IB Mathematics: Horizontal and Vertical Translations of Function Graphs | IB数学：函数图像的平移变换

📚 IB Mathematics: Horizontal and Vertical Translations of Function Graphs | IB数学：函数图像的平移变换

In IB Mathematics, mastering graph transformations is a core algebraic skill. A translation moves every point of a graph by the same vector, so the graph appears in a new location but keeps its exact shape. This article explains the two simplest translations – horizontal shifts and vertical shifts – with worked examples and exam tips.

在IB数学中，掌握图像变换是一项核心代数技能。平移变换使图像上每一点都按照同一向量移动，因此图像只改变位置而保持原有形状不变。本文将详细讲解两类最基本的平移——水平平移与垂直平移，并配备例题和考试技巧。

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1. What Is a Translation? | 什么是平移变换？

A translation is a transformation that adds a constant to the x-coordinates, the y-coordinates, or both. For a function y = f(x), a vertical translation adds a constant outside the function, while a horizontal translation adds a constant inside the function.

平移是一种对 x 坐标、y 坐标或两者同时加上常数的变换。对于函数 y = f(x)，垂直平移是在函数外部加常数，而水平平移则是在函数内部加常数。

Key idea:

关键思路：

• Outside changes: y = f(x) + c moves the graph up or down.

  函数外部的变化：y = f(x) + c 使图像上下移动。

• Inside changes: y = f(x – a) moves the graph left or right.

  函数内部的变化：y = f(x – a) 使图像左右移动。

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2. Vertical Translation: y = f(x) + c | 垂直平移：y = f(x) + c

For a constant c, the graph of y = f(x) + c is obtained from y = f(x) by adding c to every y-coordinate.

对常数 c，函数 y = f(x) + c 的图像可由 y = f(x) 的图像通过给每个 y 坐标加上 c 得到。

If c > 0, the graph shifts upward by c units. If c < 0, it shifts downward by |c| units.

若 c > 0，图像向上平移 c 个单位；若 c < 0，图像向下平移 |c| 个单位。

Point mapping: every point (x, y) becomes (x, y + c). The domain remains the same, and the range is translated by c.

点映射：每一点 (x, y) 变成 (x, y + c)。定义域保持不变，值域整体平移 c 个单位。

y = f(x) + c

Example: Let f(x) = x². Then g(x) = x² + 3 has its vertex at (0, 3), moved 3 units up from (0, 0).

例如：设 f(x) = x²，则 g(x) = x² + 3 的顶点位于 (0, 3)，即从 (0, 0) 向上平移 3 个单位。

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3. Horizontal Translation: y = f(x – a) | 水平平移：y = f(x – a)

For a constant a, the graph of y = f(x – a) is obtained by replacing x with x – a in the function f.

对常数 a，函数 y = f(x – a) 的图像是通过在函数 f 中把 x 替换为 x – a 得到的。

If a > 0, the graph shifts to the right by a units. If a < 0, it shifts to the left by |a| units.

若 a > 0，图像向右平移 a 个单位；若 a < 0，图像向左平移 |a| 个单位。

Why right? The point where the expression inside equals zero is x = a. So every feature of the original graph moves from x = 0 to x = a.

为什么是向右？因为使内部表达式为零的点是 x = a。因此原图像的每个特征都从 x = 0 移动到 x = a。

Point mapping: every point (x, y) becomes (x + a, y). The domain is translated by a, while the range remains unchanged.

点映射：每一点 (x, y) 变成 (x + a, y)。定义域平移 a 个单位，值域保持不变。

y = f(x – a)

Example: Let f(x) = √x. Then g(x) = √(x – 2) has its starting point at x = 2, because the radicand x – 2 must be non-negative. This is a shift 2 units to the right.

例如：设 f(x) = √x，则 g(x) = √(x – 2) 的起点在 x = 2，因为被开方数 x – 2 必须非负。这就是向右平移 2 个单位。

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4. The Vertex Form of a Quadratic | 二次函数的顶点式

A quadratic written as y = a(x – h)² + k is called the vertex form. Its vertex is (h, k).

将二次函数写成 y = a(x – h)² + k 的形式称为顶点式，其顶点为 (h, k)。

This equation is exactly y = x² translated horizontally by h and vertically by k. The coefficient a controls the vertical stretch or reflection, not the translation.

该方程正是 y =

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