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.
放大中心在哪里？
- 一个标志围绕原点以外的某点进行放大。仅对其坐标进行乘法运算会导致其位置错误。
- 本课学习位似中心：从该点出发，每个点的位移均乘以一个比例因子。
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.
选择数学结构
- 平移增加一个向量。反射反转相对于镜面线的有符号垂直距离。旋转需要中心、角度和方向。对于以C为中心的放大变换，使用新坐标P=C+k(P-C)。
- 计算前请先明确允许的输入项和单位。方程应表达关系本身，而不仅仅是记录计算器按键过程。
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, 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).
通过验证案例进行推导
- 将结果与初始量进行比对。代入原始关系式，或在适当情况下对比图表与数值答案。
当C=(1,1), P=(3,2), k=2时，P-C=(2,1)，因此新P=(1,1)+2(2,1)=(5,3)。点(3,2)关于y轴的反射结果为(-3,2)。绕原点逆时针旋转90°后得到(-2,3)。
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.
检验一个诱人的捷径
- 旋转不仅需要角度，还需要中心和方向。负的放大倍数会使像位于中心的另一侧。平移不会改变方向或大小。
- 当捷径失效时，找出其违背的假设。保留精确值直到题目要求的最终舍入步骤。
“除非比例因子为负，否则所有放大变换均以原点为中心。”此说法错误。请说明它违反了哪一定义或假设。
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.
解读新情境
- 在作图前完整描述变换过程。检查对应距离和角度。对于复合变换，按指定顺序依次应用；它们通常不满足交换律。
- 完整解答需给出数学结果并解释其含义。检查其在所述背景下是否可行。
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
- 4MA1 · Higher · 5. 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.
将其应用于你的课程
- 4MA1 · 高级 · 5. 在分配拓展内容前，请匹配目标层级和教学大纲要求。
- 在给出最终答案前先展示解题方法，并遵循试卷的计算器及公式使用规则。通过定位第一个无效步骤来复盘错误答案。
从该点出发，每个点的位移均乘以一个比例因子。选择合适的关系式，展示计算步骤，验证其适用假设并解释结果。