Transformations of Trigonometric Graphs | 三角函数图像的变换

📚 Transformations of Trigonometric Graphs | 三角函数图像的变换

In A-Level Mathematics, understanding how transformations affect the graphs of trigonometric functions is a fundamental skill. This topic builds on your knowledge of y = sin x, y = cos x, and y = tan x, and extends it to more complex forms such as y = a sin(bx + c) + d. Mastering these transformations allows you to sketch graphs quickly, interpret equations physically, and solve problems with confidence.

在 A-Level 数学中，理解变换如何影响三角函数图像是一项基本技能。本主题建立在对 y = sin x、y = cos x 和 y = tan x 的认识之上，并将其推广到 y = a sin(bx + c) + d 等更复杂的形式。掌握这些变换，你就能快速绘制草图、从物理角度解读方程，并自信地解题。

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

Before applying transformations, we must recall the key features of the basic graphs. For y = sin x, the domain is all real numbers, the range is [-1, 1], the period is 360° (or 2π radians), and the amplitude is 1. The graph passes through (0, 0), reaches a maximum of 1 at 90°, returns to 0 at 180°, reaches a minimum of -1 at 270°, and completes one full cycle at 360°.

在应用变换之前，我们必须回顾基本图像的关键特征。对于 y = sin x，定义域为全体实数，值域为 [-1, 1]，周期为 360°（即 2π 弧度），振幅为 1。图像经过 (0, 0)，在 90° 处达到最大值 1，在 180° 处回到 0，在 270° 处达到最小值 -1，并在 360° 处完成一个完整周期。

For y = cos x, the graph has the same amplitude, period, and range, but it starts at its maximum value of 1 when x = 0. It reaches 0 at 90°, -1 at 180°, 0 at 270°, and returns to 1 at 360°. In fact, the cosine graph is simply the sine graph shifted left by 90°.

对于 y = cos x，图像具有相同的振幅、周期和值域，但在 x = 0 时从最大值 1 开始。它在 90° 处达到 0，180° 处达到 -1，270° 处回到 0，并在 360° 处回到 1。事实上，余弦图像就是将正弦图像向左平移 90° 得到的。

For y = tan x, the graph is quite different. It has vertical asymptotes at x = 90°, 270°, etc., a period of 180°, and a range of all real numbers. The graph passes through the origin and increases without bound near the asymptotes.

对于 y = tan x，图像则完全不同。它在 x = 90°、270° 等位置有垂直渐近线，周期为 180°，值域为全体实数。图像经过原点，并在渐近线附近无限增大。

2. Vertical Stretch — Amplitude Change | 垂直伸缩——振幅变化

The transformation y = a sin x, where a > 0, produces a vertical stretch by a factor of a. The amplitude of the graph changes from 1 to a, while the period and the position of the x-intercepts remain unchanged. For example, y = 3 sin x has a maximum value of 3 and a minimum value of -3; its range becomes [-3, 3].

变换 y = a sin x（其中 a > 0）产生纵向拉伸，拉伸因子为 a。图像的振幅从 1 变为 a，而周期和 x 截距的位置保持不变。例如，y = 3 sin x 的最大值为 3，最小值为 -3；其值域变为 [-3, 3]。

If a is negative, such as y = -2 sin x, the graph is stretched by a factor of 2 and then reflected in the x-axis. The amplitude, defined as the distance from the midline to the maximum, is always positive; for y = -2 sin x, the amplitude is still 2, but the graph is inverted.

如果 a 为负数，例如 y = -2 sin x，则图像先被拉伸 2 倍，然后关于 x 轴反射。振幅定义为从中线到最大值或最小值的距离，始终为正；对于 y = -2 sin x，振幅仍为 2，但图像被翻转。

y = a sin x → amplitude = |a|, range = [-|a|, |a|]

3. Horizontal Stretch — Period Change | 水平伸缩——周期变化

The transformation y = sin(bx), where b > 0, produces a horizontal stretch by a factor of 1/b. The period of the function changes from 360° to 360°/b (in degrees) or from 2π to 2π/b (in radians). For instance, y = sin(2x) has a period of 180°, meaning the graph completes two full cycles within the span of 360°.

变换 y = sin(bx)（其中 b > 0）产生水平伸缩，伸缩因子为 1/b。函数的周期从 360° 变为 360°/b（角度制），或从 2π 变为 2π/b（弧度制）。例如，y = sin(2x) 的周期为 180°，意味着图像在 360° 范围内完成两个完整周期。

When b is less than 1, for example y = sin(½x), the period becomes 720°, so the graph is stretched horizontally. When b is greater than 1, the graph is compressed horizontally. The amplitude remains unchanged in this type of transformation.

当 b 小于 1 时，例如 y = sin(½x)，周期变为 720°，因此图像在水平方向被拉伸。当 b 大于 1 时，图像在水平方向被压缩。此类变换中振幅保持不变。

y = sin(bx) → period = 360°/b (degrees) = 2π/b (radians)

4. Vertical Translation — The Constant d | 垂直平移——常数 d

The transformation y = sin x + d shifts the entire graph vertically by d units. If d > 0, the graph moves upward; if d < 0, it moves downward. This shift changes the midline (also called the principal axis) from y = 0 to y = d. The range of y = sin x + d becomes [d - 1, d + 1].

变换 y = sin x + d 将整个图像垂直平移 d 个单位。若 d > 0，图像向上移动；若 d < 0，图像向下移动。此平移改变了中线（亦称主轴）的位置，从 y = 0 变为 y = d。y = sin x + d 的值域变为 [d - 1, d + 1]。

For example, y = sin x – 2 has a midline at y = -2, a maximum at -1, and a minimum at -3. The period and amplitude are unaffected. This type of transformation is often combined with amplitude changes to model real-world phenomena such as tidal heights or alternating current.

例如，y = sin x – 2 的中线在 y = -2，最大值为 -1，最小值为 -3。周期和振幅不受影响。此类变换常与振幅变化结合，用于模拟潮汐高度或交流电等真实世界现象。

y = sin x + d → midline: y = d, range: [d – 1, d + 1]

5. Horizontal Translation — Phase Shift | 水平平移——相位移动

The transformation y = sin(x + c) shifts the graph horizontally. If c > 0, the graph shifts to the left by c units; if c < 0, the graph shifts to the right by |c| units. This is known as a phase shift. For example, y = sin(x - 90°) is the sine graph shifted 90° to the right, which gives the same graph as y = cos x.

变换 y = sin(x + c) 会使图像水平平移。若 c > 0，图像向左平移 c 个单位；若 c < 0，图像向右平移 |c| 个单位。这称为相位移动。例如，y = sin(x - 90°) 是正弦图像向右平移 90°，得到的图像与 y = cos x 相同。

Be careful with the direction: the transformation y = sin(x + c) with c > 0 shifts the graph left. This is because a point at x = 0 in the original graph now occurs at x = -c in the transformed graph. It is a common mistake to think that a positive c moves the graph right.

注意方向：变换 y = sin(x + c) 当 c > 0 时将图像向左平移。这是因为原图像中 x = 0 的点，在新图像中出现在 x = -c 处。认为正的 c 使图像向右移动是一个常见错误。

y = sin(x + c) → phase shift = -c units (right if c < 0, left if c > 0)

6. Reflections | 反射变换

There are two types of reflections to consider. The reflection y = -f(x) reflects the graph in the x-axis. Each point (x, y) on the original graph is mapped to (x, -y). For y = -cos x, the graph is inverted; where the original had a maximum, the reflected graph has a minimum, and vice versa.

需要考虑两种反射。反射 y = -f(x) 将图像关于 x 轴翻转。原图像上的每个点 (x, y) 映射到 (x, -y)。对于 y = -cos x，图像被翻转；原图像的最大值处变为反射图像的最小值处，反之亦然。

The reflection y = f(-x) reflects the graph in the y-axis. Each point (x, y) maps to (-x, y). Since sine is an odd function, y = sin(-x) = -sin x, which is the same as reflecting the sine graph in the x-axis. Since cosine is an even function, y = cos(-x) = cos x, so the graph is unchanged.

反射 y = f(-x) 将图像关于 y 轴翻转。每个点 (x, y) 映射到 (-x, y)。由于正弦函数是奇函数，y = sin(-x) = -sin x，这与将正弦图像关于 x 轴反射相同。由于余弦函数是偶函数，y = cos(-x) = cos x，因此图像不变。

7. The General Form y = a sin(bx + c) + d | 一般形式 y = a sin(bx + c) + d

The most general sine function encountered in A-Level is y = a sin(bx + c) + d. In this form, the four parameters each control a specific aspect of the graph: a controls amplitude and reflection, b controls period, c controls phase shift, and d controls vertical translation.

A-Level 中最一般的正弦函数形式为 y = a sin(bx + c) + d。在此形式中，四个参数各自控制图像的一个特定方面：a 控制振幅和反射，b 控制周期，c 控制相位移，d 控制垂直平移。

To find the phase shift, the expression inside the sine function must be factored. Writing y = a sin(b(x + c/b)) + d makes it clear that the phase shift is -c/b. For example, in y = 3 sin(2x – 60°) + 1, we rewrite it as y = 3 sin(2(x – 30°)) + 1. Here, the amplitude is 3, the period is 360°/2 = 180°, the phase shift is 30° to the right, and the midline is y = 1.

要找到相位移，必须对正弦函数内的表达式进行因式分解。将 y = a sin(bx + c) + d 写成 y = a sin(b(x + c/b)) + d 即可清楚地看出相位移为 -c/b。例如，在 y = 3 sin(2x – 60°) + 1 中，我们改写为 y = 3 sin(2(x – 30°)) + 1。这里，振幅为 3，周期为 360°/2 = 180°，相位移为向右 30°，中线为 y = 1。

The range of this general form is [d – |a|, d + |a|]. The maximum value is d + |a| and the minimum value is d – |a|. The amplitude is always |a|, regardless of the other parameters.

此一般形式的值域为 [d – |a|, d + |a|]。最大值为 d + |a|，最小值为 d – |a|。振幅始终为 |a|，无论其他参数如何。

y = a sin(bx + c) + d → amplitude = |a|, period = 360°/b, phase shift = -c/b, midline = d

8. Order of Transformations | 变换的顺序

When multiple transformations are applied, the order matters, especially when both horizontal stretch and horizontal translation are present. The safest approach is to start from the basic graph and apply transformations in the following sequence: horizontal stretch/compression first, then horizontal translation, then vertical stretch, then vertical translation.

当应用多个变换时，顺序很重要，尤其是当水平伸缩和水平平移同时存在时。最稳妥的方法是从基本图像出发，按以下顺序应用变换：先水平伸缩，再水平平移，再垂直伸缩，最后垂直平移。

To illustrate, consider y = 2 sin(3x + 90°) – 1. Begin with y = sin x. Apply the horizontal compression by a factor of 1/3 to get y = sin(3x). Then shift left by 90°/3 = 30° to get y = sin(3(x + 30°)) = sin(3x + 90°). Next, apply the vertical stretch by a factor of 2 to get y = 2 sin(3x + 90°). Finally, translate down by 1 unit to obtain y = 2 sin(3x + 90°) – 1.

举例说明，考虑 y = 2 sin(3x + 90°) – 1。从 y = sin x 开始。先进行水平压缩，因子为 1/3，得到 y = sin(3x)。然后向左平移 90°/3 = 30°，得到 y = sin(3(x + 30°)) = sin(3x + 90°)。接着做垂直拉伸，因子为 2，得到 y = 2 sin(3x + 90°)。最后向下平移 1 个单位，得到 y = 2 sin(3x + 90°) – 1。

Alternatively, for horizontal transformations, it is better to factor out the coefficient of x first. This reveals the true phase shift and avoids the common mistake of applying the translation before the stretch.

或者，对于水平方向的变换，最好的方法是先将 x 的系数提取出来。这将揭示真正的相位移，并避免先平移后伸缩的常见错误。

9. Sketching Strategy for Transformed Graphs | 绘制变换图像的策略

When sketching a transformed trigonometric graph, follow these steps systematically. Step 1: Identify the parameters a, b, c, and d by comparing with the general form. Step 2: Calculate the amplitude |a|, the period 360°/b, the phase shift -c/b, and the midline y = d. Step 3: Mark the midline on the graph as a dashed horizontal line. Step 4: Mark the maximum and minimum lines at y = d + |a| and y = d – |a|.

绘制变换后的三角函数图像时，请按以下步骤系统操作。第一步：通过与一般形式比较识别参数 a、b、c 和 d。第二步：计算振幅 |a|、周期 360°/b、相位移 -c/b 和中线 y = d。第三步：在图像上用虚线标出中线。第四步：在 y = d + |a| 和 y = d – |a| 处标出最大值线和最小值线。

Step 5: Determine the starting point of one cycle. For y = a sin(b(x + c/b)) + d, the cycle starts at x = -c/b and ends at x = -c/b + 360°/b. Step 6: Divide the period into four equal intervals corresponding to the quarter points of the sine or cosine cycle. Step 7: Plot the five key points: start, maximum, midpoint, minimum, and end. Step 8: Draw a smooth curve through these points, extending the pattern as needed.

第五步：确定一个周期的起点。对于 y = a sin(b(x + c/b)) + d，周期从 x = -c/b 开始，到 x = -c/b + 360°/b 结束。第六步：将周期分成四等份，对应于正弦或余弦周期的四个四分点。第七步：标出五个关键点：起点、最大值点、中点、最小值点和终点。第八步：用平滑曲线连接这些点，并根据需要延展图像。

10. Worked Example | 实例讲解

Let us work through a typical exam-style question. Sketch the graph of y = 2 cos(½x – 45°) + 1 for 0° ≤ x ≤ 720°.

让我们解决一道典型的考试风格题目。在 0° ≤ x ≤ 720° 范围内绘制 y = 2 cos(½x – 45°) + 1 的图像。

First, factor the expression: y = 2 cos(½(x – 90°)) + 1. Now identify the parameters: a = 2, b = ½, phase shift = +90° (to the right), d = 1. The amplitude is 2, the period is 360°/(½) = 720°, and the midline is y = 1.

首先，对表达式进行因式分解：y = 2 cos(½(x – 90°)) + 1。现在识别参数：a = 2，b = ½，相位移 = +90°（向右），d = 1。振幅为 2，周期为 360°/(½) = 720°，中线为 y = 1。

The maximum value is 1 + 2 = 3, occurring at x = 90° (where cos(0) = 1) and at x = 90° + 720° = 810° (outside our interval). Within the interval, the maximum also occurs at x = 90°. The minimum value is 1 – 2 = -1, occurring at x = 90° + 360° = 450°. The graph crosses the midline y = 1 at x = 90° + 180° = 270° and at x = 90° – 180° = -90° (outside the interval).

最大值为 1 + 2 = 3，出现在 x = 90° 处（此时 cos(0) = 1），也在 x = 90° + 720° = 810° 处（在我们的区间之外）。在区间内，最大值也出现在 x = 90° 处。最小值为 1 – 2 = -1，出现在 x = 90° + 360° = 450° 处。图像在 x = 90° + 180° = 270° 处穿越中线 y = 1，也在 x = 90° – 180° = -90° 处（在区间之外）。

Plot the key points: maximum at (90°, 3), midline crossing at (270°, 1), minimum at (450°, -1), and midline crossing again at (630°, 1). Connect these with a smooth U-shaped cosine curve. The graph starts at (0, 2), reaches its maximum at 90°, descends to the midline at 270°, reaches a minimum at 450°, returns to the midline at 630°, and ends at approximately (720°, 2).

标出关键点：最大值点 (90°, 3)，中线交点 (270°, 1)，最小值点 (450°, -1)，以及再次穿越中线的点 (630°, 1)。用平滑的余弦 U 形曲线连接这些点。图像从 (0, 2) 开始，在 90° 达到最大值，下降到 270° 处的 midline，在 450° 达到最小值，在 630° 回到中线，并在大约 (720°, 2) 结束。

11. Exam Tips and Common Pitfalls | 考试要点与常见陷阱

Several pitfalls frequently appear in examinations. First, students often confuse the direction of horizontal shifts: remember that y = sin(x + c) shifts left when c > 0. Second, students forget to factor out the coefficient of x before determining the phase shift; always rewrite bx + c as b(x + c/b) first.

考试中经常出现几个陷阱。第一，学生常混淆水平平移的方向：记住当 c > 0 时 y = sin(x + c) 向左平移。第二，学生在确定相位移之前忘记提取 x 的系数；始终先将 bx + c 改写为 b(x + c/b)。

Third, when dealing with y = a sin(bx) + d, students sometimes confuse the amplitude with the maximum value. The maximum is d + |a|, not |a|. Fourth, do not forget that the period of tan x is 180°, not 360°; for y = tan(bx), the period is 180°/b. Finally, always check whether the exam expects answers in degrees or radians.

第三，在处理 y = a sin(bx) + d 时，学生有时会将振幅与最大值混淆。最大值是 d + |a|，而不是 |a|。第四，不要忘记 tan x 的周期是 180° 而不是 360°；对于 y = tan(bx)，周期为 180°/b。最后，始终检查考试要求使用角度制还是弧度制。

Another useful tip is to verify your sketch by checking specific values. Substitute x = 0 into the transformed function and confirm that the plotted y-value matches. Also, check that the number of complete cycles visible in the given interval equals the interval length divided by the period.

另一个有用的技巧是通过代入特定值来验证你的草图。将 x = 0 代入变换后的函数，确认绘制的 y 值是否匹配。还要检查给定区间内可见的完整周期数是否等于区间长度除以周期。

12. Connection to Real-World Applications | 联系实际应用

Transformations of trigonometric graphs are not merely abstract exercises; they appear throughout physics and engineering. Simple harmonic motion, for instance, is described by equations of the form x = A sin(ωt + φ), where A is the amplitude, ω determines the period, and φ represents the phase. The vertical shift d appears when modelling the motion of an object relative to a non-zero equilibrium position.

三角函数图像的变换不仅仅是抽象练习；它们贯穿于物理和工程学中。例如，简谐运动由 x = A sin(ωt + φ) 形式的方程描述，其中 A 是振幅，ω 决定周期，φ 表示相位。垂直平移 d 在模拟物体相对于非零平衡位置的运动时出现。

In alternating current theory, voltage is expressed as V = V₀ sin(ωt + φ) + V_dc, where the vertical shift represents a DC offset. In oceanography, tidal heights follow a sinusoidal pattern where the amplitude corresponds to the tide’s magnitude, the period to the tidal cycle of approximately 12 hours, and the vertical translation to the mean sea level.

在交流电理论中，电压表示为 V = V₀ sin(ωt + φ) + V_dc，其中垂直平移表示直流偏置。在海洋学中，潮汐高度遵循正弦模式，振幅对应潮汐幅度，周期对应约 12 小时的潮汐循环，垂直平移对应平均海平面。

Understanding these transformations also helps in solving trigonometric equations graphically. The number of solutions to an equation such as 2 sin x = 1 in a given interval can be counted by sketching y = 2 sin x and the line y = 1, then counting their intersections. This graphical approach is often quicker and less error-prone than purely algebraic methods.

理解这些变换也有助于通过图像法求解三角方程。例如，在给定区间内，方程 2 sin x = 1 的解的个数可以通过绘制 y = 2 sin x 和直线 y = 1 并计算交点来数出。这种图像法通常比纯代数方法更快、更不容易出错。

Mastering the transformations of trigonometric graphs gives you a powerful toolkit for sketching, analysing, and applying trigonometric functions. Once you can mentally visualise how each parameter alters the graph, you will find that even complex questions become manageable.

掌握三角函数图像的变换，为你提供了一个强大的工具箱，用于绘制、分析和应用三角函数。一旦你能在脑海中想象每个参数如何改变图像，你会发现即使复杂的问题也变得容易处理。

Summary of key formulas: For y = a sin(bx + c) + d, the amplitude is |a|, the period is 360°/b, the phase shift is -c/b, and the equation of the midline is y = d. Always factor out b before reading the phase shift, apply horizontal stretch before horizontal translation, and verify your sketch with key points.

关键公式总结：对于 y = a sin(bx + c) + d，振幅为 |a|，周期为 360°/b，相位移为 -c/b，中线方程为 y = d。在读取相位移之前务必提取出 b，先进行水平伸缩再进行水平平移，并用关键点验证你的草图。

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Transformations of Trigonometric Graphs | 三角函数图像的变换