📚 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：
- |a| is the amplitude.
- |a| 是振幅。
- 2π/b is the period for sine and cosine.
- 2π/b 是正弦和余弦的周期。
- h is the horizontal shift, also called the phase shift.
- h 是水平移动，也称相位移动。
- d is the vertical shift, so the midline is y = d.
- d 是垂直移动，因此中线为 y = d。
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。
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。
- Rewrite the function in the form y = a sin[b(x − h)] + d, or the equivalent form for cos or tan.
- 将函数改写为 y = a sin[b(x − h)] + d，余弦或正切同理。
- Draw the midline y = d.
- 画出中线 y = d。
- Mark the maximum and minimum values y = d + |a| and y = d − |a|.
- 标出最大值与最小值 y = d + |a| 和 y = d − |a|。
- Calculate the period and mark one full cycle from x = h to x = h + period.
- 计算周期，并标出一个完整周期从 x = h 到 x = h + 周期。
- Plot the key points using the shape of sine or cosine, then draw a smooth curve.
- 根据正弦或余弦的形状标出关键点，然后画出光滑曲线。
- For tan, first mark the vertical asymptotes by solving the angle equal to π/2 + kπ.
- 对于正切函数，先令角度等于 π/2 + kπ，解出垂直渐近线。
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：
- Amplitude = 2.
- 振幅 = 2。
- Midline = −1.
- 中线 = −1。
- Maximum = 1 and minimum = −3.
- 最大值 = 1，最小值 = −3。
- Period = 2π/3.
- 周期 = 2π/3。
- Phase shift = −π/12, so the graph starts at x = −π/12.
- 相位移动 = −π/12，因此图像从 x = −π/12 开始。
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。
- Using a calculator when not allowed: always memorise the periods of sin, cos and tan.
- 在允许使用计算器的情况下仍要记住 sin、cos 和 tan 的周期；考试中有时不允许使用计算器。
- Confusing the direction of horizontal shift: sin(x + π/3) shifts left by π/3, not right.
- 混淆水平移动方向：sin(x + π/3) 向左移动 π/3，而不是向右。
- Forgetting to factor b before reading the phase shift: rewrite bx + c as b(x + c/b) first.
- 在读取相位移动前忘记提取 b：应先将 bx + c 改写为 b(x + c/b)。
- Using the wrong period for tan: the period of y = tan(bx) is π/b, not 2π/b.
- 对正切使用错误的周期：y = tan(bx) 的周期是 π/b，而不是 2π/b
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