Transforming Trigonometric Graphs | 三角函数的图像变换

📚 Transforming Trigonometric Graphs | 三角函数的图像变换

Understanding how to transform the graphs of sine, cosine, tangent and their reciprocal functions is a core skill in A-Level Edexcel Mathematics. By applying translations, stretches and reflections, you can sketch unfamiliar trigonometric curves with confidence and solve modelling problems. This guide walks through every transformation type, from amplitude and period changes to phase shifts, and shows you how to combine them using function notation.

掌握正弦、余弦、正切及其倒数函数图像的变换是A-Level Edexcel数学的核心技能。通过应用平移、伸缩和反射，你能够自信地绘制不熟悉的三角函数曲线并解决建模问题。本指南将逐一讲解每一种变换类型——从振幅和周期变化到相位平移，并展示如何用函数符号将它们组合起来。

1. The Standard Sine Graph and Its Key Features | 标准正弦曲线及其关键特征

values of 1 occur at x = π/2 + 2nπ, and its minimum values of −1 occur at x = 3π/2 + 2nπ.

母函数 y = sin x 的振幅为 1，周期为 2π 弧度（或 360°），并在 π 的整数倍处穿过 x 轴。其最大值 1 出现在 x = π/2 + 2nπ，最小值 −1 出现在 x = 3π/2 + 2nπ。

These features form the baseline for all sine transformations. When you apply a vertical stretch, the amplitude changes; when you apply a horizontal stretch, the period changes. Always refer back to the five key points (intercepts, maxima and minima) when sketching transformed curves.

这些特征是所有正弦变换的基准。应用垂直伸缩时，振幅会改变；应用水平伸缩时，周期会改变。绘制变换后的曲线时，务必回溯到五个关键点（截距、最大值和最小值）。

2. Vertical Stretch: Amplitude Change (y = a sin x) | 垂直伸缩：振幅变化 (y = a sin x)

ing sin x by a constant a gives y = a sin x. The graph vertically stretches by a factor of |a|. The amplitude becomes |a|, while the period and intercepts with the x-axis remain unchanged.

将 sin x 乘以常数 a 得到 y = a sin x。图像沿纵轴方向以 |a| 的倍数伸缩。振幅变为 |a|，而周期和与 x 轴的交点保持不变。

−2 sin x has amplitude 2 and reflects every point vertically, so a maximum of 2 becomes a minimum of −2 at the same x-coordinate.

如果 a 为负，图像同时关于 x 轴反射。例如，y = −2 sin x 的振幅为 2，且垂直反射每个点，因此原来在 2 处的最大值在相同 x 坐标处变为最小值 −2。

3. Horizontal Stretch: Period Change (y = sin(bx)) | 水平伸缩：周期变化 (y = sin(bx))

Replacing x with bx in y = sin(bx) compresses the graph horizontally by a factor of 1/|b|. The period of sin(bx) is 2π/|b| rad or 360°/|b| s period π rad, completing two full cycles in the same interval where sin x completes one.

在 y = sin(bx) 中用 bx 替代 x 会将图像沿水平方向以 1/|b| 的倍数压缩。sin(bx) 的周期为 2π/|b| 弧度或 360°/|b|。例如，y = sin(2x) 的周期为 π 弧度，在 sin x 完成一个完整周期的同一区间内完成两个完整周期。

/2) has period 4π rad. If b is negative, the graph also reflects in the y-axis, producing a mirror image that can often be simplified using odd/even identities.

当 b 是分数时，例如 ½，图像会水平拉伸。y = sin(x/2) 的周期为 4π 弧度。如果 b 为负，图像还会关于 y 轴反射，产生镜像，通常可以利用奇偶恒等式进行简化。

4. Phase Shift (Horizontal Translation): y = sin(x + c) | 相位平移（水平位移）：y = sin(x + c)

sin(x + c) as y = sin(x − (−c)). The graph moves left by c if c is positive, and right by |c| c is negative. For example, y = sin(x − π/3) shifts the standard sine curve right by π/3.

括号中的常数 c 会将图像水平移动。将 y = sin(x + c) 写成 y = sin(x − (−c))。若 c 为正，图像向左移动 c；若 c 为负，图像向右移动 |c|。例如，y = sin(x − π/3) 将标准正弦曲线右移 π/3。

)]. The true phase shift is −c/b. Many exam mistakes occur when students forget to factor.

当与水平伸缩结合时，务必先提取因子 b：y = sin(bx + c) = sin[b(x + c/b)]。真正的相位平移量是 −c/b。很多考试错误都源于学生忘记提取因子。

5. Vertical Translation: y = sin x + d | 垂直位移：y = sin x + d

The midline moves from y = 0 to y = d. All maxima and minima are elevated or lowered by d.

在正弦函数外部加上常数 d 会将整个图像垂直移动 d 个单位。中线从 y = 0 移动到 y = d。所有最大值和最小值都随之升降 d。

and the x-intercepts disappear unless d = 0. Always label the new midline when sketching.

例如，y = sin x + 2 在 3 和 1 之间振荡。振幅仍为 1，且除非 d = 0，否则图像与 x 轴不再有交点。绘图时务必标注新的中线。

6. Reflections of the Sine Graph | 正弦图像的反射

y a negative b inside the argument: y = sin(−x) produces a mirror image across the y-axis.

当系数 a 为负时，产生关于 x 轴的反射：y = −sin x 将图像上下翻转。当括号内 b 为负时，产生关于 y 轴的反射：y = sin(−x) 生成横跨 y 轴的镜像。

xis. For cosine (an even function), cos(−x) = cos x, so a y-axis reflection leaves the graph unchanged.

由于正弦是奇函数，sin(−x) = −sin x。这个恒等式意味着关于 y 轴的反射与关于 x 轴的反射得到相同的图像。对于余弦（偶函数），cos(−x) = cos x，因此关于 y 轴的反射不改变图像。

7. Combining Transformations: y = a sin(bx + c) + d | 组合变换：y = a sin(bx + c) + d

When multiple parameters are present, apply transformations in the order: horizontal first (stretch by 1/b, then translate by −c/b), then vertical (stretch by a, then translate by d). Always rewrite the bracket as b(x + c/b) to identify the phase shift correctly.

当存在多个参数时，按以下顺序进行变换：先水平（以 1/b 伸缩，再平移 −c/b），再垂直（以 a 伸缩，再平移 d）。务必先将括号改写为 b(x + c/b)，以正确识别相位平移。

de 3, vertical shift −1. The midline is y = −1, and the graph oscillates between 2 and −4.

例如，y = 3 sin(2x + π/4) − 1：b = 2 得周期 π，相位平移 = −(π/4)/2 = −π/8（左移 π/8），振幅 3，垂直位移 −1。中线为 y = −1，图像在 2 与 −4 之间振荡。

y = a sin(bx + c) + d summary:

广告 / Sponsored

Feature	Formula
Amplitude	|a|
Period	2π/|b| (rad) or 360°/|b|
Phase shift	−c/b
Vertical shift	d
Midline	y = d

8. Transformations of the Cosine Graph | 余弦图像的变换

ne y = 0. Transformations y = a cos(bx + c) + d follow exactly the same rules as for sine.

余弦图像 y = cos x 与正弦图形状相同，但从最大值 (0,1) 开始。其周期为 2π，振幅为 1，中线为 y = 0。变换式 y = a cos(bx + c) + d 遵循与正弦完全相同的规则。

ching transformed cosine graphs, identify the starting point (phase-shifted maximum) first, then mark successive key points at quarter-period intervals.

相位平移 π/2 可在两者之间转换：cos(x − π/2) = sin x。绘制变换后的余弦图像时，首先确定起点（经相位平移后的最大值），然后以四分之一周期为间隔标记后续关键点。

9. Transformations of the Tangent Graph | 正切图像的变换

. Transformations of the form y = a tan(bx + c) + d alter the period, asymptote positions and steepness.

正切函数 y = tan x 的周期为 π 弧度（180°），并在 x = π/2 + nπ 处有垂直渐近线。因其值域为全体实数，故没有振幅。形如 y = a tan(bx + c) + d 的变换会改变周期、渐近线位置和陡峭程度。

The period of y = tan(bx + c) + d is π/|b| mptote lines. A vertical shift d moves the central symmetry point from (0,0) to (−c/b, d).

y = tan(bx + c) + d 的周期为 π/|b|。渐近线根据相位 −c/b 进行水平移动。垂直伸缩 a 会使曲线变陡，但不改变渐近线位置。垂直位移 d 将中心对称点从 (0,0) 移至 (−c/b, d)。

− π/3) has period π/2, and the asymptote near the origin is at 2x − π/3 = π/2 → x = 5π/12.

始终通过求解 bx + c = π/2 + nπ 来确定新渐近线。例如，y = tan(2x − π/3) 的周期为 π/2，原点附近的渐近线在 2x − π/3 = π/2 → x = 5π/12。

10. Transformations of Reciprocal Trigonometric Functions | 倒数三角函数的图像变换

, 1/sin x and 1/tan x respectively. They have vertical asymptotes where the original functions are zero. Transformations apply similarly, but you must first sketch the base reciprocal graph.

倒数函数 y = sec x、y = csc x 和 y = cot x 分别定义为 1/cos x、1/sin x 和 1/tan x。它们在原函数为零的位置有垂直渐近线。变换方式类似，但必须先画出基本的倒数图像。

For y = a sec(bx + c) + d, the period is 2π/|b| (or π/|b| ertically, and d shifts the whole graph up or down, affecting the local minima and maxima.

对于 y = a sec(bx + c) + d，周期为 2π/|b|（cot 为 π/|b|），垂直渐近线随相位 −c/b 移动。参数 a 垂直拉伸各分支，d 将整个图像上下平移，影响局部最小值和最大值。

hes start at y = 1 + 2 = 3 and max branches at y = 1 − 2 = −1. Asymptotes at x = π/4 + nπ.

例如，y = 2 csc(x − π/4) + 1：周期 2π，相位右移 π/4，垂直拉伸 2，中线 y = 1。下方分支从 y = 3 开始，上方分支从 y = −1 开始。渐近线在 x = π/4 + nπ。

11. Using Transformations to Solve Equations and Sketch Graphs | 利用变换解方程与绘制草图

, to solve 3 sin(2x − 60°) + 1 = 2.5, sketch y = 3 sin(2x − 60°) + 1 and line y = 2.5, then read off intersection points using periodicity.

变换知识让你能够通过将方程与变换后的图像联系起来，在给定区间内求解三角方程。例如，要求解 3 sin(2x − 60°) + 1 = 2.5，绘制 y = 3 sin(2x − 60°) + 1 和直线 y = 2.5，然后利用周期性读出交点。

When sketching combined transformations, mark the midline, new amplitude, period and phase shift. Plot the five key points for a sine/cosine cycle, or for tan, mark the central point and asymptotes. Always check if the question expects solutions in degrees or radians.

绘制组合变换图像时，标出中线、新振幅、周期和相位平移。对正弦/余弦周期绘制五个关键点，对正切则标出中心点和渐近线。务必检查题目要求使用度数还是弧度。

12. Common Exam Pitfalls and Final Tips | 常见考试陷阱与最终建议

Mistakes often arise from incorrect factoring of the bracket, mixing up the order of transformations, or forgetting to apply absolute values for period and amplitude. Always write the function as a sin[b(x + c/b)] + d before identifying the phase shift.

常见错误源于括号提取不当、混淆变换顺序，或在周期和振幅上忘记取绝对值。务必先将函数写成 a sin[b(x + c/b)] + d 的形式，再识别相位平移。

When dealing with tan, remember the period is π/|b|, not 2π/|b|. For reciprocal functions, sketch the sine or cosine graph lightly first to locate asymptotes. Practise sketching without a calculator, using key intervals like 0, π/2, π, 3π/2, 2π for one cycle.

处理正切时，切记周期是 π/|b| 而非 2π/|b|。对于倒数函数，先淡描正弦或余弦图以定位渐近线。在不借助计算器的情况下练习绘图，使用 0、π/2、π、3π/2、2π 等关键区间绘制一个周期。

Finally, link transformations back to real-world contexts such as modelling tides, temperature or sound waves. Contextual questions in Edexcel papers often ask you to interpret amplitude, period and vertical shift in practical terms.

最后，将变换结合到真实情境中，如潮汐、温度或声波建模。Edexcel 试卷中的情境题常要求以实际含义解释振幅、周期和垂直位移。

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