Reflections, rotations and enlargements · Higher · 反射、旋转与位似 · 进阶
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| centre of enlargement/ˈsentə ɒv enˈlɑːdʒmənt/ | 位似中心 | wèi shì zhōng xīn |
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.
Which description correctly defines centre of enlargement? · 哪个描述正确定义了位似中心？
The point from which each point's displacement is multiplied by a scale factor. · 每个点的位移按缩放因子进行乘法的原点。
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 · 反射、旋转和位似变换。
A translation adds a vector · 平移是加上一个向量。
Explain why scaling P directly would give a different enlargement. · 解释为什么直接对P进行缩放会得到不同的放大效果。
For C=(1,1),P=(3,2),k=2, find the image x-coordinate. · 对于C=(1,1)，P=(3,2)，k=2，求像的x坐标。
Image=C+2(P-C). Its x-coordinate is 1+2(3-1)=5. · 像=C+2(P-C)。其x坐标为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. · 对于该位似变换，求像的y坐标。
Its y-coordinate is 1+2(2-1)=3. · 其y坐标为1+2(2-1)=3。
Every enlargement is centred at the origin unless its scale factor is negative. · 除非其比例因子为负数，否则每个位似变换都以原点为中心。
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. · 旋转不仅需要角度，还需要中心和方向。负的位似因子会使像位于中心的另一侧。平移不会改变方向或大小。
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. · 将(3,2)关于y轴反射。求新的x坐标。
Reflection in the vertical axis negates x: 3 becomes -3. · 关于垂直轴的反射使x取反：3变为-3。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
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.