📚 Transforming Trigonometric Graphs | 三角函数的图像变换
Transforming trigonometric graphs is one of the most practical and high-yield topics in the Edexcel A-Level Mathematics syllabus. You need to understand how the parameters in y = a sin(bx + c) + d and the equivalent cosine and tangent forms change amplitude, period, phase shift and vertical position.
三角函数图像的变换是 Edexcel A-Level 数学大纲中非常实用且高频的考点。你需要理解 y = a sin(bx + c) + d 以及对应的余弦、正切形式中的参数如何改变振幅、周期、相位平移和垂直位置。
1. The Parent Trigonometric Graphs | 基础三角函数图像
Before applying transformations, recall the three parent graphs: y = sin x, y = cos x and y = tan x. Sine and cosine have period 360° and range [-1, 1], while tangent has period 180° and vertical asymptotes at x = 90° + 180°n.
在应用变换之前，先回顾三个基础图像：y = sin x、y = cos x 和 y = tan x。正弦和余弦的周期为 360°，值域为 [-1, 1]；正切的周期为 180°，并且在 x = 90° + 180°n 处有竖直渐近线。
Sine starts at the origin and rises to a maximum at 90°. Cosine starts at its maximum at 0°. Tangent passes through the origin and increases between each pair of asymptotes.
正弦从原点出发，在 90° 时达到最大值。余弦在 0° 时从最大值开始。正切穿过原点，并在每两条渐近线之间递增。
2. Vertical Stretch: y = a sin x | 垂直伸缩：y = a sin x
The coefficient a changes the amplitude of the graph. For y = a sin x and y = a cos x, the amplitude is |a|. If a is greater than 1, the graph is stretched vertically; if 0 < a < 1, it is squashed vertically.
系数 a 会改变图像的振幅。对于 y = a sin x 和 y = a cos x，振幅为 |a|。如果 a 大于 1，图像被纵向拉伸；如果 0 < a < 1，图像被纵向压缩。
When a is negative, the graph is also reflected in the x-axis. The new range becomes [-|a|, |a|], so the graph oscillates between these two y-values.
当 a 为负数时，图像同时关于 x 轴反射。新的值域变为 [-|a|, |a|]，因此图像在这两个 y 值之间振荡。
Amplitude = |a|, Range = [-|a|, |a|] for y = a sin x
3. Vertical Translation: y = sin x + d | 垂直平移：y = sin x + d
Adding d moves the entire graph vertically. A positive d shifts the graph upward by d units, while a negative d shifts it downward. The horizontal axis or midline moves from y = 0 to y = d.
加上 d 会使整条图像垂直移动。d 为正时图像向上平移 d 个单位，d 为负时向下平移。水平轴或中线从 y = 0 移动到 y = d。
For y = sin x + d, the range becomes [d – 1, d + 1]. In the general model y = a sin(bx + c) + d, the midline is y = d and the range is [d – |a|, d + |a|].
对于 y = sin x + d，值域变为 [d – 1, d + 1]。在一般模型 y = a sin(bx + c) + d 中，中线为 y = d，值域为 [d – |a|, d + |a|]。
Vertical shift = d, Midline: y = d
4. Horizontal Stretch: y = sin(bx) | 水平伸缩：y = sin(bx)
The coefficient b changes the period of the graph. For y = sin(bx) and y = cos(bx), the new period is 360° / |b|. If b is greater than 1, the graph is compressed horizontally; if 0 < b < 1, it is stretched horizontally.
系数 b 会改变图像的周期。对于 y = sin(bx) 和 y = cos(bx)，新周期为 360° / |b|。如果 b 大于 1，图像被水平压缩；如果 0 < b < 1，图像被水平拉伸。
For tangent, the parent period is 180°, so y = tan(bx) has period 180° / |b|. Always divide the original period by the absolute value of b.
正切的基础周期是 180°，所以 y = tan(bx) 的周期为 180° / |b|。始终用原周期除以 |b| 来求新周期。
Period = 360° / |b| for sine and cosine, Period = 180° / |b| for tangent
5. Horizontal Translation: y = sin(x + c) | 水平平移：y = sin(x + c)
The constant c inside the bracket produces a phase shift. The graph of y = sin(x + c) is the graph of y = sin x shifted to the left by c units, because x + c = 0 gives the starting value x = -c.
括号内的常数 c 会产生相位平移。y = sin(x + c) 的图像是将 y = sin x 向左平移 c 个单位，因为 x + c = 0 可得起点 x = -c。
Similarly, y = sin(x – c) shifts the graph to the right by c units. Many students reverse this sign, so always test the transformation using a key point such as the first maximum or the origin.
类似地，y = sin(x – c) 将图像向右平移 c 个单位。很多学生会把这个符号弄反，因此一定要用第一个最大值或原点等关键点来检验变换。
6. Combined Transformations: y = a sin(bx + c) + d | 组合变换
The general sine model is y = a sin(bx + c) + d, where a affects amplitude, b affects period, c affects horizontal position, and d affects vertical position. However, you must factorise the bracket first when identifying the horizontal shift.
一般正弦模型为 y = a sin(bx + c) + d，其中 a 影响振幅，b 影响周期，c 影响水平位置，d 影响垂直位置。但是在识别水平平移时，必须先对括号进行因式分解。
Write y = a sin[b(x + c/b)] + d to reveal the true phase shift. The horizontal translation is c/b units to the left, or -c/b units to the right.
将表达式写成 y = a sin[b(x + c/b)] + d，才能显示出真正的相位平移。水平平移为向左 c/b 个单位，或写成向右 -c/b 个单位。
Amplitude = |a|, Period = 360° / |b|, Phase shift = -c/b, Vertical shift = d
7. Reflections in the Axes | 关于坐标轴的反射
Replacing sin x with -sin x reflects the graph in the x-axis. Replacing x with -x in y = sin(-x) reflects the graph in the y-axis.
把 sin x 换成 -sin x 会使图像关于 x 轴反射。把 x 换成 -x，即 y = sin(-x)，会使图像关于 y 轴反射。
Since sine is an odd function, sin(-x) = -sin x, so the two reflections can look identical for sine. Cosine is even, so cos(-x) = cos x, which means reflection in the y-axis leaves the cosine graph unchanged.
因为正弦是奇函数，sin(-x) = -sin x，所以对正弦来说这两种反射可能看起来相同。余弦是偶函数，cos(-x) = cos x，因此关于 y 轴的反射不会改变余弦图像。
8. Worked Example 1: Identifying Transformations | 例题 1：识别变换
Given y = 3 sin(2x – 60°) + 1, first factorise the bracket to y = 3 sin[2(x – 30°)] + 1. The amplitude is 3, the period is 180°, the phase shift is 30° to the right, and the vertical shift is 1 unit upward.
已知 y = 3 sin(2x – 60°) + 1，先将括号写成 y = 3 sin[2(x – 30°)] + 1。振幅为 3，周期为 180°，相位平移为向右 30°，垂直平移为向上 1 个单位。
The midline is y = 1, and the maximum and minimum values are 4 and -2 respectively. One full cycle can be drawn from x = 30° to x = 210°.
中线为 y = 1，最大值和最小值分别为 4 和 -2。完整的一个周期可以从 x = 30° 画到 x = 210°。
y = 3 sin[2(x – 30°)] + 1: amplitude 3, period 180°, shift 30° right, shift 1 up
9. Worked Example 2: Sketching a Transformed Graph | 例题 2：画出变换后的图像
Sketch y = 2 cos(3x + 90°) – 1. Start by rewriting the bracket as y = 2 cos[3(x + 30°)] – 1. The amplitude is 2, the period is 120°, the phase shift is 30° to the left, and the vertical shift is 1 unit downward.
画出 y = 2 cos(3x + 90°) – 1。先把括号改写为 y = 2 cos[3(x + 30°)] – 1。振幅为 2，周期为 120°，相位平移为向左 30°，垂直平移为向下 1 个单位。
Begin with cos x, compress horizontally by a factor of 3, shift 30° left, stretch vertically by a factor of 2, then move down 1. The maximum occurs at x = -30° with y = 1, and the minimum occurs at x = 30° with y = -3.
从 cos x 开始，水平压缩为原来的 1/3，向左平移 30°，纵向拉伸为 2 倍，再向下平移 1。最大值出现在 x = -30°，y = 1；最小值出现在 x = 30°，y = -3。
A complete cycle runs from x = -30° to x = 90°, which confirms the period of 120°. Marking the midline y = -1 helps you place the turning points accurately.
完整周期从 x = -30° 到 x = 90°，这验证了周期为 120°。标出中线 y = -1 有助于准确定位极值点。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
The most common error is misreading the phase shift because the factor b is not separated from c. For example, y = sin(2x + 60°) must become y = sin[2(x + 30°)] before you state the shift is 30° left, not 60° left.
最常见的错误是因为没有把系数 b 和 c 分离而导致相位平移判断错误。例如，y = sin(2x + 60°) 必须先变成 y = sin[2(x + 30°)]，才能判断平移是向左 30°，而不是向左 60°。
Another common mistake is mixing degrees and radians. Edexcel questions may use either unit, so write the period using the same unit as the angle: 360° / |b| in degrees or 2π / |b| in radians.
另一个常见错误是角度制和弧度制混用。Edexcel 题目可能使用任一单位，因此周期要使用与角度相同的单位：角度制为 360° / |b|，弧度制为 2π / |b|。
When reading a graph, use the formula amplitude = (max – min) / 2 and vertical shift = (max + min) / 2. This helps you recover the values of a and d before finding b and c from the period and phase shift.
根据图像读取参数时，使用公式：振幅 = (最大值 – 最小值) / 2，垂直平移 = (最大值 + 最小值) / 2。这样可以先确定 a 和 d，再由周期和相位平移确定 b 和 c。
11. Summary Table of Transformations | 变换总结表
Use this table as a quick reference when revising common transformations from y = sin x.
复习时可以使用下面的表格快速查阅 y = sin x 的常见变换。
| Transformation | Effect | 变换 | 效果 |
|---|---|---|---|
| y = a sin x | Amplitude |a|; reflection if a < 0 | y = a sin x | 振幅 |a|；a < 0 时反射 |
| y = sin(bx) | Period 360° / |b| | y = sin(bx) | 周期 360° / |b| |
| y = sin(x + c) | Shift left by c units | y = sin(x + c) | 向左平移 c 个单位 |
| y = sin x + d | Shift up by d units; midline y = d | y = sin x + d | 向上平移 d 个单位；中线
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