📚 Functions Graph Symmetry Transformations | 函数图像的对称变换
In IB Mathematics, understanding how function graphs transform under symmetry operations is essential for analyzing complex functions without plotting every point. This guide covers all reflection-based transformations you need for both Analysis & Approaches and Applications & Interpretation.
在IB数学中，理解函数图像在对称变换下的变化规律，是高效分析复杂函数、避免逐点画图的核心技能。本指南涵盖分析与方法（AA）以及应用与解释（AI）两个方向所需的全部反射变换考点。
1. Symmetry About the y-Axis | 关于y轴对称（偶函数）
A graph is symmetric about the y-axis if reflecting it across the y-axis produces the original graph. The algebraic condition is f(−x) = f(x) for all x in the domain.
如果一个函数图像沿y轴反射后与原图像完全重合，则该图像关于y轴对称。其代数条件是：对于定义域内所有x，都有 f(−x) = f(x)。
Key point: Any point (x, y) on the graph maps to (−x, y) under this reflection.
要点：图像上的任意点 (x, y) 经过该反射后变为 (−x, y)。
Example: f(x) = x², f(x) = cos(x), f(x) = |x| are all even functions.
示例： f(x) = x²、f(x) = cos(x)、f(x) = |x| 均为偶函数。
2. Symmetry About the Origin | 关于原点对称（奇函数）
A graph is symmetric about the origin if rotating it 180° about the origin produces the original graph. The algebraic condition is f(−x) = −f(x) for all x in the domain.
如果一个函数图像绕原点旋转180°后与原图像完全重合，则该图像关于原点对称。其代数条件是：对于定义域内所有x，都有 f(−x) = −f(x)。
Key point: Any point (x, y) on the graph maps to (−x, −y) under this rotation-reflection.
要点：图像上的任意点 (x, y) 经过该中心对称变换后变为 (−x, −y)。
Example: f(x) = x³, f(x) = sin(x), f(x) = 1/x are all odd functions.
示例： f(x) = x³、f(x) = sin(x)、f(x) = 1/x 均为奇函数。
3. Reflection in the x-Axis: y = −f(x) | 关于x轴的反射：y = −f(x)
To reflect a function’s graph in the x-axis, multiply the entire function by −1. Every point (x, y) transforms to (x, −y).
要将函数图像关于x轴反射，只需将整个函数乘以−1。每个点 (x, y) 变为 (x, −y)。
Important properties:
- The x-intercepts remain unchanged, because y = 0 maps to y = 0.
- The y-intercept changes sign, from f(0) to −f(0).
- Maximum points become minimum points, and vice versa.
重要性质：
- x轴截距保持不变，因为 y = 0 反射后仍为 y = 0。
- y轴截距改变符号，从 f(0) 变为 −f(0)。
- 极大值点变为极小值点，反之亦然。
Example: If f(x) = x² − 4, then y = −f(x) = −x² + 4. The vertex moves from (0, −4) to (0, 4).
示例： 若 f(x) = x² − 4，则 y = −f(x) = −x² + 4。顶点从 (0, −4) 移动到 (0, 4)。
4. Reflection in the y-Axis: y = f(−x) | 关于y轴的反射：y = f(−x)
To reflect a function’s graph in the y-axis, replace every x with −x. Every point (x, y) transforms to (−x, y).
要将函数图像关于y轴反射，需将所有 x 替换为 −x。每个点 (x, y) 变为 (−x, y)。
Important properties:
- The y-intercept remains unchanged, because x = 0 maps to x = 0.
- All x-intercepts change sign unless they are at the origin.
- The domain is reflected: if the original domain is [a, b], the new domain is [−b, −a].
重要性质：
- y轴截距保持不变，因为 x = 0 反射后仍为 x = 0。
- 所有x轴截距改变符号，除非截距在原点。
- 定义域被反射：若原定义域为 [a, b]，新定义域为 [−b, −a]。
Example: If f(x) = 2ˣ, then y = f(−x) = 2⁻ˣ = (1/2)ˣ. The exponential growth curve becomes exponential decay.
示例： 若 f(x) = 2ˣ，则 y = f(−x) = 2⁻ˣ = (1/2)ˣ。指数增长曲线变为指数衰减曲线。
5. Reflection in Both Axes: y = −f(−x) | 关于两个轴的反射：y = −f(−x)
Reflecting in both the x-axis and y-axis is equivalent to a 180° rotation about the origin. The transformation is applied as y = −f(−x).
同时关于x轴和y轴进行反射，等价于绕原点旋转180°。变换形式为 y = −f(−x)。
Key point: Every point (x, y) maps to (−x, −y). This is the same as an odd-function symmetry, but applied to any function, not just odd ones.
要点：每个点 (x, y) 变为 (−x, −y)。这与奇函数的对称性相同，但适用于任意函数，而不仅仅是奇函数。
Example: If f(x) = x² + 1, then y = −f(−x) = −(x² + 1) = −x² − 1. The upward parabola becomes a downward parabola.
示例： 若 f(x) = x² + 1，则 y = −f(−x) = −(x² + 1) = −x² − 1。开口向上的抛物线变为开口向下的抛物线。
6. Order of Transformations | 变换的顺序
When multiple transformations are applied, the order matters. For reflection combined with translation, apply reflections before horizontal translations involving the x-term, but apply vertical translations after vertical reflections.
当多个变换同时应用时，顺序至关重要。对于反射与平移的组合，应先进行反射，再进行涉及x的水平平移；而垂直平移应在垂直反射之后进行。
Worked example: Starting from f(x) = x², obtain the graph of y = −(x + 2)² + 3.
- Step 1: Reflect in the x-axis: y = −x²
- Step 2: Shift left by 2: y = −(x + 2)²
- Step 3: Shift up by 3: y = −(x + 2)² + 3
例题： 从 f(x) = x² 出发，得到 y = −(x + 2)² + 3 的图像。
- 第一步：关于x轴反射：y = −x²
- 第二步：向左平移2个单位：y = −(x + 2)²
- 第三步：向上平移3个单位：y = −(x + 2)² + 3
Table: Standard reflection rules
| Transformation | New Function | Coordinate Change |
|---|---|---|
| Reflection in x-axis | y = −f(x) | (x, y) → (x, −y) |
| Reflection in y-axis | y = f(−x) | (x, y) → (−x, y) |
| Reflection in origin | y = −f(−x) | (x, y) → (−x, −y) |
表格：标准反射规则
| 变换类型 | 新函数 | 坐标变化 |
|---|---|---|
| 关于x轴反射 | y = −f(x) | (x, y) → (x, −y) |
| 关于y轴反射 | y = f(−x) | (x, y) → (−x, y) |
| 关于原点反射 | y = −f(−x) | (x, y) → (−x, −y) |
7. Symmetry of Inverse Functions | 反函数的对称性
The graph of an inverse function f⁻¹(x) is the reflection of the graph of f(x) across the line y = x. This is a diagonal reflection, distinct from the axis reflections above.
反函数 f⁻¹(x) 的图像是 f(x) 的图像关于直线 y = x 的反射结果。这是一种斜向反射，与上述坐标轴反射不同。
Coordinate rule: Every point (a, b) on y = f(x) corresponds to point (b, a) on y = f⁻¹(x).
坐标规则： y = f(x) 上的每个点 (a, b) 对应 y = f⁻¹(x) 上的点 (b, a)。
Example: For f(x) = 2x + 1, the inverse is f⁻¹(x) = (x − 1)/2. The point (1, 3) on f corresponds to (3, 1) on f⁻¹.
示例： 对于 f(x) = 2x + 1，其反函数为 f⁻¹(x) = (x − 1)/2。f 上的点 (1, 3) 对应 f⁻¹ 上的点 (3, 1)。
8. Identifying Symmetry from Equations | 从方程判断对称性
On IB exams, you may be asked to determine whether a function is even, odd, or neither. Use these algebraic tests:
在IB考试中，你可能会被要求判断一个函数是偶函数、奇函数还是两者都不是。使用以下代数测试：
- Even: f(−x) = f(x). Example: f(x) = x⁴ + x². Check: f(−x) = x⁴ + x² = f(x). ✓
- Odd: f(−x) = −f(x). Example: f(x) = x³ + x. Check: f(−x) = −x³ − x = −(x³ + x) = −f(x). ✓
- Neither: f(−x) ≠ f(x) and f(−x) ≠ −f(x). Example: f(x) = x² + x.
- 偶函数： f(−x) = f(x)。例如：f(x) = x⁴ + x²。验证：f(−x) = x⁴ + x² = f(x)。 ✓
- 奇函数： f(−x) = −f(x)。例如：f(x) = x³ + x。验证：f(−x) = −x³ − x = −(x³ + x) = −f(x)。 ✓
- 非奇非偶： f(−x) ≠ f(x) 且 f(−x) ≠ −f(x)。例如：f(x) = x² + x。
Useful theorem: Any function can be written as the sum of an even and an odd function:
f(x) = [f(x) + f(−x)]/2 + [f(x) − f(−x)]/2
实用定理： 任何函数都可以写成一个偶函数与一个奇函数之和：
f(x) = [f(x) + f(−x)]/2 + [f(x) − f(−x)]/2
9. Even and Odd Products | 奇偶函数的乘积性质
When multiplying functions, the symmetry of the product follows simple rules. These are frequently tested in IB paper 1 questions.
当函数相乘时，乘积的对称性遵循简单规则。这些规则在IB纸笔考试中经常出现。
- even × even = even
- odd × odd = even
- even × odd = odd
- 偶 × 偶 = 偶
- 奇 × 奇 = 偶
- 偶 × 奇 = 奇
Example: f(x) = x² (even) and g(x) = sin(x) (odd). Their product h(x) = x²·sin(x) is odd, so h(−x) = −h(x).
示例： f(x) = x²（偶函数）和 g(x) = sin(x)（奇函数）。它们的乘积 h(x) = x²·sin(x) 是奇函数，因此 h(−x) = −h(x)。
Integration shortcut: For definite integrals over symmetric intervals [−a, a]:
- If f is even: ∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx
- If f is odd: ∫₋ₐᵃ f(x)dx = 0
积分快捷技巧： 对于对称区间 [−a, a] 上的定积分：
- 若 f 为偶函数：∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx
- 若 f 为奇函数：∫₋ₐᵃ f(x)dx = 0
10. Common IB Exam Pitfalls | 常见IB考试陷阱
Students frequently lose marks on symmetry transformations due to avoidable errors. Be aware of these common mistakes:
学生常在对称变换题目中因可避免的错误而失分。请注意以下常见错误：
- Mistake 1: Confusing y = f(−x) with y = −f(x). Remember: f(−x) reflects horizontally; −f(x) reflects vertically.
- Mistake 2: Forgetting that reflecting in the y-axis reverses the domain.
- Mistake 3: Applying reflections after translations in the wrong order. Always apply reflections to the base function first when the transformation is written in the form y = −f(x + a).
- Mistake 4: Assuming all symmetric graphs are either even or odd. A graph can be symmetric about the y-axis but have no defined odd/even classification if the domain is not symmetric.
- 错误1： 混淆 y = f(−x) 与 y = −f(x)。记住：f(−x) 是水平反射；−f(x) 是垂直反射。
- 错误2： 忘记关于y轴反射会反转定义域。
- 错误3： 在平移之后才进行反射，顺序错误。当变换写成 y = −f(x + a) 形式时，务必先对基础函数进行反射。
- 错误4： 假设所有对称图像必然是偶函数或奇函数。如果定义域不对称，图像可以关于y轴对称但不属于偶函数或奇函数分类。
11. Worked Exam Question | 考试真题演练
Question: The function f is defined by f(x) = x³ − 3x for x ∈ ℝ. The function g is obtained by reflecting f in the y-axis followed by reflecting in the x-axis. Find g(x) and describe the transformation.
题目： 函数 f 定义为 f(x) = x³ − 3x，x ∈ ℝ。函数 g 是将 f 先关于y轴反射，再关于x轴反射得到的。求 g(x) 并描述该变换。
Solution:
Step 1 — Reflect in y-axis: replace x with −x.
y = f(−x) = (−x)³ − 3(−x) = −x³ + 3x
Step 2 — Reflect in x-axis: multiply by −1.
g(x) = −(−x³ + 3x) = x³ − 3x
Answer: g(x) = f(x). The composition of two reflections equals a 180° rotation about the origin, but since f is an odd function (f(−x) = −f(x)), the function remains unchanged.
解答：
第一步——关于y轴反射：将 x 替换为 −x。
y = f(−x) = (−x)³ − 3(−x) = −x³ + 3x
第二步——关于x轴反射：乘以 −1。
g(x) = −(−x³ + 3x) = x³ − 3x
答案： g(x) = f(x)。两个反射的复合等价于绕原点旋转180°，但由于 f 是奇函数（f(−x) = −f(x)），函数保持不变。
12. Summary of Symmetry Transformations | 对称变换总结
Mastering symmetry transformations requires understanding both the algebraic rules and geometric intuition. Below is the complete summary for quick revision:
掌握对称变换需要同时理解代数规则与几何直觉。以下是快速复习的完整总结：
| Operation | 操作 | Function | 函数 | Effect | 效果 |
|---|---|---|
| Reflect in x-axis | 关于x轴反射 | y = −f(x) | Flips the graph vertically | 图像上下翻转 |
| Reflect in y-axis | 关于y轴反射 | y = f(−x) | Flips the graph horizontally | 图像左右翻转 |
| Reflect in origin | 关于原点反射 | y = −f(−x) | Rotates the graph 180° | 图像旋转180° |
| Reflect in y = x | 关于y = x反射 | y = f⁻¹(x) | Produces the inverse graph | 得到反函数图像 |
Remember the coordinate mappings for each transformation, and always verify your result by testing a few key points, including intercepts and vertices.
牢记每种变换的坐标映射规律，并始终通过测试几个关键点（包括截距和顶点）来验证结果。
Final tip: For composite functions like y = 3f(−2x) + 1, the reflection in the y-axis (due to −2x) affects the horizontal coordinates only, and the reflection in the x-axis (due to the − sign if present) affects the vertical coordinates. Separate the operations to avoid confusion.
最后提示： 对于复合函数如 y = 3f(−2x) + 1，由 −2x 引起的y轴反射仅影响水平坐标；由负号（若存在）引起的x轴反射仅影响垂直坐标。分开处理各项操作以避免混淆。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导