📚 Graphing and Interpreting Trigonometric Functions | 三角函数图像的绘制与解读
Trigonometric functions are among the most important mathematical tools for modelling periodic phenomena, from sound waves and tides to seasonal temperature changes. In the IB Mathematics curriculum, mastering the graphing and interpretation of sine, cosine, and tangent functions is essential for success in both analysis and applications papers.
三角函数是建模周期现象最重要的数学工具之一，从声波、潮汐到季节性温度变化无不涉及。在 IB 数学课程中，掌握正弦、余弦和正切函数的图像绘制与解读，是分析与应用两套考卷取得高分的关键基础。
1. The Basic Sine and Cosine Curves | 基本正弦与余弦曲线
The sine function y = sin x has a domain of all real numbers and a range of [-1, 1]. Its graph passes through the origin, rises to a maximum of 1 at x = π/2, returns to 0 at x = π, reaches a minimum of -1 at x = 3π/2, and completes one full cycle at x = 2π.
正弦函数 y = sin x 的定义域为全体实数，值域为 [-1, 1]。其图像经过原点，在 x = π/2 处升至最大值 1，在 x = π 处回到 0，在 x = 3π/2 处达到最小值 -1，并在 x = 2π 处完成一个完整周期。
The cosine function y = cos x has the same domain and range, but its graph is shifted horizontally. It starts at its maximum value 1 at x = 0, decreases to 0 at x = π/2, reaches -1 at x = π, returns to 0 at x = 3π/2, and completes its cycle at x = 2π. The cosine curve is a sine curve shifted left by π/2.
余弦函数 y = cos x 具有相同的定义域和值域，但图像发生了水平平移。它从 x = 0 处的最大值 1 出发，在 x = π/2 处降至 0，在 x = π 处达到 -1，在 x = 3π/2 处回到 0，并在 x = 2π 处完成周期。余弦曲线可视为正弦曲线向左平移 π/2 的结果。
cos x = sin(x + π/2), sin x = cos(x − π/2)
Both curves are periodic with period 2π. The key points to memorise are the intercepts, maximum points, and minimum points within one cycle, as these anchor the shape of any transformed graph.
两条曲线均以 2π 为周期。需要熟记的关键点是每个周期内的零点、最大值点和最小值点，它们是绘制一切变换后图像的基本锚点。
2. The Basic Tangent Curve | 基本正切曲线
The tangent function y = tan x differs significantly from sine and cosine. Its range is all real numbers, and its period is π, not 2π. The graph has vertical asymptotes at x = π/2 + kπ, where k is an integer, because cos x = 0 at these points and tan x = sin x / cos x is undefined.
正切函数 y = tan x 与正弦、余弦函数有显著差异。其值域为全体实数，周期为 π 而非 2π。由于在 x = π/2 + kπ（k 为整数）处 cos x = 0，tan x = sin x / cos x 无定义，因此图像在这些位置有垂直渐近线。
Within each interval from -π/2 to π/2, the tangent curve increases monotonically from negative infinity to positive infinity, passing through the origin. The curve is concave down on (-π/2, 0) and concave up on (0, π/2).
在每个从 -π/2 到 π/2 的区间内，正切曲线从负无穷单调递增至正无穷，并经过原点。曲线在 (-π/2, 0) 上为上凸（凹向下），在 (0, π/2) 上为下凸（凹向上）。
When sketching y = tan x, always draw the vertical asymptotes first as dashed lines, then plot the intercept at the midpoint of each branch, and finally sketch the increasing S-shaped branch between each pair of asymptotes.
绘制 y = tan x 时，务必先用虚线画出垂直渐近线，再标出每支曲线中点的零点，最后在相邻两条渐近线之间画出递增的 S 形曲线分支。
3. Amplitude Transformations | 振幅变换
For a function of the form y = a sin x or y = a cos x, the parameter |a| is called the amplitude. The amplitude represents half the distance between the maximum and minimum values of the function. If a > 0, the graph maintains its original orientation; if a < 0, the graph is reflected across the x-axis.
对于形如 y = a sin x 或 y = a cos x 的函数，参数 |a| 称为振幅。振幅表示函数最大值与最小值之间距离的一半。若 a > 0，图像保持原有方向；若 a < 0，图像关于 x 轴反射。
For example, y = 3 sin x oscillates between 3 and -3, while y = -2 cos x oscillates between 2 and -2 but begins at -2 when x = 0. The period remains unchanged at 2π because a only scales the vertical direction.
例如，y = 3 sin x 在 3 与 -3 之间振荡，而 y = -2 cos x 在 2 与 -2 之间振荡，但在 x = 0 时从 -2 开始。由于 a 仅对垂直方向进行缩放，周期仍保持为 2π 不变。
Range of y = a sin x: [−|a|, |a|]
In IB examinations, the amplitude is often stated as a positive value in the question paper, but students must recognise that a negative sign implies a reflection. Always check whether the graph opens upward or downward at its starting point.
在 IB 考试中，题目给出的振幅通常为正数，但学生必须意识到负号意味着反射。始终检查图像在其起始点处是向上还是向下开口。
4. Period Transformations | 周期变换
For y = sin(bx) or y = cos(bx), the parameter b affects the period of the function. The relationship between the period T and the coefficient b is given by:
对于 y = sin(bx) 或 y = cos(bx)，参数 b 影响函数的周期。周期 T 与系数 b 之间的关系为：
T = 2π / |b| for sine and cosine; T = π / |b| for tangent
If b > 1, the graph is horizontally compressed, completing more cycles within the same interval. If 0 < b < 1, the graph is horizontally stretched. A negative value of b causes a reflection across the y-axis, but since cosine is an even function, cos(-bx) = cos(bx), and since sine is odd, sin(-bx) = -sin(bx).
若 b > 1，图像在水平方向被压缩，在同一区间内完成更多周期。若 0 < b < 1，图像在水平方向被拉伸。b 为负值时图像关于 y 轴反射，但余弦是偶函数故 cos(-bx) = cos(bx)，正弦是奇函数故 sin(-bx) = -sin(bx)。
For example, y = sin(2x) has a period of π, meaning two complete cycles fit into the interval [0, 2π]. The key points of the standard sine curve are divided by 2 horizontally: the maximum occurs at x = π/4, and the minimum at x = 3π/4.
例如，y = sin(2x) 的周期为 π，意味着在区间 [0, 2π] 内可容纳两个完整周期。标准正弦曲线的关键点横坐标均除以 2：最大值出现在 x = π/4，最小值出现在 x = 3π/4。
When sketching, divide the standard period 2π by b to find the new period, then divide that period into four equal subintervals to locate the maximum, minimum, and two zero-crossing points.
绘图时，先将标准周期 2π 除以 b 得到新周期，再将新周期四等分以定位最大值点、最小值点和两个零点。
5. Phase Shift | 相位平移（水平位移）
For y = sin(x – c), the parameter c causes a horizontal shift called the phase shift. If c > 0, the graph shifts to the right by c units; if c < 0, the graph shifts to the left by |c| units. The general formula for phase shift is:
对于 y = sin(x – c)，参数 c 引起水平平移，称为相位平移。若 c > 0，图像向右平移 c 个单位；若 c < 0，图像向左平移 |c| 个单位。相位平移的一般公式为：
Phase shift = c / b (for y = sin(bx – c))
For y = sin(x – π/2), the entire sine curve moves π/2 units to the right: the point that was at the origin now sits at x = π/2, which is precisely the cosine curve. This confirms the identity sin(x – π/2) = -cos x.
对于 y = sin(x – π/2)，整条正弦曲线向右移动 π/2 个单位：原本在原点的点现在位于 x = π/2，这恰好就是余弦曲线。这印证了恒等式 sin(x – π/2) = -cos x。
A common student error is to confuse the direction of the shift. Remember that the expression (x – c) means the graph of y = sin x is translated so that what happened at x now happens at x + c, i.e., shifted to the right. Think of subtraction as delaying in time.
学生常见错误是混淆平移方向。记住表达式 (x – c) 意味着 y = sin x 的图像被平移，使原来在 x 处发生的事情现在发生在 x + c 处，即向右平移。可以把减法理解为时间上的延迟。
6. Vertical Shift | 垂直位移
For y = sin x + d, the parameter d shifts the entire graph vertically. If d > 0, the graph moves upward by d units; if d < 0, it moves downward. The midline of the function, also called the principal axis, moves from y = 0 to y = d.
对于 y = sin x + d，参数 d 使整个图像垂直移动。若 d > 0，图像向上移动 d 个单位；若 d < 0，则向下移动。函数的中线（也称主轴）从 y = 0 移至 y = d。
The new maximum and minimum values are d + |a| and d – |a| respectively for the general form y = a sin(bx – c) + d. The vertical shift does not affect the period, phase shift, or amplitude — it only relocates the entire graph along the y-axis.
对于一般形式 y = a sin(bx – c) + d，新的最大值和最小值分别为 d + |a| 和 d – |a|。垂直位移不影响周期、相位平移或振幅——它仅沿 y 轴重新定位整条曲线。
For example, y = cos x + 2 oscillates between 3 and 1, with a midline at y = 2. When sketching, first draw the midline as a dashed horizontal line, then plot the maximum and minimum points relative to this line.
例如，y = cos x + 2 在 3 与 1 之间振荡，中线位于 y = 2。绘图时，先画出虚线水平中线，再相对于该中线标出最大值点和最小值点。
7. The General Form | 一般形式与综合变换
Combining all transformations, the general sine and cosine functions can be written as:
综合所有变换，一般形式的正弦和余弦函数可写为：
y = a sin(bx – c) + d 或 y = a cos(bx – c) + d
The roles of each parameter are: |a| is the amplitude; b determines the period via T = 2π/b; c/b gives the phase shift to the right if positive; d is the vertical shift (midline). The standard approach in IB is to identify each parameter systematically before sketching.
各参数的作用为：|a| 是振幅；b 通过 T = 2π/b 决定周期；c/b 给出相位平移（正值向右）；d 是垂直位移（中线）。IB 的标准解题思路是在绘图前系统性地逐一识别每个参数。
Consider y = 2 sin(3x – π) + 1. Here, amplitude = 2, period = 2π/3, phase shift = π/3 to the right, and midline is y = 1. The range is [-1, 3]. To sketch, start from y = sin x, then apply horizontal compression by 3, shift right by π/3, stretch vertically by 2, and finally shift up by 1.
考虑 y = 2 sin(3x – π) + 1。这里振幅 = 2，周期 = 2π/3，相位平移向右 π/3，中线为 y = 1，值域为 [-1, 3]。绘图时，从 y = sin x 出发，先水平压缩 3 倍，再向右平移 π/3，然后垂直拉伸 2 倍，最后向上平移 1 个单位。
For tangent functions, the general form y = a tan(bx – c) + d has no amplitude, but |a| vertically stretches the branches, the period is π/b, and the vertical asymptotes shift according to the phase shift.
对于正切函数，一般形式 y = a tan(bx – c) + d 没有振幅概念，但 |a| 会垂直拉伸各分支，周期为 π/b，垂直渐近线随相位平移而移动。
8. Sketching Strategy | 五步绘图法
To sketch any transformed trigonometric graph accurately, follow a systematic five-step strategy. First, identify the basic function (sine, cosine, or tangent) and note its standard shape. Second, determine the period and quarter-period intervals; the quarter-period is T/4, which gives the x-spacing between key points.
为准确绘制任意变换后的三角函数图像，应遵循系统性的五步策略。第一步，识别基本函数（正弦、余弦或正切）并记住其标准形状。第二步，确定周期和四分之一周期间隔；四分之一周期为 T/4，这给出了关键点之间的 x 方向间距。
Third, calculate the phase shift and mark the starting point of one cycle on the x-axis. Fourth, draw the midline y = d and mark the maximum and minimum positions relative to it. Fifth, plot the five key points for sine/cosine — start, maximum, midline crossing, minimum, and end — then connect them with a smooth curve.
第三步，计算相位平移并在 x 轴上标出一个周期的起点。第四步，画出中线 y = d 并标出最大、最小值相对中线的位置。第五步，为正弦/余弦标出五个关键点——起点、最大值点、中线交点、最小值点、终点——然后用平滑曲线连接。
For y = tan x, instead of five points, locate the asymptotes (T/2 apart), the zero crossing at the midpoint of each branch, and one additional point on each side to capture the curvature. Always extend the graph beyond one period using the periodic property.
对于 y = tan x，不用五个点，而是确定渐近线（间距为 T/2）、每支中点的零点，以及每侧各一个辅助点以反映曲率。最后利用周期性将图像延伸至一个周期之外。
9. Reading Graphs in Reverse | 由图像反求函数解析式
A standard IB question gives a graph and asks for its equation. Work in reverse: read the midline from the horizontal line halfway between maximum and minimum; this is d. The amplitude is the distance from the midline to either the maximum or the minimum; this is |a|.
IB 常见考题给出图像要求反求解析式。逆向操作：从最大值与最小值之间的水平中线读出 d；中点到最大值或最小值的距离即为振幅 |a|。
Divide the horizontal distance between two consecutive maximum points (or two consecutive minimum points) by 2π to obtain 1/b, hence b = 2π / T. To find c, locate the point where one full cycle begins — this is the phase shift, and c = b × phase shift.
将相邻两个最大值点（或相邻两个最小值点）之间的水平距离除以 2π 得到 1/b，从而 b = 2π / T。为求 c，找到一个完整周期起点的 x 坐标——这就是相位平移，c = b × 相位平移。
Determine whether to use sine or cosine based on the y-value at the cycle start: if the function begins at the midline and rises, use sine; if it begins at its maximum, use cosine. Check the sign of a by observing whether the function rises immediately after the start point.
根据周期起点处的 y 值判断用正弦还是余弦：若函数从中线开始并上升，用正弦；若从最大值开始，用余弦。通过观察起点后函数是否立即上升来判断 a 的正负。
10. Applications and Exam Insights | 实际应用与考点总结
Trigonometric graphs model real-world cyclic behaviour. In IB Paper 2 applications, you may encounter tide levels, Ferris wheel heights, temperature cycles, or sound wave frequencies. These problems typically ask you to write a sine or cosine model, then interpret or predict values using the graph or the equation.
三角函数图像可以建模现实中的周期行为。在 IB Paper 2 应用题中，你可能会遇到潮汐水位、摩天轮高度、温度周期变化或声波频率等问题。这类题目通常要求建立正弦或余弦模型，再借助图像或方程解释或预测数值。
Common exam traps include: forgetting that period = 2π/|b| not 2π/b when b is negative; confusing phase shift with the sign inside the bracket; omitting the vertical shift when reading the midline from a graph; and mixing up the direction of horizontal shifts.
常见考试陷阱包括：当 b 为负数时忘记周期 = 2π/|b| 而非 2π/b；混淆括号内符号与相位平移的方向；从图像读中线时遗漏垂直位移；搞混水平平移的方向。
To avoid these errors, always write the function in standard form y = a sin(bx – c) + d first, explicitly list all four parameters with their values and signs, and check your sketch by substituting at least one known point from the question into your equation.
为避免这些错误，务必先将函数写成标准形式 y = a sin(bx – c) + d，明确列出全部四个参数的值与符号，并通过将题目中至少一个已知点代入自己的方程来检验图像。
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