📚 Graphs 7: Transformations of Graphs | 图形变换：平移、反射与缩放
In this revision article, we explore the seventh key topic in the IGCSE Graphs series: transformations of graphs. You will learn how to sketch and interpret the effect of translations, reflections and stretches on a given function, which is a core skill for the Edexcel IGCSE Mathematics examination.
在本文中，我们探讨 IGCSE 图形系列中的第七个关键主题：图形变换。你将学习如何绘制和解释平移、反射与缩放对给定函数的影响，这是 Edexcel IGCSE 数学考试的核心技能。
1. What Is a Graph Transformation? | 什么是图形变换？
A graph transformation is a rule that changes the graph of a function into a new graph. For a given function y = f(x), we can apply algebraic changes to f(x) or to x to translate, reflect or stretch the graph.
图形变换是一种将函数图形变为新图形的规则。对于给定函数 y = f(x)，我们可以对 f(x) 或 x 进行代数改变，从而平移、反射或缩放图形。
There are three main families of transformations: translations, which move the graph without changing its shape; reflections, which flip the graph over a line; and stretches or compressions, which change the graph’s size in the x- or y-direction.
变换主要有三大类：平移——移动图形而不改变形状；反射——将图形沿一条线翻转；伸缩——在 x 方向或 y 方向改变图形的大小。
Every transformation can be understood by tracking what happens to a general point (x, y) on the original curve. This point-based view is often the clearest way to avoid mistakes.
每种变换都可以通过跟踪原曲线上一个一般点 (x, y) 的变化来理解。基于点的视角通常是避免错误的最清晰方法。
2. Vertical Translation: y = f(x) + a | 垂直平移：y = f(x) + a
Adding a positive constant a to the function moves the graph upwards by a units; subtracting a positive constant moves it downwards. Every point (x, y) on the original graph becomes (x, y + a) on the new graph.
若给函数加上正常数 a，图形会向上平移 a 个单位；减去正常数则向下移动。原图形上每个点 (x, y) 都会变为新图形上的 (x, y + a)。
For example, take f(x) = x². The graph of y = f(x) + 2 is y = x² + 2. Its vertex moves from (0, 0) to (0, 2), and the parabola keeps the same width and orientation.
例如，取 f(x) = x²。函数 y = f(x) + 2 即 y = x² + 2。它的顶点从 (0, 0) 移到 (0, 2)，抛物线宽度和开口方向保持不变。
y = f(x) + a ⇒ (x, y) → (x, y + a)
In an exam, avoid the common error of thinking that f(x) + a shifts the graph horizontally. The constant outside f(x) only affects the y-coordinate.
在考试中，要避免把 f(x) + a 误认为水平平移。f(x) 外部的常数只影响 y 坐标。
3. Horizontal Translation: y = f(x − a) | 水平平移：y = f(x − a)
For y = f(x − a), the graph shifts to the right by a units when a > 0. If a < 0, the graph shifts to the left by |a| units. This is because we replace x by x − a inside the function.
对于 y = f(x − a)，当 a > 0 时图形向右平移 a 个单位；当 a < 0 时图形向左平移 |a| 个单位。这是因为我们在函数内部用 x − a 替换了 x。
Every original point (x, y) becomes (x + a, y). Notice that the y-coordinate does not change; only the horizontal position changes.
原图形上每个点 (x, y) 变为 (x + a, y)。注意 y 坐标不变，只有水平位置改变。
For example, if f(x) = x², then the graph of y = f(x − 3) = (x − 3)² is the original parabola shifted 3 units to the right, with vertex at (3, 0).
例如，若 f(x) = x²，那么 y = f(x − 3) = (x − 3)² 的图形是原抛物线向右平移 3 个单位，顶点在 (3, 0)。
y = f(x − a) ⇒ (x, y) → (x + a, y)
Many students memorise the phrase ‘f(x − a) shifts right’ but forget that for y = f(x + a) the shift is to the left. Always test with a simple point such as the vertex or y-intercept.
许多学生记住“f(x − a) 右移”，但忘记 y = f(x + a) 是左移。请始终用一个简单点（如顶点或 y 截距）来检验。
4. Reflection in the x-axis: y = −f(x) | 关于 x 轴的反射：y = −f(x)
Multiplying f(x) by −1 reflects the graph in the x-axis. Every point (x, y) becomes (x, −y). A point above the x-axis moves to a position the same distance below the x-axis.
将 f(x) 乘以 −1，图形关于 x 轴反射。每个点 (x, y) 变为 (x, −y)。x 轴上方的点会移动到距离 x 轴同样远的下方位置。
For example, f(x) = x² − 1 has a minimum point at (0, −1). The reflected graph y = −f(x) = −x² + 1 has a maximum point at (0, 1).
例如，f(x) = x² − 1 在 (0, −1) 处有最小值点。反射后的图形 y = −f(x) = −x² + 1 在 (0, 1) 处有最大值点。
y = −f(x) ⇒ (x, y) → (x, −y)
This transformation preserves the x-coordinates of all intersection points with the x-axis, because if y = 0 then −y = 0. These invariant points are useful for checking your sketch.
此变换保持所有与 x 轴交点的 x 坐标不变，因为若 y = 0，则 −y = 0。这些不变点可用于检查你的草图。
5. Reflection in the y-axis: y = f(−x) | 关于 y 轴的反射：y = f(−x)
Replacing x by −x reflects the graph in the y-axis. Every point (x, y) becomes (−x, y). The graph is mirrored horizontally around the y-axis.
用 −x 替换 x，图形关于 y 轴反射。每个点 (x, y) 变为 (−x, y)。图形绕 y 轴水平镜像。
For example, the straight line f(x) = 2x + 1 becomes f(−x) = −2x + 1. The slope changes from +2 to −2, but the y-intercept remains at 1.
例如，直线 f(x) = 2x + 1 变为 f(−x) = −2x + 1。斜率从 +2 变为 −2，但 y 截距仍为 1。
y = f(−x) ⇒ (x, y) → (−x, y)
If the original graph is symmetric about the y-axis, such as y = x², then the reflection leaves the graph unchanged. Such functions are called even functions.
如果原图形关于 y 轴对称，例如 y = x²，那么反射后图形不变。这类函数称为偶函数。
6. Vertical Stretch: y = a f(x) | 垂直伸缩：y = a f(x)
Multiplying f(x) by a positive constant a scales the graph vertically. If a > 1, the graph stretches away from the x-axis; if 0 < a < 1, it compresses towards the x-axis.
将 f(x) 乘以正常数 a，会对图形进行垂直缩放。若 a > 1，图形远离 x 轴伸展；若 0 < a < 1，图形向 x 轴压缩。
Every point (x, y) becomes (x, ay). For example, y = 2x² is the graph of y = x² stretched vertically by factor 2, so the point (1, 1) moves to (1, 2).
每个点 (x, y) 变为 (x, ay)。例如，y = 2x² 是 y = x² 垂直拉伸 2 倍的图形，所以点 (1, 1) 移动到 (1, 2)。
y = a f(x) ⇒ (x, y) → (x, ay)
Be careful: a negative value of a would combine a vertical stretch with a reflection in the x-axis. In the IGCSE course, you should handle negative multipliers as ‘reflection plus stretch’.
注意：a 为负值时，相当于垂直伸缩与 x 轴反射的组合。在 IGCSE 课程中，应将负乘数视为“反射加伸缩”。
7. Horizontal Stretch: y = f(ax) | 水平伸缩：y = f(ax)
Replacing x by ax, where a > 0, stretches or compresses the graph horizontally. If a > 1, the graph is compressed towards the y-axis; if 0 < a < 1, it stretches away from the y-axis.
当 a > 0 时，用 ax 替换 x 会水平伸缩图形。若 a > 1，图形向 y 轴压缩；若 0 < a < 1，图形远离 y 轴伸展。
Every point (x, y) becomes (x/a, y). This is the opposite of what many students expect: dividing the x-coordinate by a means the graph becomes narrower when a is large.
每个点 (x, y) 变为 (x/a, y)。这与许多学生的直觉相反：当 a 较大时，x 坐标除以 a 使得图形变得更窄。
For example, y = f(2x) compresses y = sin(x) horizontally by factor 2, so the period changes from 360° to 180°.
例如，y = f(2x) 将 y = sin(x) 水平压缩 2 倍，周期从 360° 变为 180°。
y = f(ax) ⇒ (x, y) → (x/a, y)
Note that horizontal stretches are often combined with translations inside the bracket, for example y = f(2x + 1). In that case it is safer to rewrite f(2x + 1) as f(2(x + 1/2)) before describing the transformations.
注意，水平伸缩常与括号内的平移组合，例如 y = f(2x + 1)。此时应先将 f(2x + 1) 改写为 f(2(x + 1/2))，再描述变换。
8. Combining Transformations: Order Matters | 组合变换：顺序很重要
When applying more than one transformation, the order can change the final graph. For example, y = 2f(x + 1) means first shift the graph left by 1 unit, then stretch vertically by factor 2. The horizontal shift happens before the vertical stretch.
当应用多个变换时，顺序会影响最终图形。例如，y = 2f(x + 1) 表示先向左平移 1 个单位，再垂直拉伸 2 倍。水平平移发生在垂直伸缩之前。
A useful rule is: deal with transformations inside the brackets first, then outside. Inside the bracket, horizontal translations and stretches affect x; outside, vertical stretches and translations affect y.
一条有用的规则是：先处理括号内的变换，再处理括号外的变换。括号内是水平平移和伸缩，影响 x；括号外是垂直伸缩和平移，影响 y。
For example, transform f(x) = x² to y = −f(x − 1) + 2. First shift right by 1: (x − 1)². Then reflect in the x-axis: −(x − 1)². Finally shift up by 2: −(x − 1)² + 2.
例如，将 f(x) = x² 变换为 y = −f(x − 1) + 2。先右移 1：得到 (x − 1)²；再关于 x 轴反射：得到 −(x − 1)²；最后上移 2：得到 −(x − 1)² + 2。
y = a f(x − b) + c ⇒ horizontal shift b, then vertical stretch a, then vertical shift c
If you have more than one horizontal transformation inside the bracket, always rewrite the bracket in the form f(k(x + b)) before applying the transformations. This avoids the classic mistake of misreading the horizontal shift.
如果括号内有多个水平变换，一定要先把括号改写成 f(k(x + b)) 的形式，再进行变换。这样可以避免误读水平平移的经典错误。
9. Identifying Transformations from a Graph | 从图形识别变换
In an exam question, you may be given a basic graph and a transformed graph and asked to describe the transformation. The most reliable method is to compare the coordinates of key points, such as vertices, intercepts, or turning points.
在考试题目中，你可能会得到一个基本图形和一个变换后的图形，并要求描述变换。最可靠的方法是比较关键点的坐标，例如顶点、截距或转向点。
Suppose the basic graph is y = x² and the transformed graph has vertex at (2, 3). The vertex moved from (0, 0) to (2, 3), so the transformation is a translation by vector (2, 3), giving y = (x − 2)² + 3.
假设基本图形是 y = x²，变换后图形顶点在 (2, 3)。顶点从 (0, 0) 移动到 (2, 3)，因此变换是向量 (2, 3) 的平移，得到 y = (x − 2)² + 3。
If the basic graph is y = x³ and the transformed graph opens downward, the multiplier of f(x) must be negative, indicating a reflection in the x-axis. Check one additional point to confirm whether a stretch is also present.
如果基本图形是 y = x³，而变换后开口向下，则 f(x) 的乘数必为负，说明关于 x 轴发生了反射。再检查一个额外点，以确认是否还叠加了伸缩。
To identify a stretch, compare the y-coordinates of a nontrivial point. If the original point (1, 1) becomes (1, 3) on the new graph, the vertical stretch factor is 3. If the point (2, 4) becomes (1, 4), the horizontal compression factor is 2.
要识别伸缩，可比较非平凡点的 y 坐标。如果原图上点 (1, 1) 变为 (1, 3)，则垂直伸缩因子为 3。如果点 (2, 4) 变为 (1, 4)，则水平压缩因子为 2。
10. Exam Tips and Common Mistakes | 考试技巧与常见错误
Here is a list of frequent errors students make with graph transformations, together with advice to avoid them.
以下是学生在图形变换中常犯的错误列表，以及避免这些错误的建议。
-
Mistake: Believing y = f(x + a) shifts the graph to the right. Correction: It actually shifts it to the left. Use a test point such as the y-intercept.
错误：认为 y = f(x + a) 向右平移。纠正：它实际上向左平移。可测试一个点，例如 y 截距。
-
Mistake: Confusing horizontal and vertical stretches. Correction: y = f(ax) affects x-coordinates by factor 1/a; y = a f(x) affects y-coordinates by factor a.
错误：混淆水平伸缩和垂直伸缩。纠正：y = f(ax) 以 1/a 因子影响 x 坐标；y = a f(x) 以 a 因子影响 y 坐标。
-
Mistake: Forgetting the negative sign when reflecting in the x-axis. Correction: Always write y = −f(x) explicitly before sketching.
错误：关于 x 轴反射时忘记负号。纠正：画图前总是明确写出 y = −f(x)。
-
Mistake: Applying transformations in the wrong order. Correction: For y = a f(x − b) + c, work from
Published by TutorHao | IGCSE 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