Graph Transformations: Translations | 函数图像的平移变换

📚 Graph Transformations: Translations | 函数图像的平移变换

In A-Level Mathematics, understanding graph transformations is essential for sketching functions and solving equations graphically. A translation is one of the simplest yet most important transformations: it slides a graph horizontally, vertically, or both without changing its shape or orientation.

在 A-Level 数学中，理解函数图像的变换对于绘制函数草图和通过图像求解方程至关重要。平移变换是最简单但也最重要的一种变换：它沿水平方向、垂直方向或同时沿两个方向滑动图像，而不改变其形状或朝向。

1. Vertical Translations: y = f(x) + k | 垂直平移：y = f(x) + k

A vertical translation shifts the graph of a function up or down. If we add a positive constant k to a function, the graph moves upward by k units; if k is negative, the graph moves downward by |k| units.

垂直平移将函数图像向上或向下移动。如果我们在函数上加上一个正常数 k，图像向上移动 k 个单位；如果 k 为负，则图像向下移动 |k| 个单位。

For a function y = f(x), the transformation y = f(x) + k produces a vertical shift. Every point (x, y) on the original graph becomes (x, y + k) on the new graph. Importantly, this transformation does not affect the x-coordinates of any point.

对于函数 y = f(x)，变换 y = f(x) + k 产生垂直平移。原图像上的每个点 (x, y) 在新图像上变为 (x, y + k)。重要的是，这个变换不会影响任何点的 x 坐标。

y = f(x) + k : shift upward by k units (k > 0) or downward by |k| units (k < 0)

Consider the quadratic function y = x². Its graph is a parabola with vertex at the origin (0, 0). The function y = x² + 3 has the same shape but its vertex is now at (0, 3). Similarly, y = x² − 2 has its vertex at (0, −2).

以二次函数 y = x² 为例。其图像是一条顶点在原点 (0, 0) 的抛物线。函数 y = x² + 3 具有相同的形状，但顶点现在位于 (0, 3)。类似地，y = x² − 2 的顶点位于 (0, −2)。

Key point: the graph of f(x) + k is identical to the graph of f(x), just repositioned vertically. The horizontal asymptotes, intercepts, and stationary points all shift by the same amount k.

关键点：f(x) + k 的图像与 f(x) 的图像完全相同，只是在垂直方向上重新定位。水平渐近线、截距和驻点都平移相同的量 k。

2. Horizontal Translations: y = f(x − a) | 水平平移：y = f(x − a)

Horizontal translations are often more challenging for students because the direction of the shift is counterintuitive. The function y = f(x − a) shifts the graph of y = f(x) to the right by a units when a > 0, and to the left by |a| units when a < 0.

水平平移对学生来说通常更具挑战性，因为平移的方向与直觉相反。当 a > 0 时，函数 y = f(x − a) 将 y = f(x) 的图像向右平移 a 个单位；当 a < 0 时，向左平移 |a| 个单位。

Why does x − a move the graph to the right? Consider the point where the original function has input x = 0. In the transformed function y = f(x − a), to obtain the same output, we need x − a = 0, meaning x = a. So the entire graph shifts right by a units.

为什么 x − a 使图像向右移动？考虑原函数在输入 x = 0 处的点。在变换后的函数 y = f(x − a) 中，要获得相同的输出，需要 x − a = 0，即 x = a。因此整个图像向右平移了 a 个单位。

y = f(x − a) : shift right by a units (a > 0) or left by |a| units (a < 0)

It is crucial to remember the sign convention: inside the brackets, x minus a means right, x plus a means left. This is opposite to what many students initially expect.

务必记住符号规则：在括号内部，x − a 表示向右，x + a 表示向左。这与许多学生最初的直觉相反。

For example, y = (x − 2)² has its vertex at (2, 0), which is the graph of y = x² shifted 2 units to the right. Meanwhile, y = (x + 3)² has its vertex at (−3, 0), shifted 3 units to the left.

例如，y = (x − 2)² 的顶点在 (2, 0)，这是 y = x² 的图像向右平移 2 个单位的结果。而 y = (x + 3)² 的顶点在 (−3, 0)，是向左平移 3 个单位的结果。

3. Combined Translations | 组合平移

In many problems, a graph undergoes both a horizontal and a vertical translation simultaneously. The general form is y = f(x − a) + k, which shifts the graph by a units horizontally (right if a > 0) and k units vertically (up if k > 0).

在许多问题中，图像同时经历水平平移和垂直平移。一般形式为 y = f(x − a) + k，使图像沿水平方向平移 a 个单位（a > 0 时向右）并沿垂直方向平移 k 个单位（k > 0 时向上）。

The horizontal and vertical translations are applied independently, and their order does not matter. Whether you first shift horizontally and then vertically, or vice versa, the final position is the same.

水平平移和垂直平移是独立进行的，顺序不影响最终结果。无论是先水平平移再垂直平移，还是先垂直平移再水平平移，最终位置是相同的。

For instance, consider the cubic function y = f(x) where f(x) = x³. The transformed function y = (x − 1)³ + 2 represents a shift of 1 unit to the right and 2 units upward. The point of inflection, originally at (0, 0), moves to (1, 2).

例如，考虑三次函数 y = f(x)，其中 f(x) = x³。变换后的函数 y = (x − 1)³ + 2 表示向右平移 1 个单位并向上平移 2 个单位。拐点从原来的 (0, 0) 移动到 (1, 2)。

For quadratic functions, completing the square reveals the combined translation from y = x². The expression y = (x − p)² + q clearly shows a horizontal shift p and a vertical shift q from the parent graph y = x².

对于二次函数，配方法揭示了从 y = x² 出发的组合平移。表达式 y = (x − p)² + q 清晰地显示了相对于母图 y = x² 的水平平移 p 和垂直平移 q。

4. Translating Key Features of Graphs | 平移图像的关键特征

When a graph is translated, every characteristic feature moves with it consistently. These features include intercepts, turning points, asymptotes, and intersections.

当图像被平移时，每个特征点都随之一致地移动。这些特征包括截距、转向点、渐近线和交点。

Consider y = f(x) = 2ˣ. This exponential function has a horizontal asymptote at y = 0 (the x-axis). Under the translation y = 2ˣ + 1, the asymptote moves up to y = 1. The graph never reaches y = 1 but approaches it as x → −∞.

考虑 y = f(x) = 2ˣ。这个指数函数有一条水平渐近线 y = 0（即 x 轴）。在平移 y = 2ˣ + 1 下，渐近线向上移动到 y = 1。图像永远达不到 y = 1，但随着 x → −∞ 而无限靠近它。

For the logarithmic function y = ln x, the vertical asymptote is x = 0. The transformed function y = ln(x − 2) has a vertical asymptote at x = 2, because the entire graph shifts 2 units to the right.

对于对数函数 y = ln x，其垂直渐近线为 x = 0。变换后的函数 y = ln(x − 2) 的垂直渐近线为 x = 2，因为整个图像向右平移了 2 个单位。

Turning points: the coordinates (p, q) become (p + a, q + k)
Turning points: 坐标 (p, q) 变为 (p + a, q + k)
Intercepts: substitute x = 0 to find the new y-intercept; set y = 0 to find new x-intercepts if they exist
截距：代入 x = 0 求新的 y 截距；令 y = 0 求新的 x 截距（如果存在）
Asymptotes: horizontal asymptote y = c becomes y = c + k; vertical asymptote x = d becomes x = d + a
渐近线：水平渐近线 y = c 变为 y = c + k；垂直渐近线 x = d 变为 x = d + a

5. Translation of Trigonometric Graphs | 三角函数图像的平移

Trigonometric functions are periodic, and translations produce phase shifts. For y = sin x, the transformation y = sin(x − c) shifts the graph horizontally by c units. This is called a phase shift.

三角函数是周期函数，平移会产生相位移。对于 y = sin x，变换 y = sin(x − c) 将图像水平平移 c 个单位，这被称为相位移动。

The function y = sin(x − 90°) shifts the sine wave 90° (or π/2 radians) to the right. This transformation produces a curve identical to y = cos x, which demonstrates the well-known identity sin(x − 90°) = cos x.

函数 y = sin(x − 90°) 将正弦波向右移动 90°（即 π/2 弧度）。这个变换产生的曲线与 y = cos x 完全相同，这体现了著名的恒等式 sin(x − 90°) = cos x。

Meanwhile, y = sin x + 1 shifts the graph upward by 1 unit. The maximum value becomes 2 instead of 1, and the minimum becomes 0 instead of −1. The midline of the wave moves from y = 0 to y = 1.

同时，y = sin x + 1 将图像向上平移 1 个单位。最大值从 1 变为 2，最小值从 −1 变为 0。波的中线从 y = 0 移到 y = 1。

For y = cos(x + 45°) + 2, the graph of cos x shifts 45° to the left and 2 units up. The maximum value is 3, the minimum is 1, and the period remains 360°.

对于 y = cos(x + 45°) + 2，cos x 的图像向左平移 45° 并向上平移 2 个单位。最大值为 3，最小值为 1，周期仍为 360°。

In A-Level exams, students are often asked to determine the image of a specific point, such as the maximum point of a sine curve, after a translation. Using the transformation rules directly saves time and reduces errors.

在 A-Level 考试中，经常要求学生确定特定点在平移后的位置，例如正弦曲线的最大值点。直接使用变换规则可以节省时间并减少错误。

6. Finding the New Coordinates of a Point | 求点的新坐标

Given a point (x₁, y₁) on the graph of y = f(x), its image under the translation y = f(x − a) + k can be found systematically. The x-coordinate shifts by a (with the same sign as in the bracket), and the y-coordinate shifts by k.

已知 y = f(x) 的图像上有一点 (x₁, y₁)，在平移 y = f(x − a) + k 下，其像点的坐标可以系统地求出。x 坐标平移 a（符号与括号内相反），y 坐标平移 k。

(x₁, y₁) → (x₁ + a, y₁ + k)

For example, suppose the point (−2, 5) lies on y = f(x). Under the transformation y = f(x − 3) + 4, the new point is (−2 + 3, 5 + 4) = (1, 9).

例如，假设点 (−2, 5) 位于 y = f(x) 上。在变换 y = f(x − 3) + 4 下，新点为 (−2 + 3, 5 + 4) = (1, 9)。

Be careful: the transformation y = f(x + 2) − 1 shifts the point (x₁, y₁) to (x₁ − 2, y₁ − 1). Since the bracket contains x + 2, the sign inside the bracket is opposite to the direction of the shift.

注意：变换 y = f(x + 2) − 1 将点 (x₁, y₁) 移动到 (x₁ − 2, y₁ − 1)。因为括号内是 x + 2，括号内的符号与平移方向相反。

If the problem provides a sketch rather than a formula, locate the key features of the original graph (vertex, intercept, asymptote) and apply the translation to those features. Then redraw the graph through the new feature positions.

如果问题提供的是草图而非公式，请找到原图的关键特征（顶点、截距、渐近线），并将平移应用到这些特征上。然后通过新的特征位置重新绘制图像。

7. Translations in Solving Equations | 平移在解方程中的应用

Translations can simplify certain equations. For instance, solving (x − 2)² = 9 is equivalent to finding where the parabola y = (x − 2)² intersects the horizontal line y = 9. The solutions x = 5 and x = −1 correspond to the x-coordinates of these intersection points.

平移可以简化某些方程。例如，解 (x − 2)² = 9 等价于求抛物线 y = (x − 2)² 与水平线 y = 9 的交点。解 x = 5 和 x = −1 对应于这些交点的 x 坐标。

More importantly, understanding translations helps interpret composite functions. If f(x) = x² and g(x) = (x − 3)² + 2, then g(x) is simply the graph of f(x) shifted 3 units right and 2 units up, with vertex at (3, 2).

更重要的是，理解平移有助于解释复合函数。如果 f(x) = x² 且 g(x) = (x − 3)² + 2，那么 g(x) 就是将 f(x) 的图像向右平移 3 个单位、向上平移 2 个单位，顶点在 (3, 2)。

In exam questions, you might be given the graph of y = f(x) and asked to sketch y = f(x − 1) + 3. You should identify at least two or three key points on the original graph, translate each one, and draw a curve of the same shape through the new points.

在考试题目中，你可能会被给到 y = f(x) 的图像并要求画出 y = f(x − 1) + 3 的草图。你应该在原图上确定至少两三个关键点，平移每个点，并通过新点绘制相同形状的曲线。

8. Common Mistakes and Exam Tips | 常见错误与考试提示

One of the most common mistakes is confusing the direction of horizontal translations. Students often write y = f(x + 2) and state that the graph shifts right, when in fact it shifts left. Always remember: inside the brackets, plus is left, minus is right.

最常见的错误之一是混淆水平平移的方向。学生常常写 y = f(x + 2) 并说图像向右移动，但实际上它向左移动。始终记住：括号内，加号向左，减号向右。

Another frequent error occurs when translating asymptotes and intercepts. Some students translate the x-intercept correctly but forget to apply the vertical shift to the y-intercept. Always apply both horizontal and vertical shifts to every point.

另一个常见错误出现在平移渐近线和截距时。有些学生正确地平移了 x 截距，却忘记对 y 截距应用垂直平移。务必对每个点都同时应用水平平移和垂直平移。

Tip 1: When asked for the image of a point, write the transformation rule explicitly: (x, y) → (x + a, y + k)
提示 1：当被要求求点的像时，明确写出变换规则：(x, y) → (x + a, y + k)
Tip 2: For horizontal shifts, look at what must be added to x to keep the output unchanged
提示 2：对于水平平移，思考要保持输出不变，x 需要加上什么
Tip 3: Check asymptotes separately — they follow the same translation rules
提示 3：单独检查渐近线——它们遵循相同的平移规则
Tip 4: In multiple-choice questions, test a known point (like the vertex or an intercept) rather than the whole graph
提示 4：在选择题中，测试一个已知点（如顶点或截距），而不是整个图像

In assessment, examiners award marks for correctly identifying the type of transformation, applying it to key coordinates, and producing a sketch with the right shape and position. Show your working clearly by labelling the coordinates of at least two points before and after the transformation.

在考试中，考官会根据你正确识别变换类型、将其应用于关键坐标，并画出具有正确形状和位置的草图来给分。请清晰展示步骤，标注变换前后至少两个点的坐标。

9. Worked Example: Full Solution | 完整例题解答

The graph of y = x² − 4x is translated by the vector (3, −2). Find the equation of the resulting graph in the form y = ax² + bx + c.

函数 y = x² − 4x 的图像按向量 (3, −2) 平移。求所得图像的方程，形式为 y = ax² + bx + c。

Step 1: Recognise that the vector (3, −2) means a = 3 (shift right) and k = −2 (shift down). The resulting equation is y = (x − 3)² − 4(x − 3) − 2.

第一步：认识到向量 (3, −2) 表示 a = 3（向右平移）和 k = −2（向下平移）。所得方程为 y = (x − 3)² − 4(x − 3) − 2。

Step 2: Expand and simplify.

第二步：展开并化简。

y = (x² − 6x + 9) − 4x + 12 − 2
y = x² − 10x + 19

Step 3: Check using the vertex. The original function y = x² − 4x can be written as y = (x − 2)² − 4, so its vertex is at (2, −4). After the translation, the vertex should be at (5, −6). Substitute x = 5 into y = x² − 10x + 19: 25 − 50 + 19 = −6. This confirms the answer is correct.

第三步：用顶点验证。原函数 y = x² − 4x 可写成 y = (x − 2)² − 4，所以其顶点在 (2, −4)。平移后，顶点应在 (5, −6)。将 x = 5 代入 y = x² − 10x + 19：25 − 50 + 19 = −6。这确认了答案是正确的。

This example demonstrates the general approach: substitute (x − a) for every x in the original expression to achieve a horizontal shift, then add k for the vertical shift.

这个例子展示了一般方法：将原表达式中的每个 x 替换为 (x − a) 以实现水平平移，然后加上 k 以实现垂直平移。

Mastering translations of function graphs is a fundamental skill for A-Level Mathematics. By understanding the rules y = f(x) + k for vertical shifts and y = f(x − a) for horizontal shifts, you can confidently sketch transformed graphs, locate key points, and solve related problems in examinations.

掌握函数图像的平移变换是 A-Level 数学的基本技能。通过理解垂直平移的规则 y = f(x) + k 和水平平移的规则 y = f(x − a)，你可以自信地绘制变换后的图像、定位关键点并在考试中解决相关问题。

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Graph Transformations: Translations | 函数图像的平移变换