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A-Level Maths: Trigonometric Graph Transformations | A-Level数学：三角函数图像变换技巧

2026-08-30 23:16 · 本文部分内容由AI辅助创作

📚 A-Level Maths: Trigonometric Graph Transformations | A-Level数学：三角函数图像变换技巧

Trigonometric graphs are a central topic in Edexcel A-Level Mathematics. Transformations allow you to sketch functions such as y = 3sin(2x − π/3) + 1 without calculating dozens of points. Understanding amplitude, period, phase shift and vertical shift also helps you interpret a graph and write down its equation.

三角函数图像是 Edexcel A-Level 数学的核心内容。掌握图像变换规则，你可以快速绘制如 y = 3sin(2x − π/3) + 1 的函数图像，而无需计算大量散点。理解振幅、周期、相位移动和垂直平移，还能帮助你解读图像并写出对应方程。

1. The Basic Trigonometric Graphs | 基本三角函数图像

Before transforming graphs, you must know the key features of y = sin x, y = cos x and y = tan x. The sine and cosine graphs repeat every 2π, while the tangent graph repeats every π.

在讨论变换之前，你必须熟悉 y = sin x、y = cos x 和 y = tan x 的基本图像特征。正弦和余弦图像每 2π 重复一次，而正切图像每 π 重复一次。

Function Period Range Key Features
y = sin x 2π −1 ≤ y ≤ 1 Roots at x = kπ; maximum at x = π/2 + 2kπ; minimum at x = −π/2 + 2kπ
y = cos x 2π −1 ≤ y ≤ 1 Roots at x = π/2 + kπ; maximum at x = 2kπ; minimum at x = π + 2kπ
y = tan x π All real numbers Roots at x = kπ; vertical asymptotes at x = π/2 + kπ

Remember that the tangent graph has vertical asymptotes. These are not part of the graph but they show where the function is undefined.

请记住，正切图像具有垂直渐近线。渐近线不是图像的一部分，但它们表明了函数无定义的位置。

2. Amplitude Transformations | 振幅变换

If you multiply a sine or cosine function by a constant a, you change its amplitude. The new function is y = a sin x or y = a cos x.

如果将一个正弦或余弦函数乘以常数 a，就会改变其振幅。新函数为 y = a sin x 或 y = a cos x。

Amplitude = |a|

For example, y = 3 sin x has amplitude 3 and range −3 ≤ y ≤ 3. If a is negative, the graph is also reflected in the x-axis.

例如，y = 3 sin x 的振幅为 3，值域为 −3 ≤ y ≤ 3。若 a 为负数，图像还会沿 x 轴翻转。

y = −2 sin x has amplitude 2 but starts by falling from the origin rather than rising.

This vertical stretch does not affect the period or the position of the roots for y = sin x and y = cos x.

这种垂直伸缩不会影响 y = sin x 和 y = cos x 的周期和零点的位置。

3. Vertical Translations | 垂直平移

Adding a constant d outside the function shifts the whole graph vertically:

在函数外侧加上常数 d 会使整个图像垂直平移：

y = sin x + d

If d > 0, the graph moves upward by d units. If d < 0, the graph moves downward.

若 d > 0，图像向上移动 d 个单位；若 d < 0，图像向下移动。

The central horizontal line becomes y = d. This line is often called the midline or principal axis. For example, y = sin x + 2 has a midline of y = 2 and a range from 1 to 3.

图像的中轴线变为 y = d。这条水平线通常称为中线或主轴。例如，y = sin x + 2 的中线为 y = 2，值域为 1 到 3。

The graph of y = tan x + d has the same vertical asymptotes as y = tan x, but every branch is shifted up or down by d.

y = tan x + d 的垂直渐近线与 y = tan x 相同，但每一支图像都会上下平移 d 个单位。

4. Period Transformations | 周期变换

Changing the coefficient of x inside the sine or cosine function changes the period. For y = sin(bx) and y = cos(bx), the period is:

改变正弦或余弦函数中 x 的系数会改变周期。对于 y = sin(bx) 和 y = cos(bx)，周期为：

Period = 2π / b

For y = tan(bx), the period becomes:

对于 y = tan(bx)，周期变为：

Period = π / b

Here b is usually positive in Edexcel questions. If b > 1, the graph is compressed horizontally, so the waves appear more frequently. If 0 < b < 1, the graph is stretched horizontally, so the waves appear less frequently.

在 Edexcel 考试中，b 通常为正数。若 b > 1，图像被水平压缩，波形更密集；若 0 < b < 1，图像被水平拉伸，波形更稀疏。

For example, y = sin(2x) has period π. The graph completes one full cycle from x = 0 to x = π.

例如，y = sin(2x) 的周期为 π。图像从 x = 0 到 x = π 完成一个完整周期。

5. Phase Shifts | 相位移动

A horizontal translation inside the angle is called a phase shift. For y = sin(x + c):

对角度内部进行水平平移称为相位移动。对于 y = sin(x + c)：

y = sin(x + c) shifts left by c units when c > 0.

y = sin(x − c) shifts right by c units when c > 0.

Be careful: adding a positive number inside the bracket moves the graph to the left, not the right.

要特别注意：括号内加上正数会使图像向左移动，而不是向右移动。

When there is also a coefficient of x, factor it out first:

当 x 前还有系数时，必须先提取系数：

y = sin(bx + c) = sin[b(x + c/b)]

So the phase shift is −c/b. For y = sin(2x − π/3), rewrite as y = sin[2(x − π/6)], so the graph shifts right by π/6.

因此相位移动为 −c/b。对于 y = sin(2x − π/3)，可改写为 y = sin[2(x − π/6)]，所以图像向右移动 π/6。

6. Combining Transformations | 组合变换

The general form for a transformed sine or cosine graph is:

变换后的正弦或余弦图像的一般形式为：

y = a sin[b(x − h)] + d

In this form:

在这种形式下：

If a is negative, the graph is reflected in the x-axis. This affects where the function starts but not its period or midline.

若 a 为负数，图像沿 x 轴翻转。这会影响函数的起始位置，但不会影响周期或中线。

Always rewrite expressions like y = a sin(bx + c) + d in the bracketed form before reading off the phase shift.

在读取相位移动之前，务必将 y = a sin(bx + c) + d 改写为带括号的标准形式。

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7. Transformations of y = tan x | 正切函数的变换

The tangent graph has no amplitude, but it does have a period, vertical asymptotes and roots. The transformed form is:

正切图像没有振幅，但它有周期、垂直渐近线和零点。变换后的形式为：

y = a tan[b(x − h)] + d

The constant a vertically stretches or compresses the branches and, if negative, reflects the graph in the x-axis. The constant d shifts the branches vertically.

常数 a 会对各支图像进行垂直拉伸或压缩；若为负数，则沿 x 轴翻转。常数 d 使各支图像垂直平移。

For y = tan(2x), the period is π/2 and the vertical asymptotes are at x = π/4 + kπ/2.

对于 y = tan(2x)，周期为 π/2，垂直渐近线位于 x = π/4 + kπ/2。

For y = tan(2x − π/3), rewrite as y = tan[2(x − π/6)]. The asymptotes shift right by π/6, so they become x = 5π/12 + kπ/2.

对于 y = tan(2x − π/3)，改写为 y = tan[2(x − π/6)]。渐近线向右移动 π/6，因此变为 x = 5π/12 + kπ/2。

8. A Step-by-Step Sketching Strategy | 作图的逐步策略

Use a consistent routine when sketching transformed trig graphs.

绘制变换后的三角函数图像时，使用一套固定的步骤。

For a sine curve starting at the midline, one cycle passes through midline, maximum, midline, minimum, midline. For a cosine curve starting at a maximum, one cycle passes through maximum, midline, minimum, midline, maximum.

对于从中线开始的正弦曲线，一个周期依次经过中线、最大值、中线、最小值、中线。对于从最大值开始的余弦曲线，一个周期依次经过最大值、中线、最小值、中线、最大值。

9. Worked Example: Sketch y = 2cos(3x + π/4) − 1 | 例题：绘制 y = 2cos(3x + π/4) − 1

First rewrite the expression by factoring out the coefficient of x.

首先提取 x 的系数，改写表达式。

y = 2cos[3(x + π/12)] − 1

From this form:

由此可以得到：

The start of the first cosine cycle is at x = −π/12. Since the coefficient a is positive, the graph begins at the maximum value y = 1.

第一个余弦周期的起点在 x = −π/12。由于系数 a 为正数，图像从最大值 y = 1 开始。

The cycle finishes at x = −π/12 + 2π/3 = 7π/12. Between these points, the cosine falls to the midline, reaches a minimum, rises back to the midline, and returns to the maximum.

该周期结束于 x = −π/12 + 2π/3 = 7π/12。在这两点之间，余弦曲线下降到中线，到达最小值，再回到中线，最后回到最大值。

10. Worked Example: Finding the Equation from a Graph | 例题：从图像求方程

If a graph has maximum value M and minimum value m, the amplitude and vertical shift can be found directly:

若图像的最大值为 M，最小值为 m，则可以直接求出振幅和垂直位移：

Amplitude = (M − m) / 2

Midline = (M + m) / 2

For example, suppose a periodic graph has maximum 6 and minimum −2. Then amplitude = (6 − (−2))/2 = 4 and midline = (6 + (−2))/2 = 2.

例如，某个周期图像的最大值为 6，最小值为 −2。则振幅 = (6 − (−2))/2 = 4，中线 = (6 + (−2))/2 = 2。

If the period is π and the graph looks like a cosine curve with a maximum at x = 0, then b = 2π/π = 2. The equation is y = 4cos(2x) + 2.

若周期为 π，且图像在 x = 0 处取得最大值，形状类似余弦曲线，则 b = 2π/π = 2。因此方程为 y = 4cos(2x) + 2。

If the graph passes through the midline at x = 0 and rises, use y = 4sin(2x) + 2 instead.

若图像在 x = 0 处经过中线并上升，则应使用 y = 4sin(2x) + 2。

11. Common Mistakes and Exam Tips | 常见错误与考试技巧

Here are the most common errors students make in the exam, and how to avoid them.

以下是学生在考试中最常见的错误，以及如何避免它们。