for: A rotation needs its centre and direction, not just an angle.