Sketching Graphs: A Guide for IB Mathematics | 图像绘制：IB数学指南

📚 Sketching Graphs: A Guide for IB Mathematics | 图像绘制：IB数学指南

Graph sketching is a fundamental skill in IB Mathematics, allowing you to visualise functions and understand their behaviour without relying solely on technology. It combines transformations, calculus, and algebraic insight to produce accurate representations of linear, polynomial, rational, exponential, logarithmic, and trigonometric functions. This article provides a comprehensive, step-by-step approach to sketching graphs, covering essential techniques and common pitfalls, all aligned with the IB Analysis & Approaches and Applications & Interpretation syllabi.

图像绘制是IB数学的一项基本技能，它能帮助你直观地理解函数行为，而不仅仅依赖计算器。图像绘制结合了变换、微积分和代数洞察，能够准确描绘一次、多项式、有理、指数、对数及三角函数。本文提供一套全面、逐步的图像绘制方法，涵盖核心技巧与常见误区，完全契合IB分析与方法和应用与解释课程大纲要求。

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1. Introduction to Graph Sketching | 图像绘制简介

Graph sketching is not the same as plotting points precisely. Instead, it is about capturing the key features of a function: intercepts, asymptotes, turning points, symmetry, and end behaviour. In IB exams, you are often asked to sketch a curve after applying transformations or after finding stationary points using calculus. A clear, well-labelled sketch shows understanding and earns method marks.

图像绘制并非逐点精确绘图，而是抓住函数的关键特征：截距、渐近线、拐点、对称性和末端行为。在IB考试中，常要求你在进行变换后或用微积分求出驻点后画出函数草图。一张标注清晰、特征明确的草图不仅能展示你的理解，还能赢得过程分。

A successful sketch usually includes axes, labelled intercepts, asymptotes (if any), the general shape, and coordinates of turning points where relevant. You do not need to calculate every point; instead, use the function’s structure and derivatives to infer its behaviour.

一张成功的草图通常包含坐标轴、标出截距、渐近线（如有）、大致形状以及相关拐点的坐标。你无需计算每一个点，而是借助函数的结构和导数来推断其走势。

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2. Linear and Quadratic Graphs | 一次与二次函数图像

Linear functions of the form y = mx + c are the simplest to sketch. The constant c gives the y-intercept, and m is the gradient. A positive m yields an increasing line, while a negative m gives a decreasing line. Horizontal lines occur when m = 0, and vertical lines have the equation x = k.

形如 y = mx + c 的一次函数是最易绘制的。常数 c 代表 y 轴截距，m 为斜率。m 为正时直线上升，为负时直线下降；m=0 时得到水平线，而竖直线方程为 x = k。

Quadratic functions y = ax² + bx + c have a parabolic shape. The sign of a determines the opening direction: if a > 0, the parabola opens upwards and has a minimum turning point; if a < 0, it opens downwards and has a maximum. The vertex can be found by completing the square to obtain y = a(x − h)² + k, where (h, k) is the vertex. Alternatively, use x = −b/(2a) for the axis of symmetry.

二次函数 y = ax² + bx + c 的图像为抛物线。a 的符号决定开口方向：a > 0 时开口向上，有极小值拐点；a < 0 时开口向下，有极大值拐点。顶点可通过配方法得到 y = a(x − h)² + k，其中 (h, k) 即为顶点，或用对称轴公式 x = −b/(2a) 来定位。

The discriminant Δ = b² − 4ac tells us about x-intercepts: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root (the graph touches the x-axis), and Δ < 0 means no real roots (the graph does not cross the x-axis). Always mark the y-intercept (0, c) and the vertex on your sketch.

判别式 Δ = b² − 4ac 能揭示 x 轴截距情况：Δ > 0 有两个相异实根，Δ = 0 有一个重根（图像与 x 轴相切），Δ < 0 则无实根（图像不与 x 轴相交）。画草图时务必标出 y 轴截距 (0, c) 和顶点。

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3. Polynomial Functions: End Behaviour and Turning Points | 多项式函数：末端行为与拐点

For higher-degree polynomial functions like y = axⁿ + … (with integer n ≥ 2), the end behaviour depends on the sign of the leading coefficient a and whether the degree n is even or odd. If n is even and a > 0, the graph rises on both ends; if a < 0, it falls on both ends. If n is odd, a > 0 means the graph falls to the left and rises to the right, while a < 0 gives the opposite behaviour.

对于高次多项式函数，如 y = axⁿ + …（整数 n ≥ 2），其末端行为取决于首项系数 a 的符号以及次数 n 的奇偶性。若 n 为偶数且 a > 0，图像两端均向上；a < 0 则两端向下。若 n 为奇数，a > 0 时图像左降右升，a < 0 时左升右降。

A polynomial of degree n can have up to n − 1 turning points, though it may have fewer. For example, a cubic function y = x³ − 3x has degree 3 and up to 2 turning points. To locate these points precisely, we use differentiation (covered later). When sketching by hand, factorising the polynomial helps find x-intercepts and understand the shape between roots.

一个 n 次多项式最多可有 n − 1 个拐点，但实际可能更少。例如三次函数 y = x³ − 3x 的次数为 3，最多有 2 个拐点。要精确定位这些点，需要借助微分（后文将介绍）。手绘草图时，因式分解有助于找出 x 轴截距并理解各根之间的形状。

Repeated roots also affect the graph: a root with even multiplicity (e.g. (x − 2)²) causes the graph to touch the x-axis and turn back, whereas an odd multiplicity root (e.g. (x + 1)³) means the graph crosses the x-axis with a flattening effect.

重根也会影响图像：偶重根（如 (x − 2)²）使图像与 x 轴相切后折返；奇重根（如 (x + 1)³）则图像穿过 x 轴，且在交点处变平。

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4. Rational Functions: Asymptotes and Intercepts | 有理函数：渐近线与截距

Rational functions are ratios of two polynomials, e.g. f(x) = P(x)/Q(x). The key features to identify are vertical asymptotes (where Q(x) = 0 and the numerator is non-zero at that point), horizontal or oblique asymptotes (determined by comparing degrees of P and Q), and intercepts. Always check whether a common factor can be cancelled, as this creates a hole in the graph rather than a vertical asymptote.

有理函数是两个多项式之比，例如 f(x) = P(x)/Q(x)。需要识别的主要特征包括垂直渐近线（在 Q(x)=0 且分子在该点非零处）、水平或斜渐近线（通过比较分子和分母的次数确定）以及截距。务必检查能否约去公因式，若能则会产生一个可去间断点（图像中的“洞”）而非垂直渐近线。

• If degree of P < degree of Q, horizontal asymptote is y = 0.
• If degree of P equals degree of Q, horizontal asymptote is y = (leading coefficient of P)/(leading coefficient of Q).
• If degree of P = degree of Q + 1, there is an oblique (slant) asymptote found by polynomial long division.

• 若分子次数 < 分母次数，水平渐近线为 y = 0。
• 若分子次数等于分母次数，水平渐近线为 y = (分子首项系数)/(分母首项系数)。
• 若分子次数 = 分母次数 + 1，存在斜渐近线，可通过多项式长除法求得。

To sketch a rational function, first find the x- and y-intercepts, then draw the asymptotes as dashed lines. Test the sign of f(x) in each region created by vertical asymptotes and x-intercepts to see whether the graph approaches +∞ or −∞ near the asymptotes.

绘制有理函数草图时，先求出 x、y 截距，再用虚线画出渐近线。然后检验各垂直渐近线与 x 截距划分出的区间内 f(x) 的正负号，以判断图像在渐近线附近是趋向 +∞ 还是 −∞。

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5. Exponential and Logarithmic Graphs | 指数与对数函数图像

Exponential functions of the form y = aˣ with a > 0, a ≠ 1, have a horizontal asymptote at y = 0, pass through (0, 1), and are always positive. For a > 1, the graph increases rapidly; for 0 < a < 1, it decreases. The natural exponential function y = eˣ is particularly important in calculus because its derivative is itself.

形如 y = aˣ（a > 0，a ≠ 1）的指数函数具有水平渐近线 y = 0，经过点 (0, 1) 且函数值恒正。当 a > 1 时图像急速上升；当 0 < a < 1 时图像衰减。自然指数函数 y = eˣ 在微积分中格外重要，因为其导数等于自身。

Logarithmic functions y = logₐ x are inverses of exponentials. Their domain is x > 0, they have a vertical asymptote at x = 0, and they pass through (1, 0). The graph of y = ln x (natural log) is the reflection of y = eˣ in the line y = x. Transformations of these functions follow the same rules as others: y = 2ˣ⁻¹ + 3 involves a horizontal shift right by 1, vertical shift up by 3.

对数函数 y = logₐ x 是指数函数的反函数。其定义域为 x > 0，有垂直渐近线 x = 0，且经过 (1, 0)。y = ln x（自然对数）的图像是 y = eˣ 关于直线 y = x 的反射。这些函数的变换规则与其他函数一致：如 y = 2ˣ⁻¹ + 3 表示右移 1 单位、上移 3 单位。

When sketching, always show the asymptote and at least two key points. For exponential growth or decay models, consider the long-term behaviour and any horizontal asymptote that describes equilibrium.

绘图时务必画出渐近线及至少两个关键点。对于指数增长或衰减模型，需考虑长期行为以及描述平衡态的水平渐近线。

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6. Trigonometric Graphs and Transformations | 三角函数图像及其变换

The three primary trigonometric functions are sine, cosine, and tangent. The graphs of y = sin x and y = cos x have period 2π, amplitude 1, and oscillate between -1 and 1. The sine graph starts at the origin, while the cosine graph starts at (0, 1). The tangent function y = tan x has period π and vertical asymptotes at x = π/2 + nπ.

三个基本三角函数是正弦、余弦和正切。y = sin x 与 y = cos x 的图像周期为 2π，振幅为 1，在 -1 与 1 之间振荡。正弦图从原点出发，余弦图则始于 (0, 1)。正切函数 y = tan x 周期为 π，在 x = π/2 + nπ 处有垂直渐近线。

General sinusoidal functions can be written as y = A sin(B(x − C)) + D or y = A cos(B(x − C)) + D. Here, |A| is the amplitude, the period is 2π/|B|, C represents the horizontal shift (phase shift), and D is the vertical shift (equilibrium line). For tangent, the period is π/|B|.

一般正弦型函数可写作 y = A sin(B(x − C)) + D 或 y = A cos(B(x − C)) + D。其中 |A| 为振幅，周期为 2π/|B|，C 表示水平位移（相位移动），D 表示垂直位移（平衡线）。对正切函数，周期为 π/|B|。

Transformation (变换)	Effect on y = sin x (对 y = sin x 的影响)
y = 2 sin x	Vertical stretch, amplitude = 2 (垂直拉伸，振幅为2)
y = sin(x) + 1	Shift up by 1 (上移1单位)
y = sin(x − π/2)	Shift right by π/2 (右移π/2)
y = sin(2x)	Horizontal compression, period = π (水平压缩，周期为π)

Always mark the maximum, minimum, intercepts, and asymptotes clearly. When sketching transformed trigonometric functions, it helps to apply transformations step by step, starting from the basic parent function.

绘图时务必清晰标出最大值、最小值、截距和渐近线。在绘制经变换的三角函数时，从基本母函数出发逐步施加变换会很有帮助。

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7. Transformations of Functions: Translations, Reflections, Stretches | 函数变换：平移、反射与伸缩

Transformations allow you to sketch a new graph from a known parent function f(x). The main types are:

• Vertical translation: y = f(x) + k – moves the graph up (k > 0) or down (k < 0).
• Horizontal translation: y = f(x − h) – moves the graph right (h > 0) or left (h < 0). Note the sign inside the bracket is opposite to the direction of shift.
• Reflection in x-axis: y = -f(x) – flips the graph vertically.
• Reflection in y-axis: y = f(-x) – flips the graph horizontally.
• Vertical stretch/compression: y = a f(x) – if |a| > 1, the graph stretches vertically; if 0 < |a| < 1, it compresses. A negative a also reflects.
• Horizontal stretch/compression: y = f(bx) – if |b| > 1, the graph compresses horizontally; if 0 < |b| < 1, it stretches.

变换使你能从已知的母函数 f(x) 画出新图像。主要类型包括：

• 垂直平移： y = f(x) + k – 图像上移（k > 0）或下移（k < 0）。
• 水平平移： y = f(x − h) – 图像右移（h > 0）或左移（h < 0）。注意括号内的符号与移动方向相反。
• 关于 x 轴反射： y = -f(x) – 垂直翻转图像。
• 关于 y 轴反射： y = f(-x) – 水平翻转图像。
• 垂直伸缩： y = a f(x) – 若 |a| > 1，垂直拉伸；若 0 < |a| < 1，垂直压缩。a 为负时同时反射。
• 水平伸缩： y = f(bx) – 若 |b| > 1，水平压缩；若 0 < |b| < 1，水平拉伸。

It is vital to apply transformations in the correct order when multiple are combined. The standard convention is to perform horizontal shifts and stretches first (working with the input x), then reflections, and finally vertical stretches and shifts. However, sometimes rewriting the function in a fully factored form makes the order clearer.

当多种变换组合时，按正确顺序施加变换至关重要。通常的习惯是先进行水平移动和伸缩（处理输入 x），再反射，最后进行垂直伸缩和移动。不过，有时将函数改写成完全因式分解的形式能使顺序更清晰。

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8. Using Derivatives: Critical Points and Concavity | 利用导数：临界点与凹凸性

Calculus greatly enhances graph sketching by pinpointing turning points, intervals of increase/decrease, and concavity. For a function f(x), the first derivative f'(x) gives the slope of the tangent. Setting f'(x) = 0 yields stationary points (maxima, minima, or points of inflection). The sign of f'(x) tells us where the function is increasing (f'(x) > 0) or decreasing (f'(x) < 0).

微积分能极大地提升图像绘制的精度，通过确定拐点、单调区间和凹凸性。对于函数 f(x)，一阶导数 f'(x) 给出切线斜率。令 f'(x) = 0 可求出驻点（极大值点、极小值点或驻点拐点）。f'(x) 的正负号则表明函数在何处递增（f'(x) > 0）或递减（f'(x) < 0）。

The second derivative f”(x) indicates concavity: if f”(x) > 0, the graph is concave up (like a cup); if f”(x) < 0, it is concave down. Points where concavity changes are points of inflection, provided f''(x) changes sign. You can use the second derivative test: if f'(c) = 0 and f''(c) > 0, then f has a local minimum at x = c; if f”(c) < 0, a local maximum.

二阶导数 f”(x) 表示凹凸性：若 f”(x) > 0，图像凹向上（似杯形）；若 f”(x) < 0，则凹向下。凹凸性改变的点即为拐点，前提是 f''(x) 变号。你可以使用二阶导数检验：若 f'(c) = 0 且 f''(c) > 0，则 f 在 x = c 处有局部极小值；若 f”(c) < 0，则为局部极大值。

Example: f(x) = x³ − 3x. Then f'(x) = 3x² − 3 = 3(x − 1)(x + 1). Stationary points at x = -1 and x = 1. f”(x) = 6x; f”(-1) = -6 (< 0, local max), f''(1) = 6 (> 0, local min). The graph crosses the origin, has a maximum at (-1, 2), and a minimum at (1, -2). Including this information gives an accurate sketch.

示例：f(x) = x³ − 3x。则 f'(x) = 3x² − 3 = 3(x − 1)(x + 1)。驻点在 x = -1 和 x = 1。f”(x) = 6x；f”(-1) = -6（< 0，局部极大），f''(1) = 6（> 0，局部极小）。图像穿过原点，在 (-1, 2) 处有极大值，在 (1, -2) 处有极小值。纳入这些信息即可画出准确草图。

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9. Sketching Trigonometric Functions with Shifts | 带平移的三角函数草图

Consider y = 2 sin(x − π/3) + 1. Start from y = sin x. Step 1: horizontal shift right by π/3 ⇒ y = sin(x − π/3). Step 2: vertical stretch by factor 2 ⇒ y = 2 sin(x − π/3). Step 3: vertical shift up by 1 ⇒ y = 2 sin(x − π/3) + 1. The amplitude becomes 2, the period remains 2π, the midline moves to y = 1, and the phase shift is π/3 to the right.

考虑 y = 2 sin(x − π/3) + 1。从 y = sin x 出发。步骤1：水平右移 π/3 ⇒ y = sin(x − π/3)。步骤2：垂直拉伸 2 倍 ⇒ y = 2 sin(x − π/3)。步骤3：垂直上移 1 ⇒ y = 2 sin(x − π/3) + 1。振幅变为 2，周期仍为 2π，中线上移至 y = 1，相位移动为向右 π/3。

Mark key points: the maximum will be at height 3, minimum at -1. The graph now oscillates around y = 1. To find where maxima occur, solve sin(x − π/3) = 1 → x − π/3 = π/2 + 2nπ ⇒ x = 5π/6 + 2nπ. Label these on the sketch. The same approach works for cosine and tangent functions with shifts.

标出关键点：最大值高度为 3，最小值为 -1。图像现在围绕 y = 1 振荡。要找到最大值的位置，解 sin(x − π/3) = 1 → x − π/3 = π/2 + 2nπ ⇒ x = 5π/6 + 2nπ。在草图上标注这些点。同样的方法也适用于带平移的余弦和正切函数。

Be careful with combinations: y = cos(2x + π) can be rewritten as y = cos(2(x + π/2)). This shows a horizontal compression by factor 1/2 (period = π) and a left shift of π/2. Always factor the coefficient of x first when applying horizontal transformations.

处理组合变换时需谨慎：y = cos(2x + π) 可改写为 y = cos(2(x + π/2))。这表明先有水平压缩为原来的 1/2（周期 = π），再向左平移 π/2。施加水平变换时，务必将 x 的系数提取出来。

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10. Reciprocal and Absolute Value Graphs | 倒数与绝对值函数图像

The reciprocal transformation y = 1/f(x) takes the original graph and inverts all y-values. Key consequences: wherever f(x) = 0, 1/f(x) has a vertical asymptote. As f(x) → ±∞, 1/f(x) → 0, creating a horizontal asymptote at y = 0. The sign of 1/f(x) matches the sign of f(x). Large |f(x)| values become small, and small non-zero values become large. Turning points of f correspond to turning points on the reciprocal graph, but a maximum on f may become a minimum on 1/f.

倒数变换 y = 1/f(x) 将原图像的所有 y 值取倒数。主要结果如下：当 f(x) = 0 时，1/f(x) 存在垂直渐近线；当 f(x) → ±∞ 时，1/f(x) → 0，产生水平渐近线 y = 0。1/f(x) 的正负号与 f(x) 一致。|f(x)| 较大的值会变小，而小的非零值会变大。f 的拐点在倒数图像上也对应拐点，但 f 上的极大值点可能在 1/f 上变为极小值点。

The absolute value transformation y = |f(x)| keeps the parts of the graph where f(x) ≥ 0 unchanged, and reflects the negative parts (where f(x) < 0) across the x-axis to become positive. This results in a graph that never dips below the x-axis. Any x-intercept remains, but negative y-values are flipped upwards. This is particularly useful for functions like |x|, |x² − 4|, or |sin x|.

绝对值变换 y = |f(x)| 保留 f(x) ≥ 0 的部分不变，并将负值部分（f(x) < 0）沿 x 轴向上反射，使其变为正。所得图像永远不会落到 x 轴下方。任何 x 轴截距保留，但负的 y 值被翻折上去。这对 |x|、|x² − 4| 或 |sin x| 等函数尤为有用。

When sketching y = |f(x)|, first lightly draw y = f(x), then erase the portions below the x-axis and mirror them above. For piecewise functions, apply the definition |u| = u if u ≥ 0, and -u if u < 0, to find expressions for different intervals.

绘制 y = |f(x)| 时，可先轻轻画出 y = f(x)，然后擦去 x 轴以下的部分，并将其镜像翻折至上方。对于分段函数，可根据定义 |u| = u （若 u ≥ 0），以及 -u（若 u < 0）来求得各区间上的表达式。

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11. Combining Transformations: A Step-by-Step Approach | 组合变换：逐步法

When a function involves several transformations, such as y = -2f(3x + 6) – 1, a systematic method prevents errors. Rewrite the expression to reveal the horizontal factor: y = -2f(3(x + 2)) – 1. Now apply transformations in this order: (1) Horizontal shift left by 2 (inside bracket). (2) Horizontal compression by factor 1/3 (multiply x-values by 1/3). (3) Reflection in the x-axis and vertical stretch by factor 2 (multiply y-values by -2). (4) Vertical shift down by 1.

当一个函数包含多种变换时，如 y = -2f(3x + 6) – 1，采用系统化方法可避免错误。改写表达式以揭示水平因子：y = -2f(3(x + 2)) – 1。然后按以下顺序施加变换：(1) 水平左移 2 单位（括号内）。(2) 水平压缩为 1/3（x 值乘以 1/3）。(3) 关于 x

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