IB Mathematics: Graphs and Properties of General Trigonometric Functions | IB数学：一般三角函数的图像与性质

📚 IB Mathematics: Graphs and Properties of General Trigonometric Functions | IB数学：一般三角函数的图像与性质

The study of general trigonometric functions extends the basic sine, cosine and tangent graphs by introducing vertical stretches, horizontal compressions, phase shifts and vertical translations. In IB mathematics, a solid command of these transformations is essential for solving equations, modelling periodic phenomena and interpreting graphs.

一般三角函数的研究，通过在基本正弦、余弦和正切图像的基础上引入纵向伸缩、水平压缩、相移与垂直平移，使我们可以描述更丰富的周期现象。在 IB 数学中，熟练掌握这些变换，是解方程、建立周期模型以及解读图像的关键。

1. General Form and Parameters | 一般形式与参数

The general sine function is written as y = A sin(Bx + C) + D, where A, B, C and D are real constants. Each parameter has a clear geometric effect: A controls the vertical stretch, B controls the horizontal stretch, C produces a horizontal shift, and D produces a vertical shift. The same structure applies to cosine and tangent, although the tangent function has no finite amplitude.

一般正弦函数写作 y = A sin(Bx + C) + D，其中 A、B、C、D 为实常数。每个参数都有明确的几何作用：A 控制纵向伸缩，B 控制水平伸缩，C 产生水平平移，D 产生垂直平移。类似的框架也适用于余弦和正切函数，但正切函数没有有限的振幅。

y = A sin(Bx + C) + D

2. Amplitude | 振幅

The amplitude of y = A sin(Bx + C) + D is |A|. It measures half the vertical distance between the maximum and minimum values of the function. If A > 0, the graph oscillates between the horizontal lines y = D + A and y = D – A; if A < 0, the graph is reflected across the midline but the amplitude is still |A|.

函数 y = A sin(Bx + C) + D 的振幅为 |A|，它衡量最大值与最小值之间垂直距离的一半。若 A > 0，图像在水平直线 y = D + A 与 y = D – A 之间上下震荡；若 A < 0，图像关于中线翻转，但振幅仍为 |A|。

For example, in y = 3 sin(2x + π/3) + 1, the amplitude is 3. The midline is y = 1, so the maximum value is 4 and the minimum value is -2. In IB questions, the amplitude is often found by taking half the difference between the maximum and minimum y-values.

例如，在 y = 3 sin(2x + π/3) + 1 中，振幅为 3。中线为 y = 1，因此最大值为 4，最小值为 -2。在 IB 考试中，通常取最大 y 值与最小 y 值之差的一半来求振幅。

3. Period | 周期

The period of a trigonometric function is the smallest positive horizontal distance after which the graph repeats itself. For y = A sin(Bx + C) + D and y = A cos(Bx + C) + D, the period is P = 2π/B. For y = A tan(Bx + C) + D, the period is P = π/B.

三角函数的周期，是图像重复自身所需的最短正水平距离。对于 y = A sin(Bx + C) + D 和 y = A cos(Bx + C) + D，周期为 P = 2π/B；对于 y = A tan(Bx + C) + D，周期为 P = π/B。

This formula can be derived from the base function. Since sin x has period 2π, replacing x by Bx + C compresses the graph by a factor of B. Therefore, when x increases by P, the expression Bx + C increases by 2π, giving BP = 2π, so P = 2π/B.

该公式可由基本函数推导得出。由于 sin x 的周期为 2π，将 x 替换为 Bx + C 后，图像在水平方向压缩了 B 倍。因此，当 x 增加 P 时，表达式 Bx + C 增加 2π，即 BP = 2π，故 P = 2π/B。

P = 2π / B, P = π / B (tan)

4. Phase Shift | 相移

The phase shift is the horizontal displacement of the graph relative to a reference curve. To read it clearly, rewrite the function as y = A sin(B(x – h)) + k. Then h is the phase shift: the graph moves right by h units if h > 0, and left by |h| units if h < 0. Comparing this with y = A sin(Bx + C) + D gives h = -C/B and k = D.

相移是图像相对于参考曲线的水平位移。为了清楚地读出相移，可将函数改写为 y = A sin(B(x – h)) + k。此时 h 即为相移：若 h > 0，图像向右移动 h 个单位；若 h < 0，则向左移动 |h| 个单位。将它与 y = A sin(Bx + C) + D 比较，可得 h = -C/B，k = D。

For example, y = 3 sin(2x – π/3) + 1 can be written as y = 3 sin(2(x – π/6)) + 1. The phase shift is π/6 to the right. Notice that the sign of C alone is not enough; the phase shift always depends on the ratio -C/B.

例如，y = 3 sin(2x – π/3) + 1 可改写为 y = 3 sin(2(x – π/6)) + 1，其相移为向右 π/6。注意不能只看 C 的符号，相移始终取决于比值 -C/B。

5. Vertical Shift | 垂直位移

The vertical shift D moves the entire graph up or down without changing its shape. The horizontal midline, originally y = 0 for y = sin x, becomes y = D. Consequently, the maximum value of the transformed function is D + |A| and the minimum value is D – |A|.

垂直位移 D 使整个图像在不改变形状的前提下上下移动。原本 y = sin x 的水平中线为 y = 0，变换后变为 y = D。因此，变换后函数的最大值为 D + |A|，最小值为 D – |A|。

On a graph, the midline can always be located as the average of the maximum and minimum values. This is a fast and reliable way to determine D from a sketch or a set of data points.

在图像上，中线总可以取最大值与最小值的平均值来确定。这是从图像或数据点中快速确定 D 的可靠方法。

6. Sine Function: Graph and Properties | 正弦函数：图像与性质

The graph of y = sin x is a smooth wave that starts at the origin, rises to 1, falls back to 0, reaches -1, and then returns to 0. Its domain is all real numbers, its range is [-1, 1], and its period is 2π. The sine function is odd, meaning sin(-x) = -sin x.

函数 y = sin x 的图像是平滑波形，从原点出发，上升至 1，回到 0，再下降至 -1，最后回到 0。其定义域为全体实数，值域为 [-1, 1]，周期为 2π。正弦函数是奇函数，即 sin(-x) = -sin x。