📚 How to Sketch Graphs of Functions: A Step-by-Step Guide | 函数图像绘制方法步骤详解
Sketching the graph of a function is an essential skill in IB Mathematics. It combines algebraic analysis with geometric intuition, allowing you to visualise behaviour without plotting every single point. This guide presents a systematic method for drawing accurate graphs by hand.
绘制函数图像是IB数学中的核心技能。它结合了代数分析与几何直觉，使您无需逐点描摹即可直观理解函数行为。本指南提供一套系统化的手工绘图方法，帮助您准确画出函数图像。
1. Determine the Domain and Range | 确定定义域和值域
Before drawing anything, identify the set of all valid input values \(x\) — the domain. Look for restrictions such as denominators, square roots, logarithms, and inverse trigonometric functions. For rational functions, exclude values that make the denominator zero. For even roots, require the radicand to be non-negative. For logarithms, require the argument to be positive.
在绘图之前，首先确定所有有效输入值 \(x\) 的集合，即定义域。注意分母、平方根、对数及反三角函数等限制条件。对于有理函数，排除使分母为零的值；对于偶次根式，要求被开方数非负；对于对数，要求真数为正。
Determine the range by considering the possible output values. For quadratics, find the vertex. For transformed functions, apply vertical shifts and stretches to the base function’s range. If the function is continuous, use limits at domain boundaries to help infer the range.
通过考虑可能的输出值来确定值域。对于二次函数，找到顶点；对于变换后的函数，在基底函数值域基础上应用垂直平移和伸缩。若函数连续，可利用定义域边界处的极限来推断值域。
2. Check for Symmetry | 检验对称性
Symmetry simplifies graphing considerably. If \(f(-x) = f(x)\) for all \(x\) in the domain, the function is even; its graph is symmetric about the y-axis. If \(f(-x) = -f(x)\), the function is odd; its graph is symmetric about the origin. Many functions are neither.
对称性可大幅简化绘图。若对定义域内所有 \(x\) 都有 \(f(-x) = f(x)\)，则函数为偶函数，图像关于 y 轴对称。若 \(f(-x) = -f(x)\)，则为奇函数，图像关于原点对称。许多函数两者都不是。
If symmetry exists, you only need to sketch one half and then reflect it. For example, \(y = x^2\) is even, while \(y = x^3\) is odd. Polynomials with only even powers are even; those with only odd powers are odd.
若存在对称性，只需绘制一半图像再对称翻折即可。例如 \(y = x^2\) 是偶函数，\(y = x^3\) 是奇函数。只含偶次幂的多项式是偶函数，只含奇次幂的多项式是奇函数。
3. Find Intercepts | 求截距
The y-intercept is found by setting \(x = 0\). It gives the point where the graph crosses the y-axis. The x-intercepts (roots or zeros) are found by solving \(f(x) = 0\). These points are critical for anchoring the graph.
令 \(x = 0\) 可求得 y 截距，即图像与 y 轴的交点。令 \(f(x) = 0\) 解方程可得 x 截距（根或零点）。这些点对于确定图像位置至关重要。
For rational functions, x-intercepts come from the numerator being zero, provided the denominator is not also zero at the same point. For polynomials, factorise to find roots and note their multiplicities — a root with even multiplicity touches the axis, while an odd multiplicity crosses it.
对有理函数，x 截距来自分子为零（且分母在该点不为零）。对多项式，分解因式求根并注意重数——偶重根处图像与轴相切，奇重根处图像穿过轴。
4. Analyse Asymptotes | 分析渐近线
Asymptotes describe behaviour as \(x\) approaches certain values or infinity. Vertical asymptotes occur where the function tends to ±∞, typically at zeros of the denominator that do not cancel with the numerator. Determine the one-sided limits to know whether the graph rises to +∞ or falls to −∞.
渐近线描述当 \(x\) 趋近某值或无穷时函数的行为。垂直渐近线出现在函数趋于 ±∞ 处，通常为分母为零且不与分子约去的位置。通过单侧极限判断图像是趋向 +∞ 还是 −∞。
Horizontal asymptotes are found by evaluating \(\lim_{x \to \pm\infty} f(x)\). For rational functions, compare the degrees of numerator and denominator: if degree numerator < degree denominator, asymptote \(y = 0\); if equal, \(y =\) ratio of leading coefficients; if greater, no horizontal asymptote.
水平渐近线通过求 \(\lim_{x \to \pm\infty} f(x)\) 得到。对于有理函数，比较分子分母的次数：若分子次数 < 分母次数，渐近线为 \(y = 0\)；若相等，\(y =\) 首项系数之比；若分子次数更大，则无水平渐近线。
Oblique asymptotes appear when the numerator degree is exactly one more than the denominator. Perform polynomial long division to find the linear asymptote \(y = ax + b\).
当分子次数恰好比分母多一时，存在斜渐近线。通过多项式长除法可求得线性渐近线 \(y = ax + b\)。
5. Determine Intervals of Increase and Decrease | 确定增减区间
Use the first derivative \(f'(x)\) to study monotonicity. Find critical points where \(f'(x) = 0\) or where \(f'(x)\) does not exist. These points split the domain into intervals. Test a sample point in each interval to determine the sign of \(f’\).
利用一阶导数 \(f'(x)\) 研究单调性。求出临界点（\(f'(x) = 0\) 或 \(f'(x)\) 不存在的点）。这些点将定义域分成若干区间。在每个区间取测试点判断 \(f’\) 的符号。
If \(f'(x) > 0\) on an interval, \(f\) is increasing; if \(f'(x) < 0\), \(f\) is decreasing. Record this information in a sign table. This analysis tells you the overall shape of the graph between critical points.
若在某区间内 \(f'(x) > 0\)，则 \(f\) 递增；若 \(f'(x) < 0\)，则 \(f\) 递减。将结果记录在符号表中。该分析揭示了临界点之间图像的大体走向。
6. Locate Local Extrema | 定位局部极值
At a critical point, apply the first derivative test: if \(f’\) changes from positive to negative, the point is a local maximum; if negative to positive, it is a local minimum. If the sign does not change, the point is neither.
在临界点处使用一阶导数判别法：若 \(f’\) 由正变负，该点为局部极大值；若由负变正，则为局部极小值；若符号不变，则既非极大也非极小。
Alternatively, use the second derivative test: if \(f”(x) < 0\) at a critical point, it is a local maximum; if \(f''(x) > 0\), a local minimum. When \(f”(x) = 0\), the test is inconclusive. Compute the y-coordinate to plot the exact extrema.
也可使用二阶导数判别法：若临界点处 \(f”(x) < 0\)，则为局部极大值；若 \(f''(x) > 0\)，则为局部极小值；当 \(f”(x) = 0\) 时判别法失效。计算 y 坐标以精确标出极值点。
7. Examine Concavity and Points of Inflection | 考察凹凸性与拐点
Concavity is determined by the second derivative \(f”(x)\). Where \(f”(x) > 0\), the graph is concave up (cup-shaped); where \(f”(x) < 0\), it is concave down (cap-shaped). These intervals shape the curvature of the graph.
凹凸性由二阶导数 \(f”(x)\) 决定。当 \(f”(x) > 0\) 时，图像凹向上（碗形）；当 \(f”(x) < 0\) 时，图像凹向下（帽形）。这些区间决定了图像的弯曲方向。
An inflection point occurs where the concavity changes. Find candidates by solving \(f”(x) = 0\) or where \(f”\) is undefined. Verify that \(f”\) changes sign around the candidate. Plot the point with its y-coordinate.
拐点是凹凸性发生改变的点。通过解 \(f”(x) = 0\) 或 \(f”\) 不存在的点找到候选点，验证 \(f”\) 在该点两侧变号。标出该点的 y 坐标。
8. Identify Periodicity and Other Features | 识别周期性与其他特征
For trigonometric functions, determine the period and amplitude. For \(y = a\sin(bx + c) + d\), the period is \(2\pi/|b|\), amplitude is \(|a|\), and phase shift is \(-c/b\). Sketch one period and translate it.
对于三角函数，确定周期和振幅。对于 \(y = a\sin(bx + c) + d\)，周期为 \(2\pi/|b|\)，振幅为 \(|a|\)，相位移动为 \(-c/b\)。先绘制一个周期，再进行平移。
Also note any discontinuities, cusps, or vertical tangents. Piecewise functions require separate treatment for each domain piece. For absolute value functions, consider the sign inside the absolute value to split into cases.
还需注意不连续点、尖点或垂直切线。分段函数需对每一段分别处理。对于绝对值函数，根据绝对值内部的正负分段讨论。
9. Plot Key Points and Connect Smoothly | 标出关键点并平滑连线
Bring everything together. Mark the intercepts, extrema, inflection points, and asymptotes on a coordinate plane. Draw asymptotes as dashed lines. Include the behaviour near asymptotes, using arrows to indicate ±∞.
将所有信息整合起来。在坐标平面上标出截距、极值点、拐点和渐近线。渐近线用虚线绘制。在渐近线附近标出趋向 ±∞ 的行为，用箭头表示。
Connect the points with a smooth curve, respecting monotonicity and concavity. For example, if \(f\) is increasing and concave down, draw a rising curve that bends downward. Ensure the curve approaches asymptotes correctly and does not cross them where forbidden.
用平滑曲线连接各点，确保符合单调性和凹凸性。例如，若 \(f\) 递增且凹向下，应画一条上升但向下弯曲的曲线。确保曲线正确逼近渐近线，且不在禁止处穿越。
10. Verify with Technology and Check Common Mistakes | 利用技术辅助验证并检查常见错误
After sketching, use a graphing calculator or software to confirm key features. Check that the number of turning points and inflection points matches the degree of a polynomial. Ensure asymptotes are correctly placed and that the graph matches the domain.
草图完成后，使用图形计算器或软件验证关键特征。检查多项式函数的转向点数和拐点数是否与次数相符。确保渐近线位置正确，图像与定义域一致。
Common mistakes include ignoring a hole when a factor cancels, misplacing a horizontal asymptote, or forgetting to check the sign of \(f’\) near undefined points. Always test the behaviour at domain boundaries, and do not draw curves through holes or over vertical asymptotes.
常见错误包括忽略因子约去后产生的空洞、水平渐近线位置错误、忘记检查未定义点附近的 \(f’\) 符号。始终测试定义域边界处的行为，不要将曲线穿过空洞或越过垂直渐近线。
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