📚 Sine and Cosine Transforms | 正弦和余弦变换
This article investigates the transformations of sine and cosine graphs – the core building blocks of trigonometric modelling in IB Mathematics. By exploring vertical stretches (amplitude), horizontal stretches (period), horizontal shifts (phase shift) and vertical shifts, students will develop a systematic method for sketching, analysing and constructing sinusoidal functions from real-world data. Mastering these transforms lays the groundwork for understanding oscillatory phenomena in physics, engineering and beyond.
本文探究正弦和余弦函数图像的变换——这是IB数学中三角建模的核心组成部分。通过探讨纵向伸缩（振幅）、横向伸缩（周期）、水平平移（相移）和纵向平移，学生将建立起一套系统的方法，用以绘制草图、分析并根据真实数据构造正弦型函数。掌握这些变换能为理解物理、工程及其他领域中的振荡现象打下坚实基础。
1. The Parent Graphs of Sine and Cosine | 正弦与余弦的原始图像
s a characteristic wave that starts at (0,0), reaches a maximum of 1 at π/2, crosses the x‑ rts at (0,1), drops to 0 at π/2, reaches −1 at π, returns to 0 at 3π/2 and ends the cycle at (2π,1). Both graphs are continuous, smooth and periodic with the same period, domain and amplitude.
基本函数 y = sin x 的波形特征为：从 (0,0) 出发，在 π/2 处达到最大值 1，在 π 处穿过 x 轴，在 3π/2 处达到最小值 −1，并在 2π 处完成一个完整周期。其值域为 [−1,1]，周期为 2π。余弦基本函数 y = cos x 从 (0,1) 开始，在 π/2 处降至 0，在 π 处到达 −1，在 3π/2 处回到 0，并在 (2π,1) 结束该周期。两个图像均连续、光滑且具有周期性，同时享有相同的周期、定义域和振幅。
2. Vertical Stretch – Amplitude Transformation | 纵向伸缩——振幅变换
trigonometric function, as in y = A sin x or y = A cos x, changes the amplitude from 1 to |A|. If |A| > 1, the graph is vertically stretched; if 0 < |A| 1, it is vertically compressed. A negative A additionally reflects the graph across the x‑ is always defined as the distance from the midline to a maximum or minimum, hence given by |A|. The period and midline remain unchanged under this transformation alone.
在三角函数前引入系数 A，例如 y = A sin x 或 y = A cos x，能将振幅从 1 改变为 |A|。若 |A| > 1，图像被纵向拉伸；若 0 < |A| < 1，则被纵向压缩。负的 A 还会使图像关于 x 轴反射，颠倒波峰与波谷。振幅始终定义为中点线到最大值或最小值的距离，因此由 |A| 给出。仅在此变换下，周期和中点线保持不变。
Key values for reference:
参考关键值：
| Function | Amplitude | Effect |
| y = 2 sin x | 2 | Wave peaks at 2, troughs at −2 |
| y = ½ cos x | ½ | Peaks at 0.5, troughs at −0.5 |
| y = −3 sin x | 3 | Reflected across x‑axis; amplitude 3 |
The amplitude governs the intensity of oscillation and is the first parameter to identify when modelling periodic behaviour.
振幅决定了振荡的强度，在对周期性行为建模时是应首先确定的参数。
3. Horizontal Stretch – Period Transformation | 横向伸缩——周期变换
ent, such as y = sin(Bx) or y = cos(Bx), alters the period. The new period is given by 2π/|B|. If B > 1, the graph completes a cycle faster – ine or vertical shift. The transformation is purely a horizontal dilation centred on the y‑axis.
在自变量内部引入系数 B，例如 y = sin(Bx) 或 y = cos(Bx)，会改变周期。新周期由 2π/|B| 给出。若 B > 1，图像更快完成一个周期——周期缩短。若 0 < B < 1，图像在水平方向上拉伸，周期变长。关键是，B 不会影响振幅、中点线或纵向平移。该变换纯粹是以 y 轴为中心的横向伸缩。
Period = 2π/|B|
ts, while y = cos(x/3) needs 6π to complete one cycle. When B is negative, the period uses |B| tion, although for sine and cosine this can often be absorbed by supplementary identities.
例如，y = sin(2x) 每隔 π 单位完成一个完整波形，而 y = cos(x/3) 需要 6π 才能完成一个周期。当 B 为负数时，周期使用 |B|，同时图像还经历水平反射；不过对于正弦和余弦而言，这种反射通常可通过诱导公式吸收。
4. Horizontal Translation – Phase Shift | 水平平移——相移
ays factor out B first: y = sin[B(x − C/B)], then identify the correct phase shift as C/B.
自变量中的 C 项，如 y = sin(x − C) 或 y = sin(B(x − C))，会产生水平平移。在因式分解形式 y = sin(B(x − C)) 中，当 C > 0 时图像向右平移 C 单位，C < 0 时向左平移。若函数以 y = sin(Bx − C) 的形式给出，务必先提取公因式 B：y = sin[B(x − C/B)]，然后再将真正的相移确定为 C/B。
A common mistake is to read the shift directly from the raw expression without factoring. Remember that only the fully factored argument reveals the true translation.
一个常见错误是直接从原始表达式读取平移量而不进行因式分解。请记住，只有完全因式分解后的自变量形式才能揭示真正的平移。
Phase shift = C (when argument is B(x − C))
cos(x + π/2) shifts left by π/2, placing the starting maximum at −π/2 instead of 0. Combining phase shift with period changes requires care, as a shift that looks small can represent a large change when the period is compressed.
对余弦函数同样适用：y = cos(x + π/2) 向左平移 π/2，使起始最大值从 0 移至 −π/2。将相移与周期变化相结合时需要小心，因为当周期压缩时，看似微小的平移可能对应较大的变化。
5. Vertical Translation – Midline Shift | 纵向平移——中点线移动
ion, as in y = sin x + D or y = cos x + D, shifts the entire graph vertically. The midline – the horizontal line halfway between maximum and minimum – moves from y = 0 to y = D. The amplitude stays |A|, and the new maximum becomes D + |A| while the new minimum becomes D − |A|.
在三角函数外部加上常数 D，如 y = sin x + D 或 y = cos x + D，会将整个图像沿纵向平移。中点线——即最大值与最小值之间中点的水平线——从 y = 0 移至 y = D。振幅保持为 |A|，新的最大值变为 D + |A|，新的最小值变为 D − |A|。
This transformation is independent of all others: it simply floats the wave up or down. In applications, D often represents an equilibrium value (e.g. average water depth, room temperature, baseline voltage).
该变换独立于所有其他变换：它只是将波形向上或向下浮起。在应用中，D 通常代表某个平衡值（如平均水深、室温、基准电压）。
midline is y = 3, maximum is 5 and minimum is 1. The entire cosine ridge lifts by 3 units.
对于 y = 2 cos x + 3，中点线为 y = 3，最大值为 5，最小值为 1。整个余弦波形被抬高了 3 个单位。
6. Combining All Four Transformations | 四种变换的综合
= A sin(B(x − C)) + D or y = A cos(B(x − C)) + D. To sketch it efficiently, follow a step‑by‑step approach:
一般的正弦型函数可写作 y = A sin(B(x − C)) + D 或 y = A cos(B(x − C)) + D。为高效绘制草图，请遵循以下步骤：
- Identify the midline y = D and draw a dashed line.
- 确定中点线 y = D 并画出虚线。
- Determine amplitude |A| and mark the maximum D + |A| and minimum D − |A|.
- 确定振幅 |A|，并标出最大值 D + |A| 和最小值 D − |A|。
- Calculate period 2π/|B| and mark key intervals (typically four equal divisions: period/4).
- 计算周期 2π/|B|，并标记关键区间（通常分为四个等份：period/4)。
- Identify phase shift C: the starting point for one cycle (often where the sine normally starts at 0, or cosine starts at a peak).
- 确定相移 C：作为单个周期的起始点（对于正弦通常是原本应该位于 0 的位置，余弦则是波峰）。
- Plot the five key points (beginning, peak/valley, crossing midline, peak/valley, end) and join smoothly.
- 标出五个关键点（起点、波峰/波谷、穿过中点线、波峰/波谷、终点），并平滑连接。
→ plot. This prevents the common error of applying transformations in the wrong sequence.
每次都按相同顺序练习：中点线 → 振幅 → 周期 → 相移 → 描点。这能避免因变换顺序错误而引起的常见问题。
7. Transforming the Cosine Function | 余弦函数的变换
ns at a maximum, whereas for sine it begins at the midline increasing. When a phase shift is introduced, this distinction becomes less important because you can always complete the sketch using key points. For modelling, the choice between sine and cosine often depends on the initial condition of the phenomenon.
所有变换规则对余弦基本函数同样适用。仅有的细微差别在于周期的起点：对于 y = cos x，标准周期从最大值开始，而对于正弦，则从增加的中点线开始。当引入相移后，这一区别便不再那么重要，因为总能利用关键点完成草图。在建模时，选择正弦还是余弦通常取决于所描述现象的初始条件。
A cos(Bt) + D. Conversely, a pendulum released from the central point might be better fitted by a sine function. Understanding both parents gives flexibility.
例如，在 t = 0 时处于最高点的潮汐，可以用无相移的余弦函数方便地建模：y = A cos(Bt) + D。相反，从中心点释放的摆则更适合用正弦函数拟合。同时理解两种原始函数能提供灵活性。
8. Determining a Function from a Graph | 根据图像确定函数
A typical IB question provides a graph of a sinusoidal wave and asks for the values of A, B, C, D. Follow this strategy:
典型的 IB 试题会提供一条正弦波图像，要求求出 A、B、C、D 的值。请遵循下列策略：
- Find D by locating the midline: average of maximum and minimum values.
- 通过确定中点线求出 D：最大值与最小值的平均值。
- Calculate A as half the vertical distance from maximum to minimum.
- 计算 A 为最大值到最小值垂直距离的一半。
- Measure the length of one complete cycle to obtain the period P, then compute B = 2π/P.
- 测量一个完整周期的长度以获得周期 P，然后计算 B = 2π/P。
- Identify a convenient starting point for a sine cycle (midline rising) or a cosine cycle (maximum), then read the horizontal displacement C. Check sign carefully.
- 为正弦（上升的中点）或余弦（最大值）的周期确定一个方便的起点，然后读取水平位移 C。仔细核对符号。
- If the graph is reflected compared to the parent (e.g. starting at a trough), incorporate a negative A.
- 若图像与基本函数相比存在反射（例如从波谷开始），则给 A 加上负号。
Always verify your function by evaluating it at a few key x‑values and checking against the graph.
始终通过在几个关键 x 值处计算函数值并与图像进行核对来验证你的函数。
9. Reflections and Symmetry | 反射与对称性
the midline. A negative B, as in y = sin(−x), reflects the graph horizontally across the y‑ cosine, cos(−x) = cos x, so the cosine graph remains unchanged under horizontal reflection – it exhibits even symmetry. Sine is an odd function and thus is symmetric about the origin.
当 A 或 B 取负号时会发生反射。负的 A，例如 y = −sin x，将图像关于中点线纵向反射。负的 B，如 y = sin(−x)，将图像关于 y 轴水平反射。然而，由于 sin(−x) = −sin x，正弦图像的水平反射等效于纵向反射加上一个可能的相移。对余弦而言，cos(−x) = cos x，因此余弦图像在水平反射下保持不变——它具有偶对称性。正弦是奇函数，因此关于原点对称。
These identities are powerful tools when manipulating waveforms and can simplify the determination of C in ambiguous graphs.
这些恒等式是处理波形时强有力的工具，可以在含混不清的图像中简化 C 的确定过程。
10. Modelling Periodic Phenomena | 周期现象建模
odelling cycle involves: identifying the independent variable (often time t), recognising the periodic nature, extracting amplitude, period, phase shift and vertical shift from data, and finally interpreting constants in the real‑ l H(t) = 1.2 cos(0.51t − 2) + 3.4 might represent a harbour depth with a mean depth of 3.4 m, tidal range of 2.4 m and a cycle slightly over 12 hours.
正弦模型出现在潮汐高度、日照时长、温度变化、交流电压和轮子运动等场景中。典型的建模周期包括：确定自变量（通常是时间 t），识别周期性特征，从数据中提取振幅、周期、相移和纵向平移，最后在现实情境中解读各常量。例如，潮汐模型 H(t) = 1.2 cos(0.51t − 2) + 3.4 可能表示某个港口的深度，其中平均深度 3.4 米，潮差 2.4 米，周期略长于 12 小时。
When given raw data points, students may need to compute differences between maxima and minima to find A and D, determine the time between successive maxima for B, and choose a suitable reference point for C. Real‑world models often require rounding and careful unit conversion.
当给定原始数据点时，学生可能需要计算最大值和最小值之间的差值以求出 A 和 D，确定连续最大值之间的时间差以得到 B，并选择一个合适的参照点计算 C。现实世界模型常需要进行四舍五入和谨慎的单位换算。
11. Common Pitfalls and Refinements | 常见陷阱与细节完善
Even experienced students can stumble on these subtleties:
即使有经验的学生也可能在这些细微之处出错：
- shift. Always rewrite y = sin(2x − π/3) as y = sin[2(x − π/6)] to see the true shift π/6.
- 在确定相移时忘记从括号中提取 B。始终将 y = sin(2x − π/3) 改写为 y = sin[2(x − π/6)]，以看到真正的平移量 π/6。
- f cycles per unit interval, f = 1/Period. B is the angular frequency, ω = 2πf = 2π/Period.
- 混淆周期与频率。频率是单位间隔内的周期数，f = 1/周期。B 为角频率，ω = 2πf = 2π/周期。
- Reversing the sign of C. When the graph appears shifted to the right, C is positive in (x − C). Always test with a point.
- 弄反 C 的符号。当图像看似向右平移时，(x − C) 中的 C 为正。务必选点验证。
- Missing the impact of vertical shift on y‑axis scale. A change in D can dramatically alter where the wave sits, affecting the feasibility of a model.
- 忽略纵向平移对 y 轴尺度的影响。D 的变化可能大幅改变波形的位置，从而影响模型的可行性。
Consistent methodical practice is the best defence against these errors.
保持有条不紊的练习是防止这些错误的最佳手段。
12. Summary and IB Exam Tips | 总结与 IB 考试技巧
n(B(x − C)) + D provides complete control over sine and cosine graphs. All parameters influence the shape and position independently, and their geometric meanings – amplitude, period, phase shift and midline – must be internalised. In IB exams, questions progress from simple, single‑step transformations to multi‑step analysis of graphs and real‑ ing, annotating sketches with numerical values, and checking your final function against the given data are essential habits for top marks.
变换框架 y = A sin(B(x − C)) + D 能让我们完全掌控正弦和余弦图像。所有参数各自独立地影响图像的形状和位置，而它们的几何含义——振幅、周期、相移和中点线——必须被内化于心。在 IB 考试中，题目从简单的单步变换逐步过渡到对图像进行多步分析以及现实世界建模。展示清晰的步骤、在草图上标注数值、以及将最终函数与给定的数据进行核对，都是获得高分的必要习惯。
Remember that sine and cosine are merely shifted versions of each other, so the choice of which to use can simplify your modelling. With fluent transformation skills, you can decode any sinusoidal graph and construct accurate, meaningful models – a vital competency in the IB Mathematics syllabus.
请记住，正弦和余弦彼此仅是经过平移的版本，因此选择哪一个来使用可以简化你的建模过程。具备了娴熟的变换技能后，你就能解读任何正弦型图形并构建准确、有意义的模型——这是 IB 数学课程大纲中的一项关键能力。
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