Trigonometric Graphs & Transformations | 三角函数图像与变换

📚 Trigonometric Graphs & Transformations | 三角函数图像与变换

In mathematics, the graphs of sine, cosine, and tangent functions are fundamental tools for modelling periodic behaviour. Understanding how these graphs can be stretched, reflected, and shifted allows us to describe real‑world phenomena such as sound waves, tides, and alternating current. This article focuses on the standard forms y = a sin(bx + c) + d, y = a cos(bx + c) + d, and y = a tan(bx + c) + d, explaining each parameter’s effect with clear step‑by‑step reasoning.

在数学中，正弦、余弦和正切函数的图像是描述周期性现象的基本工具。理解这些图像如何被拉伸、反射和平移，能帮助我们描述声波、潮汐和交流电等现实世界中的现象。本文重点讨论标准形式 y = a sin(bx + c) + d、y = a cos(bx + c) + d 和 y = a tan(bx + c) + d，逐步解释每个参数的作用。

1. The Parent Graphs of Sine, Cosine, and Tangent | 正弦、余弦和正切的基本图像

Before applying any transformations, it is essential to recall the basic shapes of the three trigonometric graphs. The sine function y = sin x has a maximum of 1 and a minimum of −1, with a period of 360° (2π rad). It starts at the origin, rises to a peak, returns to zero, drops to a trough, and completes one full cycle at 360°.

在进行任何变换之前，首先需要回顾三个基本三角函数的图像形状。正弦函数 y = sin x 的最大值为 1，最小值为 −1，周期为 360°（2π 弧度）。它从原点出发，上升到波峰，回到零点，下降到波谷，并在 360° 处完成一个完整周期。

The cosine function y = cos x also has an amplitude of 1 and a period of 360°, but it starts at its maximum point (0,1) and falls to the midline before reaching its minimum. The tangent function y = tan x has a period of 180° (π rad), with vertical asymptotes at x = 90°, 270°, … where the value is undefined, and it passes through the origin.

余弦函数 y = cos x 的振幅也为 1，周期为 360°，但它从最大值 (0,1) 开始，下降到中间线后再达到最小值。正切函数 y = tan x 的周期为 180°（π 弧度），在 x = 90°、270° ……处有垂直渐近线，这些点函数值未定义，并且图像经过原点。

2. Amplitude Changes: The Role of Parameter a | 振幅变化：参数 a 的作用

For sine and cosine graphs, the parameter a in y = a sin x or y = a cos x determines the amplitude, which is the vertical distance from the midline to a peak or trough. The amplitude is given by |a|. If a is negative, the graph is reflected in the x‑axis, flipping the peaks into troughs while keeping the period unchanged.

对于正弦和余弦图像，y = a sin x 或 y = a cos x 中的参数 a 决定了振幅，即从中间线到波峰或波谷的垂直距离。振幅由 |a| 给出。如果 a 为负数，图像会关于 x 轴反射，峰变成谷，而周期保持不变。

For the tangent function, a does not represent an amplitude because the range is unbounded. Instead, a acts as a vertical stretch factor, making the graph steeper near the asymptotes when |a| > 1 and shallower when 0 < |a| < 1.

对于正切函数，a 不代表振幅，因为它的值域是无界的。相反，a 起到垂直拉伸因子的作用，当 |a| > 1 时图像在渐近线附近更陡，当 0 < |a| < 1 时更平缓。

Amplitude of y = a sin x or a cos x = |a|

y = a sin x 或 a cos x 的振幅 = |a|

3. Period Changes: The Role of Parameter b | 周期变化：参数 b 的作用

The parameter b affects the period of all three trigonometric functions. For y = sin(bx) or y = cos(bx), the period becomes 360° / |b| (or 2π / |b| rad). For y = tan(bx), the period is 180° / |b| (or π / |b| rad). When |b| > 1, the graph compresses horizontally; when 0 < |b| < 1, it stretches horizontally.

参数 b 会影响所有三个三角函数的周期。对于 y = sin(bx) 或 y = cos(bx)，周期变为 360° / |b|（或 2π / |b| 弧度）。对于 y = tan(bx)，周期为 180° / |b|（或 π / |b| 弧度）。当 |b| > 1 时，图像水平压缩；当 0 < |b| < 1 时，图像水平拉伸。

A common mistake is to think that b simply multiplies the x‑values; instead, the transformation is a horizontal scaling by factor 1/b. This means that key points such as intercepts and turning points must be recalculated by solving bx = standard angle.

一个常见错误是以为 b 只是简单地乘以 x 值；实际上，这种变换是以因数 1/b 进行水平缩放。这意味着必须通过解方程 bx = 标准角来重新计算截距和转折点等关键点。

Period of sin(bx) or cos(bx) = 360° / |b| ; Period of tan(bx) = 180° / |b|

sin(bx) 或 cos(bx) 的周期 = 360° / |b| ；tan(bx) 的周期 = 180° / |b|

4. Phase Shift: The Role of Parameter c | 相位移：参数 c 的作用

The parameter c in y = sin(x + c) creates a horizontal translation, often called a phase shift. Writing the expression as (x + c), the shift is to the left by c units when c is positive, and to the right by |c| units when c is negative. In the general form y = sin(bx + c), it is safer to factorise b: y = sin[b(x + c/b)], which reveals a phase shift of −c/b.

y = sin(x + c) 中的参数 c 产生水平平移，通常称为相位移。将表达式写作 (x + c)，当 c 为正数时图像向左平移 c 个单位，当 c 为负数时向右平移 |c| 个单位。在一般形式 y = sin(bx + c) 中，更安全的方法是提取因子 b：y = sin[b(x + c/b)]，这揭示出相位移为 −c/b。

This is a critical detail for examinations: the phase shift is not simply c; you must divide by b. For example, y = sin(2x − 40°) has a phase shift of +20° to the right, because the factorised form is sin[2(x − 20°)].

这是考试中的一个关键细节：相位移并不仅仅是 c，需要除以 b。例如，y = sin(2x − 40°) 的相位移是向右 20°，因为因式分解后得到 sin[2(x − 20°)]。

5. Vertical Translation: The Role of Parameter d | 垂直平移：参数 d 的作用

The parameter d in y = sin x + d moves the entire graph vertically. The midline shifts from y = 0 to y = d. For sine and cosine, the maximum becomes d + |a| and the minimum becomes d − |a|. For tangent, the asymptotes stay in the same positions, but the whole curve moves up or down by d units.

y = sin x + d 中的参数 d 将整个图像垂直移动。中间线从 y = 0 移动到 y = d。对于正弦和余弦，最大值变为 d + |a|，最小值变为 d − |a|。对于正切，垂直渐近线保持不动，但整条曲线向上或向下移动 d 个单位。

This transformation is particularly useful when modelling data that oscillates around a non‑zero baseline, such as the average daily temperature that varies around 15 °C rather than 0 °C.

当模拟围绕非零基线波动的数据时，这种变换特别有用，例如日平均气温在 15 °C 左右变化，而非 0 °C。

6. Combining Transformations: Order of Operations | 变换组合：操作顺序

When a trigonometric function contains all four parameters, the order in which we apply the transformations matters. Starting from the basic graph y = sin x, follow these steps:

当一个三角函数包含所有四个参数时，应用变换的顺序至关重要。从基本图像 y = sin x 出发，按以下步骤进行：

Step 1: Horizontal scaling by 1/b (period change).
Step 2: Horizontal translation by −c/b (phase shift).
Step 3: Vertical scaling by a (amplitude) and reflection if a is negative.
Step 4: Vertical translation by d.

步骤 1：水平缩放 1/b（周期变化）。
步骤 2：水平平移 −c/b（相位移）。
步骤 3：垂直缩放 a（振幅），若 a 为负则进行反射。
步骤 4：垂直平移 d。

This sequence works because it follows the standard mathematical order of operations inside the function argument first. Mixing up the order leads to incorrect graphs.

这个顺序有效是因为它遵循数学中标准运算顺序，先处理函数括号内部。打乱顺序会导致图像错误。

7. Sketching Transformed Sine and Cosine Graphs | 绘制变换后的正弦和余弦图像

To sketch y = a sin(bx + c) + d accurately, begin by identifying the midline y = d and drawing it as a dashed horizontal line. Next, mark the maximum and minimum lines y = d + |a| and y = d − |a|. Determine the period P = 360° / |b| and divide the period into four equal intervals to locate key points (start, peak, midline crossing, trough, end). Then apply the phase shift to position the starting point.

要准确绘制 y = a sin(bx + c) + d，首先确定中间线 y = d 并用虚线画出。接着标出最大值线 y = d + |a| 和最小值线 y = d − |a|。计算周期 P = 360° / |b|，将周期分成四个等间距，以确定关键点（起点、峰、中间线交点、谷、终点）。然后应用相位移来定位起点。

For a negative a, reflect the shape across the midline after plotting. Finally, connect the points with a smooth sinusoidal curve, ensuring it touches the peak and trough exactly once per cycle.

若 a 为负，在绘制后关于中间线反射形状。最后，用平滑的正弦曲线连接各点，确保每个周期精确地接触一次波峰和波谷。

8. Sketching Transformed Tangent Graphs | 绘制变换后的正切图像

Tangent graphs require a slightly different approach because they have asymptotes. Start with the basic period P = 180° / |b|. The vertical asymptotes occur where bx + c = 90° + k × 180° (in degrees). Solve these equations to find the asymptote positions, then shift them vertically by d if needed.

正切图像需要稍有不同的方法，因为它有渐近线。从基本周期 P = 180° / |b| 开始。垂直渐近线出现在 bx + c = 90° + k × 180°（以度为单位）的位置。解这些方程求出渐近线位置，然后根据需要垂直移动 d。

The curve passes through the point where bx + c = 0, translated vertically by d. Apply the vertical stretch a: if |a| > 1 the curve is steeper; if 0 < |a| < 1 it is shallower. Finally, draw the typical ‘S‑shaped’ branches between successive asymptotes.

曲线经过 bx + c = 0 对应的点，并垂直平移 d。应用垂直拉伸 a：如果 |a| > 1 曲线更陡；如果 0 < |a| < 1 更平缓。最后，在相邻渐近线之间画出典型的“S 形”分支。

9. Writing Equations from Given Graphs | 根据给定图像写出方程

When given a trigonometric graph, the equation can be deduced by inspecting amplitude, period, phase shift, and vertical shift. Measure the vertical distance from midline to peak for |a|; the sign is positive if the graph starts upward from the midline for sine or at a maximum for cosine, otherwise negative.

当给定一个三角函数图像时，可以通过观察振幅、周期、相位移和垂直位移来推导方程。测量从中间线到波峰的垂直距离得到 |a|；对于正弦，若从中间线向上起始 a 为正，对于余弦，若从最大值起始 a 为正，否则为负。

Find the period by measuring the distance between two successive peaks, then set P = 360° / |b| to solve for b. The phase shift can be read from the x‑coordinate of a key point, and d is the midline y‑value.

通过测量两个连续波峰之间的距离确定周期，然后设 P = 360° / |b| 解出 b。相位移可以从关键点的 x 坐标读取，d 是中间线的 y 值。

Always verify the equation by checking a couple of points to ensure the sign of a and the phase shift value are correct, especially when multiple forms could fit the same curve.

始终通过检查一两个点来验证方程，确保 a 的符号和相位移值正确，尤其当多个形式可能匹配同一条曲线时。

10. Real‑World Applications and Modelling | 现实世界应用与建模

Transformed trigonometric functions appear extensively in physics and engineering. For example, the depth of water in a harbour can be modelled as D(t) = 5 sin(30t − 15)° + 12, where t is time in hours. Here the amplitude 5 m represents the tidal variation, the midline 12 m is the average depth, and the phase shift accounts for the time of the first high tide.

变换后的三角函数广泛出现在物理学和工程学中。例如，港口水深可以用 D(t) = 5 sin(30t − 15)° + 12 建模，其中 t 是以小时为单位的时间。这里振幅 5 米代表潮汐变化，中间线 12 米是平均水深，相位移则体现了首次高潮的时间。

In electricity, the voltage of an AC circuit is often given by V = V₀ sin(ωt + φ), where V₀ is the peak voltage, ω relates to frequency, and φ is the phase angle. Understanding these parameters is vital for analysing circuit behaviour.

在电学中，交流电路的电压常用 V = V₀ sin(ωt + φ) 表示，其中 V₀ 是峰值电压，ω 与频率有关，φ 是相位角。理解这些参数对于分析电路行为至关重要。

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Trigonometric Graphs & Transformations | 三角函数图像与变换