📚 Graphs and Transformations | 图形与变换
Graphs and transformations form a cornerstone of A-Level mathematics, linking algebraic expressions to their geometric visualisations. Mastering this topic enables you to sketch complex curves quickly, solve equations graphically, and understand how changes to a function’s equation affect its graph.
图形与变换是 A-Level 数学的核心内容之一，它将代数表达式与其几何图像紧密联系起来。掌握这一主题，你便能快速绘制复杂曲线、以图解方式求解方程，并理解函数解析式变化对图像的影响。
1. Cubic Graphs | 三次函数图像
A cubic function has the general form y = ax³ + bx² + cx + d, where a ≠ 0. The graph of a cubic has one of two fundamental shapes depending on the sign of a.
三次函数的一般形式为 y = ax³ + bx² + cx + d，其中 a ≠ 0。三次函数的图像根据 a 的符号呈现两种基本形状之一。
When a > 0, the curve rises from the bottom-left and exits towards the top-right, resembling an elongated ‘S’ shape. When a < 0, the curve falls from the top-left and exits towards the bottom-right. Cubic graphs may have up to two turning points and cross the x-axis at up to three points.
当 a > 0 时，曲线从左下方向上攀升，向右上方延伸，形似拉长的 “S” 形；当 a < 0 时，曲线从左上方向下倾斜，向右下方延伸。三次函数图像最多可有两个极值点，并且最多与 x 轴有三个交点。
To sketch a cubic graph accurately, identify the key features below:
若要准确绘制三次函数图像，需确定以下关键特征：
- Roots: solve ax³ + bx² + cx + d = 0, for example by factorising or using the factor theorem.
- y-intercept: evaluate y when x = 0.
- End behaviour: determined by the sign of a.
- 零点：通过因式分解或因式定理求解 ax³ + bx² + cx + d = 0。
- y 截距：令 x = 0，计算 y 的值。
- 端部趋势：由 a 的符号决定。
Example: y = x³ − 4x = x(x − 2)(x + 2)
The graph crosses the x-axis at x = −2, 0 and 2, and the y-intercept is 0. Since a = 1 > 0, the curve rises as x → ∞ and falls as x → −∞. The graph has two turning points, located between consecutive roots.
该图像在 x = −2、0 和 2 处与 x 轴相交，y 截距为 0。由于 a = 1 > 0，曲线在 x → ∞ 时上升，在 x → −∞ 时下降。图像有两个极值点，位于相邻零点之间。
2. Quartic Graphs | 四次函数图像
A quartic function has the general form y = ax⁴ + bx³ + cx² + dx + e, where a ≠ 0. Quartic graphs are more varied than cubics: they can have up to three turning points and up to four real roots.
四次函数的一般形式为 y = ax⁴ + bx³ + cx² + dx + e，其中 a ≠ 0。四次函数图像比三次函数更加多样：最多可有三个极值点，并且最多有四个实数零点。
When a > 0, both ends of the curve point upwards; when a < 0, both ends point downwards. A common special case is y = x⁴, which has a single minimum point at the origin. Another typical case is y = x⁴ − 5x² + 4 = (x² − 1)(x² − 4), which crosses the x-axis at x = ±1 and x = ±2.
当 a > 0 时，曲线两端均向上延伸；当 a < 0 时，曲线两端均向下延伸。一个常见的特例是 y = x⁴，它在原点处有唯一的极小值点。另一典型例子是 y = x⁴ − 5x² + 4 = (x² − 1)(x² − 4)，它在 x = ±1 和 x = ±2 处与 x 轴相交。
When sketching quartics, always check whether the equation can be treated as a quadratic in x². This substitution often reveals symmetry about the y-axis.
绘制四次函数图像时，务必检查方程是否可以视为关于 x² 的二次方程。这种代换往往能揭示关于 y 轴的对称性。
| Function | Shape | Number of roots (max) |
|---|---|---|
| y = ax³ + bx² + cx + d | One ‘S’ curve | 3 |
| y = ax⁴ + bx³ + cx² + dx + e | W or M shaped | 4 |
| 函数类型 | 图像特征 | 最多零点数 |
|---|---|---|
| y = ax³ + bx² + cx + d | 一条 “S” 形曲线 | 3 |
| y = ax⁴ + bx³ + cx² + dx + e | W 形或 M 形 | 4 |
3. Reciprocal Graphs | 反比例函数图像
Reciprocal graphs are essential in the Edexcel specification. The simplest form is y = k/x, where k is a constant. Its graph consists of two branches in opposite quadrants, with the coordinate axes acting as asymptotes.
反比例函数图像是 Edexcel 考纲的重要内容。最简单的形式是 y = k/x，其中 k 为常数。其图像由位于相对象限的两条分支组成，坐标轴充当渐近线。
When k > 0, the branches lie in the first and third quadrants; when k < 0, they lie in the second and fourth quadrants. The curve never touches the axes; as x → 0⁺, y → +∞ when k > 0, and as x → ∞, y → 0.
当 k > 0 时，两条分支位于第一和第三象限；当 k < 0 时，位于第二和第四象限。曲线永远不会接触坐标轴；当 x → 0⁺ 时，若 k > 0，则 y → +∞；当 x → ∞ 时，y → 0。
y = k/x² | y = k/x²（k > 0）
Another important reciprocal graph is y = k/x². Both branches lie above the x-axis when k > 0, and the y-axis is a vertical asymptote. The x-axis is a horizontal asymptote in all reciprocal graphs.
另一个重要的反比例函数图像是 y = k/x²。当 k > 0 时，两条分支均位于 x 轴上方，y 轴为垂直渐近线。在所有反比例函数图像中，x 轴均为水平渐近线。
You must also be able to sketch transformed reciprocals such as y = 1/(x − 3) + 2. Here the vertical asymptote shifts to x = 3, and the horizontal asymptote shifts to y = 2.
你还必须能够绘制经过变换的反比例函数图像，例如 y = 1/(x − 3) + 2。此时垂直渐近线移至 x = 3，水平渐近线移至 y = 2。
4. Translations | 平移变换
A translation moves a graph without changing its shape or orientation. There are two basic types of translation applied to a function y = f(x).
平移变换在不改变图形形状和方向的前提下移动图像。对于函数 y = f(x)，有两种基本的平移方式。
Vertical translation: y = f(x) + a shifts the graph upwards by a units when a > 0, and downwards by |a| units when a < 0. Every point (x, y) moves to (x, y + a).
垂直平移：y = f(x) + a，当 a > 0 时图像向上移动 a 个单位；当 a < 0 时向下移动 |a| 个单位。每个点 (x, y) 移动到 (x, y + a)。
Horizontal translation: y = f(x − a) shifts the graph to the right by a units when a > 0, and to the left by |a| units when a < 0. Every point (x, y) moves to (x + a, y). Note carefully that y = f(x − 2) moves the graph right, not left.
水平平移：y = f(x − a)，当 a > 0 时图像向右移动 a 个单位；当 a < 0 时向左移动 |a| 个单位。每个点 (x, y) 移动到 (x + a, y)。请注意：y = f(x − 2) 是向右移动图像，而不是向左。
y = f(x) + a → vertical shift | y = f(x − a) → horizontal shift
In vector notation, the translation can be written as (0, a) for vertical shifts and (a, 0) for horizontal shifts. For a combined translation, use the column vector (a, b) to represent a shift of a units right and b units up.
用向量记号表示：垂直平移为 (0, a)，水平平移为 (a, 0)。对于复合平移，可用列向量 (a, b) 表示向右移动 a 个单位、向上移动 b 个单位。
5. Reflections | 反射变换
Reflections flip a graph over a given line. Two reflections are examined in A-Level mathematics: reflection in the x-axis and reflection in the y-axis.
反射变换将图像关于某条直线翻转。A-Level 数学中考查两种反射：关于 x 轴的反射和关于 y 轴的反射。
Reflection in the x-axis: y = −f(x). Every point (x, y) maps to (x, −y). This flips the graph vertically; any point above the x-axis moves below it, and vice versa.
关于 x 轴的反射：y = −f(x)。每个点 (x, y) 映射到 (x, −y)。这会将图像垂直翻转；x 轴上方的点移动到下方，反之亦然。
Reflection in the y-axis: y = f(−x). Every point (x, y) maps to (−x, y). This flips the graph horizontally. For example, if f(x) = x³ + 2, then f(−x) = (−x)³ + 2 = −x³ + 2.
关于 y 轴的反射：y = f(−x)。每个点 (x, y) 映射到 (−x, y)。这会将图像水平翻转。例如，若 f(x) = x³ + 2，则 f(−x) = (−x)³ + 2 = −x³ + 2。
A useful consequence: if f(x) is an even function, then f(−x) = f(x) and reflection in the y-axis leaves the graph unchanged. If f(x) is odd, then f(−x) = −f(x) and reflection in the y-axis is equivalent to reflection in the x-axis followed by a 180° rotation about the origin.
一个有用的结论：若 f(x) 是偶函数，则 f(−x) = f(x)，关于 y 轴的反射不改变图像。若 f(x) 是奇函数，则 f(−x) = −f(x)，关于 y 轴的反射等价于关于 x 轴的反射后再绕原点旋转 180°。
6. Stretches | 拉伸变换
Stretches alter the scale of a graph along one axis while leaving the other axis unchanged. There are two types: vertical stretches and horizontal stretches.
拉伸变换沿某一坐标轴改变图像的比例，同时保持另一坐标轴不变。拉伸分为两类：垂直拉伸和水平拉伸。
Vertical stretch: y = k·f(x) with k > 0. Every point (x, y) maps to (x, k·y). If k > 1, the graph is stretched away from the x-axis; if 0 < k < 1, it is compressed towards the x-axis. Points on the x-axis (where y = 0) remain fixed.
垂直拉伸：y = k·f(x)，其中 k > 0。每个点 (x, y) 映射到 (x, k·y)。若 k > 1，图像远离 x 轴拉伸；若 0 < k < 1，图像向 x 轴压缩。x 轴上的点（y = 0）保持不变。
Horizontal stretch: y = f(kx) with k > 0. Every point (x, y) maps to (x/k, y). If k > 1, the graph is compressed towards the y-axis; if 0 < k < 1, it is stretched away from the y-axis. Points on the y-axis (where x = 0) remain fixed.
水平拉伸：y = f(kx)，其中 k > 0。每个点 (x, y) 映射到 (x/k, y)。若 k > 1，图像向 y 轴压缩；若 0 < k < 1，图像远离 y 轴拉伸。y 轴上的点（x = 0）保持不变。
Be careful with negative scale factors: y = −3f(x) combines a vertical stretch by factor 3 with a reflection in the x-axis. The order of these two operations does not matter for a single-axis transformation.
注意负的伸缩因子：y = −3f(x) 是垂直拉伸 3 倍与关于 x 轴反射的组合。对于单轴变换，这两个操作的顺序无关紧要。
7. Combining Transformations | 复合变换
When multiple transformations are applied to a function, the order matters. The Edexcel syllabus requires you to handle combinations of translations, reflections and stretches systematically.
当对函数施加多个变换时，顺序至关重要。Edexcel 考纲要求你系统地处理平移、反射和拉伸的组合。
For transformations applied to the y-coordinate, such as y = 2f(x) + 3, apply the operations in the order they appear algebraically: first stretch by 2 (y = 2f(x)), then translate up by 3 (y = 2f(x) + 3).
对于作用于 y 坐标的变换，例如 y = 2f(x) + 3，按代数式中出现的顺序操作：先拉伸 2 倍（y = 2f(x)），再向上平移 3（y = 2f(x) + 3）。
For transformations applied to the x-coordinate, such as y = f(2x − 4), rewrite as y = f(2(x − 2)). Then apply the horizontal stretch by factor 1/2 first, followed by a translation right by 2 units. A common mistake is to translate first; that produces the wrong graph.
对于作用于 x 坐标的变换，例如 y = f(2x − 4)，应先改写为 y = f(2(x − 2))。然后先进行水平拉伸（伸缩因子为 1/2），再向右平移 2 个单位。常见错误是先平移后拉伸，这会得到错误的图像。
A reliable strategy is to track a single key point, such as a turning point or a root, through each transformation stage. Verify your result by checking one or two points at the end.
一个可靠的策略是追踪一个关键点（如极值点或零点）经过每个变换阶段的位置。最后通过检查一两个点来验证结果。
8. Transforming Key Points | 关键点的变换
Exam questions often give you the graph of y = f(x) with labelled points and ask you to sketch a transformed version. The most efficient method is to transform each labelled point individually.
考试题通常会给出 y = f(x) 的图像并标注若干点，要求你绘制变换后的图像。最有效的方法是逐一变换每个标注点。
For a point (x, y) on y = f(x):
对于 y = f(x) 上的点 (x, y)：
- y = f(x) + a maps (x, y) → (x, y + a)
- y = f(x − a) maps (x, y) → (x + a, y)
- y = −f(x) maps (x, y) → (x, −y)
- y = f(−x) maps (x, y) → (−x, y)
- y = k·f(x) maps (x, y) → (x, k·y)
- y = f(kx) maps (x, y) → (x/k, y)
- y = f(x) + a 将 (x, y) → (x, y + a)
- y = f(x − a) 将 (x, y) → (x + a, y)
- y = −f(x) 将 (x, y) → (x, −y)
- y = f(−x) 将 (x, y) → (−x, y)
- y = k·f(x) 将 (x, y) → (x, k·y)
- y = f(kx) 将 (x, y) → (x/k, y)
When multiple transformations act together, apply the x-transformations to the x-coordinate and the y-transformations to the y-coordinate independently. Use the rewritten form to decide the correct order.
当多个变换同时作用时，分别对 x 坐标施加 x 方向的变换、对 y 坐标施加 y 方向的变换。利用改写后的表达式确定正确的顺序。
9. Finding the Equation from a Transformed Graph | 由变换后的图像求解析式
You may be given a graph and asked to identify which transformation produced it from a known base function. Compare the features of the new graph with the original: the position of asymptotes, the locations of turning points, and the intercepts.
题目可能给出一个图像，要求你判断它是由哪个已知基本函数经过何种变换得到的。将新图像与原图像的特征进行对比：渐近线的位置、极值点的位置和截距。
For example, if the graph of y = x² has its vertex moved from (0, 0) to (3, −4), the equation becomes y = (x − 3)² − 4. This combines a horizontal translation right by 3 and a vertical translation down by 4.
例如，若 y = x² 的顶点从 (0, 0) 移动到 (3, −4)，则方程为 y = (x − 3)² − 4。这是向右平移 3 个单位与向下平移 4 个单位的复合。
If a graph is reflected and stretched, compare the end behaviour and the y-intercept to determine the scale factor and the sign. Always write your final equation in a simplified, expanded form if requested by the question.
若图像经过反射和拉伸，可通过比较端部趋势和 y 截距来确定伸缩因子和符号。若题目要求，务必写出化简后展开形式的最终方程。
10. Sketching Strategy | 作图策略
A systematic approach to sketching transformed graphs will save time in exams and reduce errors. Follow these steps in order.
系统化的作图方法能为你节省考试时间并减少错误。请按以下步骤依次进行。
Step 1: Identify the base function and write down its key features: asymptotes, intercepts, turning points.
第一步：确定基本函数，并记录其关键特征：渐近线、截距、极值点。
Step 2: Rewrite the transformed equation in the form y = a·f(b(x − c)) + d, where a, b, c and d are constants. This reveals the order of transformations clearly.
第二步：将变换后的方程改写为 y = a·f(b(x − c)) + d 的形式，其中 a、b、c、d 为常数。这样能清晰地显示变换顺序。
Step 3: Apply the x-transformations to the x-coordinates of all key points, then the y-transformations to the y-coordinates. Draw asymptotes first if they exist.
第三步：先将 x 方向的变换作用于所有关键点的 x 坐标，再将 y 方向的变换作用于 y 坐标。若存在渐近线，先画出渐近线。
Step 4: Sketch the curve smoothly, ensuring it approaches the asymptotes correctly and passes through all transformed points. Label the axes, intercepts, turning points and asymptotes.
第四步：平滑地绘制曲线，确保曲线正确趋近渐近线并经过所有变换后的点。标注坐标轴、截距、极值点和渐近线。
Finally, always check your sketch by substituting x = 0 to verify the y-intercept, and by considering the behaviour as x → ±∞. These two quick checks catch most common errors.
最后，务必通过代入 x = 0 来验证 y 截距，并考虑 x → ±∞ 时的趋势来检查草图。这两个快速检查能捕捉到大多数常见错误。
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