📚 Example 4.5.1: Graph Transformations and Translations | 示例 4.5.1：图像变换与平移
of rules. Example 4.5.1 uses the general form y = f(x − a) + b to translate the graph of a basic cubic function, and then applies the same reasoning to intercepts, turning points and points of inflection.
图像变换是 AQA A-Level 数学中最容易稳定得分的知识点之一，因为几乎每道题都遵循同样的一小组规则。示例 4.5.1 使用一般形式 y = f(x − a) + b，对基本三次函数的图像进行平移，并把同样的推理应用到截距、拐点与对称点上。
This article explains the method step by step and shows how to apply it in exam-style questions, including the sign traps that cost candidates the most marks.
本文将逐步解释方法，并展示如何在考试类型题目中应用，包括最容易让考生丢分的符号陷阱。
1. Why Transformations Matter | 为什么变换很重要
Transformations allow you to sketch complicated functions quickly by recognising the relationship between a target equation and a simpler parent function.
变换能让你通过识别目标方程与更简单母函数之间的关系，快速绘制复杂函数图像。
In AQA exam papers, transformation questions often ask for the new coordinates of turning points, intercepts, or asymptotes after a translation, stretch, or reflection. The examiner is testing geometry, not algebra, so long calculations are rarely needed.
在 AQA 考试中，变换题常要求给出平移、拉伸或对称后拐点、截距或渐近线的新坐标。考官考查的是几何直观，而不是代数运算，所以通常不需要冗长的计算。
aph has a clear S-shape and a point of inflection at the origin. Once you know this shape, you can translate it instead of plotting many points.
对于 y = x³ 这样的三次函数，母函数图像具有清晰的 S 形，并在原点有一个拐点。一旦掌握这个形状，就可以直接平移，而无需绘制大量点。
The fastest way to see this is to read an equation as a recipe rather than a string of symbols: f(x) tells you the shape, the number inside the bracket tells you how far the graph moves along the x-axis, and the number outside the bracket tells you how far it moves along the y-axis.
理解这一点最快的方式，是把方程看作一份配方而不是一串符号：f(x) 告诉你形状，括号内的数字告诉你图像沿 x 轴移动多少，括号外的数字告诉你图像沿 y 轴移动多少。
2. The General Form y = f(x − a) + b | 一般形式 y = f(x − a) + b
Every pure translation of a graph can be written in the single compact form below, where a and b are real constants.
图像的每一个纯平移都可以写成下面这一个紧凑形式，其中 a 与 b 为实数常数。
y = f(x − a) + b
The parameter a controls horizontal movement and b controls vertical movement. Crucially, the two effects are independent: changing b never affects the x-coordinates of a point, and changing a never affects the y-coordinates.
参数 a 控制水平移动，b 控制竖直移动。关键在于，这两个效果相互独立：改变 b 不会影响点的 x 坐标，改变 a 也不会影响点的 y 坐标。
ent to the point (x + a, y + b). The whole graph is translated by the column vector (a, b)ᵀ, and the shape, orientation and steepness of the curve are completely unchanged.
从几何上看，曲线 y = f(x) 上的每个点 (x, y) 都被送到点 (x + a, y + b)。整条曲线按列向量 (a, b)ᵀ 平移，曲线的形状、朝向与陡峭程度完全不变。
| Expression | 表达式 | Translation | 平移 |
| y = f(x) + b, b > 0 | Up b units | 向上平移 b 个单位 |
| y = f(x) + b, b < 0 | Down |b| units | 向下平移 |b| 个单位 |
| y = f(x − a), a > 0 | Right a units | 向右平移 a 个单位 |
| y = f(x − a), a < 0 | Left |a| units | 向左平移 |a| 个单位 |
the vertex of y = f(x − a) + b is at (p + a, q + b). Nothing else about the curve changes.
一个有用的检验方法是追踪某一个具体点。若 y = f(x) 的顶点在 (p, q)，则 y = f(x − a) + b 的顶点在 (p + a, q + b)。曲线其他任何性质都不改变。
3. Vertical Translation: y = f(x) + a | 竖直平移：y = f(x) + a
If you add a constant outside the function, the whole graph moves vertically.
如果在函数外部加上常数，整个图像会竖直移动。
y = f(x) + a
en y = x² + 3 shifts the parabola up by 3 units because every y-coordinate increases by 3.
例如，若 f(x) = x²，则 y = x² + 3 将抛物线向上平移 3 个单位，因为每个 y 坐标都增加了 3。
in y = x² − 5, the graph shifts down by 5 units because every y-coordinate decreases by 5.
如果 a 为负，例如 y = x² − 5，图像向下平移 5 个单位，因为每个 y 坐标都减少了 5。
- a > 0: shift upward by a units. a > 0：向上平移 a 个单位。
- a < 0: shift downward by |a| units. a < 0：向下平移 |a| 个单位。
This rule is direct: the sign of a matches the direction of movement on the y-axis.
这个规则很直接：a 的符号与在 y 轴上的移动方向一致。
Because vertical translation changes every y-coordinate by the same amount, it moves the y-intercept but leaves the x-intercepts' x-values shifted only in the sense that the whole curve rises or falls; the roots themselves change because the curve no longer crosses the axis at the same places.
由于竖直平移让每个 y 坐标改变相同的量，它会改变 y 轴截距；而曲线整体上升或下降后，零点位置也会随之改变，因为曲线不再在原来的位置穿过 x 轴。
4. Horizontal Translation: y = f(x − a) | 水平平移：y = f(x − a)
Horizontal translations use the opposite sign from the one inside the bracket.
水平平移的符号与括号内的符号相反。
y = f(x − a)
n y = (x − 3)² shifts the vertex from (0, 0) to (3, 0), moving the graph right by 3 units.
若 f(x) = x²，则 y = (x − 3)² 将顶点从 (0, 0) 移到 (3, 0)，图像右移 3 个单位。
Similarly, y = (x + 2)² shifts the graph left by 2 units because x + 2 = x − (−2).
类似地，y = (x + 2)² 将图像左移 2 个单位，因为 x + 2 = x − (−2)。
input to equal 5, so x = 8. Every feature therefore appears 3 units further to the right.
符号相反的原因值得理解一次。为了得到 f 在 x = 5 处产生的同一个 y 值，新函数 f(x − 3) 要求括号内的量等于 5，即 x = 8。因此每一个特征点都向右移动了 3 个单位。
- f(x − a) with a > 0: shift right. f(x − a) 且 a > 0：向右平移。
- f(x + a) with a > 0: shift left, since f(x + a) = f(x − (−a)). f(x + a) 且 a > 0：向左平移，因为 f(x + a) = f(x − (−a))。
The most common exam error is to move the curve in the direction of the visible sign. Always rewrite the bracket in the form (x − a) first, then read off a.
最常见的考试错误是按照括号里可见的符号方向移动曲线。务必先把括号写成 (x − a) 的形式，再读出 a。
5. Combining Both Translations | 组合两种平移
When a question contains both an inside change and an outside change, apply them together as one single translation rather than two separate sketchings.
当题目同时包含括号内和括号外的变化时，应当把它们当作一次平移来处理，而不是分两次画图。
y = f(x − 2) + 1, so a = 2 and b = 1. The point of inflection moves from (0, 0) to (2, 1).
对于 f(x) = x³ 与方程 y = (x − 2)³ + 1，改写为 y = f(x − 2) + 1，于是 a = 2、b = 1。拐点从 (0, 0) 移动到 (2, 1)。
The order in which you imagine the two translations happening does not matter. Moving right 2 then up 1 gives the same final curve as moving up 1 then right 2, because vector addition is commutative.
想象两种平移发生的先后顺序并不重要。先右移 2 再上移 1，与先上移 1 再右移 2，得到完全相同的最终曲线，因为向量加法满足交换律。
What does matter is that you do not mix up which constant belongs to which axis. A constant added inside the bracket affects x-coordinates only; a constant added outside affects y-coordinates only.
真正重要的是不要混淆哪个常数对应哪条轴。加在括号内的常数只影响 x 坐标；加在括号外的常数只影响 y 坐标。
6. Tracking Key Points, Intercepts and Asymptotes | 追踪关键点、截距与渐近线
You never need to transform an entire curve point by point. In an exam, identify three or four defining features of the parent graph and translate each of them.
你完全不需要逐点变换整条曲线。在考试中，只需找出母函数图像的三到四个决定性特征，并逐个平移它们。
The features usually worth tracking are turning points (maxima and minima), points of inflection, axis intercepts, endpoints of a restricted domain, and any asymptotes.
通常值得追踪的特征包括：拐点（极大值与极小值）、对称中心、坐标轴截距、定义域端点和渐近线。
point is simple: the point (x, y) on y = f(x) becomes (x + a, y + b) on y = f(x − a) + b.
点的规则很简单：y = f(x) 上的点 (x, y) 在 y = f(x − a) + b 上变为 (x + a, y + b)。
| Point on y = f(x) | y = f(x) 上的点 | Point on y = f(x − 3) − 2 | y = f(x − 3) − 2 上的点 |
| (0, 0) | (3, −2) |
| (1, 1) | (4, −1) |
| (2, 4) | (5, 2) |
ptote y = c becomes y = c + b, because the y-coordinate shifts. Neither asymptote changes direction.
渐近线的行为与普通点不同。竖直渐近线 x = k 变为 x = k + a，因为 x 坐标发生了平移。水平渐近线 y = c 变为 y = c + b，因为 y 坐标发生了平移。两条渐近线的方向都不改变。
To find where the image crosses the y-axis, set x = 0 in the new equation and solve for y.
截距必须重新计算，而不能简单平移，因为截距的定义是某个坐标为 0。要求像曲线与 y 轴的交点，就在新方程中令 x = 0 并解出 y。
7. Worked Example: A Translated Cubic | 例题：平移后的三次函数
urve y = (x − 1)³ − 4. Rewrite it as y = f(x − 1) − 4 with f(x) = x³, so a = 1 and b = −4.
考虑曲线 y = (x − 1)³ − 4。改写为 y = f(x − 1) − 4，其中 f(x) = x³，所以 a = 1、b = −4。
The parent cubic has its point of inflection at (0, 0), so the new point of inflection is at (1, −4). The S-shape and the sign of the gradient are unchanged.
母三次函数的对称中心在 (0, 0)，因此新的对称中心为 (1, −4)。S 形与梯度的符号都不改变。
pt, set x = 0: y = (0 − 1)³ − 4 = −1 − 4 = −5, so the curve crosses the y-axis at (0, −5).
求 y 轴截距：令 x = 0，得 y = (0 − 1)³ − 4 = −1 − 4 = −5，所以曲线与 y 轴交于 (0, −5)。
To find the x-intercept, set y = 0: (x − 1)³ = 4, so x − 1 = ∛ and x ≈ 2.587. The single real root is consistent with a cubic that has no turning points.
求 x 轴截距：令 y = 0，得 (x − 1)³ = 4，于是 x − 1 = ∛4 ≈ 1.587，x ≈ 2.587。只有一个实根，这与没有极值点的三次函数相符。
−∞. Translation never reverses this end behaviour, because no reflection has taken place.
最后检查两端的走向。当 x → +∞ 时，y → +∞；当 x → −∞ 时，y → −∞。平移永远不会改变这种两端行为，因为这里没有发生对称（翻折）。
8. Common Mistakes and Exam Traps | 常见错误与考试陷阱
The single biggest error is moving the graph in the wrong horizontal direction. Candidates see (x + 2) and move right; correct work moves left.
最大的错误是水平方向移反。考生看到 (x + 2) 就向右移，而正确的做法是向左移。
The second trap is treating the transformation as a stretch. Adding a constant translates the graph; multiplying by a constant stretches or reflects it. These are different operations and the marks are not interchangeable.
第二个陷阱是把变换当成拉伸。加上常数是平移图像；乘以常数才是拉伸或翻折。二者是不同的运算，得分点不能互换。
A third mistake is applying the horizontal translation to the y-intercept's y-value. The point (0, 5) moves to (a, 5 + b), not to (0, 5 + b).
第三个错误是把水平平移错误地作用到 y 轴截距的 y 值上。点 (0, 5) 应移到 (a, 5 + b)，而不是 (0, 5 + b)。
- Always rewrite the bracket as (x − a) before reading a. | 读 a 之前，先把括号改写成 (x − a)。
- Check the point of inflection or vertex first; it anchors the whole sketch. | 先确定对称中心或顶点，它锚定整张草图。
- Do not assume the y-intercept stays fixed; recalculate with x = 0. | 不要假设 y 轴截距不变，用 x = 0 重新计算。
- Keep the shape identical; translation alone never changes curvature. | 保持形状完全相同，仅平移不会改变曲率。
Finally, remember that a translation of the graph is not the same as solving a translated equation. Sketching gives you the geometry quickly, but exact roots still require algebra.
最后要记住，平移图像并不等于解平移后的方程。画图能快速给出几何信息，但精确的根仍然需要代数运算。
9. Exam Method Template | 考试解题模板
Use the following six-step routine whenever a transformation question appears. It takes less than a minute and prevents sign errors.
每当遇到变换题，就使用下面这六步流程。它耗时不到一分钟，却能避免符号错误。
Step 1: Identify the parent function f(x). 步骤 1：确定母函数 f(x)。
Step 2: Rewrite the given equation in the form y = f(x − a) + b. 步骤 2：把给定方程改写成 y = f(x − a) + b 的形式。
Step 3: Read off a and b, including their signs. 步骤 3：读出 a 与 b，连同符号一起。
Step 4: Mark the key points of the parent graph. 步骤 4：标出母函数图像的关键点。
Step 5: Translate each point by (a, b) and redraw the same shape. 步骤 5：把每个点按 (a, b) 平移，并画出相同形状。
Step 6: State the new coordinates clearly as your final answer. 步骤 6：清晰地写出新坐标作为最终答案。
If the question asks only for coordinates, steps 1 to 3 followed by a direct calculation on one point may be enough. If it asks you to sketch, you must also label all intercepts and key features.
如果题目只要求坐标，完成步骤 1 到 3，再对某一点直接计算即可。如果要求画图，则还必须标注所有截距与关键特征。
10. Practice Questions with Solutions | 练习题与解答
Question 1: The curve y = f(x) passes through (4, 7). Find the corresponding point on y = f(x
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