📚 IB Mathematics: Techniques for Sketching Function Graphs | IB数学：函数图像绘制技巧
A clear and accurate graph is a powerful tool for understanding a function's behaviour. In IB Mathematics, sketching graphs without a calculator is a key skill that tests your grasp of transformations, asymptotes, symmetry and calculus.
清晰而准确的函数图像是理解函数行为的有力工具。在IB数学中，不使用计算器绘制函数图像是一项关键技能，它考查你对变换、渐近线、对称性以及微积分的掌握程度。
1. Domain, Range and Intercepts | 定义域、值域与截距
Before drawing any graph, identify the set of input values for which the function is defined. This avoids plotting points outside the domain.
在绘制任何图像之前，先确定函数有定义的输入值集合，避免在定义域之外描点。
ude values that make the denominator zero. For example, f(x) = 1/(x − 2) has domain x ≠ 2.
对于有理函数，排除使分母为零的值。例如，f(x) = 1/(x − 2) 的定义域为 x ≠ 2。
For even roots, the radicand must be non-negative. Thus g(x) = √x − 1 requires x ≥ 1.
对于偶次根式，被开方数必须非负。因此 g(x) = √x − 1 要求 x ≥ 1。
or the y-intercept and f(x) = 0 for x-intercepts. These provide anchor points on the axes.
令 x = 0 得 y 截距，令 f(x) = 0 得 x 截距。这些是图像在坐标轴上的锚点。
y-intercept: f(0) | x-intercepts: solve f(x) = 0
2. Symmetry: Even and Odd Functions | 对称性：偶函数与奇函数
Recognising symmetry halves the work needed to complete a graph and helps verify the shape.
识别对称性能将绘制图像的工作量减半，并有助于验证图像形状。
). Its graph is symmetric about the y-axis. Typical examples include y = x² and y = cos x.
偶函数满足 f(−x) = f(x)，其图像关于 y 轴对称。典型例子包括 y = x² 和 y = cos x。
h has rotational symmetry of 180° about the origin. Examples include y = x³ and y = sin x.
奇函数满足 f(−x) = −f(x)，其图像关于原点具有 180° 旋转对称性。例子包括 y = x³ 和 y = sin x。
look for other symmetries such as reflections across the line y = x for inverse functions.
若函数既非偶也非奇，可寻找其他对称性，例如反函数图像关于直线 y = x 的反射对称。
Even: f(−x) = f(x) (y-axis) | Odd: f(−x) = −f(x) (origin)
3. Asymptotes and Limiting Behaviour | 渐近线与极限行为
Vertical and horizontal asymptotes describe where a graph tends to infinity or settles near a constant. Oblique asymptotes appear when the degree of the numerator is one more than that of the denominator.
垂直渐近线和水平渐近线描述了图像趋于无穷或逼近常数的行为。当分子次数比分母次数高一次时，会出现斜渐近线。
-sided limits are infinite. For h(x) = 2/(x + 3), the line x = −3 is a vertical asymptote.
当有理函数在某点无定义且单侧极限为无穷时，该处出现垂直渐近线。对 h(x) = 2/(x + 3)，直线 x = −3 是一条垂直渐近线。
. If the degrees are equal, the horizontal asymptote is the ratio of leading coefficients.
水平渐近线通过令 x → ±∞ 取极限求得。若分子分母次数相同，水平渐近线为最高次项系数之比。
ned by polynomial division. The graph approaches this line at extremes, but may cross it for finite x.
斜渐近线 y = mx + c 通过多项式除法得到。图像在无穷远处逼近该直线，但在有限的 x 处可能穿过它。
| Function | Vertical Asymptote | Horizontal/Oblique Asymptote |
| f(x) = 1/(x − 2) | x = 2 | y = 0 |
| g(x) = (2x² + 1)/(x² − 1) | x = ±1 | y = 2 |
4. Translation Transformations | 平移变换
Translations shift the graph horizontally or vertically without changing its shape. They follow directly from changes in the function's equation.
平移使图像在水平或垂直方向上移动而不改变其形状，它直接由函数方程的变化决定。
y = f(x) + a shifts the graph upward by a units for a > 0, and downward by |a| units if a < 0.
y = f(x) + a 将图像向上平移 a 个单位（a > 0），若 a < 0 则向下平移 |a| 个单位。
raph to the right by a units, because x must be a units larger to produce the same output.
y = f(x − a) 将图像向右平移 a 个单位，因为 x 必须增大 a 才能得到相同的输出值。
Be careful: y = f(x + a) shifts left by a units. Many students confuse this sign.
注意：y = f(x + a) 是向左平移 a 个单位，许多学生容易混淆这里的符号。
y = f(x − h) + k → translate by vector (h, k)
5. Scaling and Stretching | 伸缩变换
Scaling changes the graph's steepness or spread along the axes. Vertical and horizontal stretches are governed by coefficients placed outside or inside the function.
伸缩改变图像沿坐标轴的陡峭程度或展开范围。垂直与水平伸缩分别由函数外或函数内的系数控制。
a > 1, stretches the graph vertically by factor a. If 0 < a < 1, it compresses vertically.
y = a·f(x)，当 a > 1 时，图像在垂直方向拉伸 a 倍；当 0 < a < 1 时，垂直压缩。
compresses the graph horizontally by factor 1/b. If 0 < b < 1, it stretches horizontally.
y = f(bx)，当 b > 1 时，图像在水平方向压缩为原来的 1/b；当 0 < b < 1 时，水平拉伸。
The vertical scale factor affects y-coordinates, while the horizontal scale factor affects x-coordinates; they operate independently.
垂直伸缩因子作用于 y 坐标，水平伸缩因子作用于 x 坐标，二者相互独立。
y = a·f(b(x − h)) + k | order: scale, then translate
6. Reflections | 反射变换
Reflections produce mirror images across an axis or a line. They are simple to apply but essential for understanding inverse functions.
反射产生关于坐标轴或某条直线的镜像。它们应用简单，但对理解反函数至关重要。
y = −f(x) reflects the graph across the x-axis, changing the sign of every y-coordinate.
y = −f(x) 将图像关于 x 轴反射，每个 y 坐标都改变符号。
y = f(−x) reflects the graph across the y-axis, changing the sign of every x-coordinate.
y = f(−x) 将图像关于 y 轴反射，每个 x 坐标都改变符号。
⁻¹(x), reflect the original graph across the line y = x. The domain and range are swapped.
绘制反函数 y = f⁻¹(x) 时，将原图像关于直线 y = x 反射，定义域与值域互换。
y = −f(x): x-axis | y = f(−x): y-axis | y = f⁻¹(x): line y = x
7. Absolute Value and Piecewise Functions | 绝对值与分段函数
Graphs of |f(x)| and piecewise functions require careful attention to where the expression changes sign.
绘制 |f(x)| 和分段函数的图像时，需要仔细关注表达式改变符号的位置。
For y = |f(x)| on of f(x) that lies below the x-axis upward. The parts above the x-axis remain unchanged.
对 y = |f(x)|，将 f(x) 位于 x 轴下方的部分向上反射，x 轴上方的部分保持不变。
For y = f(|x| keep the graph for x ≥ 0 and reflect it across the y-axis. This creates an even function.
对 y = f(|x|)，保留 x ≥ 0 的部分并关于 y 轴反射，从而得到一个偶函数。
A piecewise function is drawn by plotting each branch over its specified interval. Check endpoints and open/closed dots carefully.
分段函数在每个指定区间上分别绘制对应分支。注意端点以及空心点与实心点的使用。
8. Using Derivatives: Monotonicity and Extrema | 利用导数分析单调性与极值
When a function is differentiable, the sign of its first derivative reveals where it increases or decreases, and where local maxima or minima occur.
当函数可导时，一阶导数的符号揭示了函数的增减区间以及局部极大值或极小值的位置。
If f'(x) > 0 on an interval, then f is increasing there. If f'(x) < 0, then f is decreasing.
若在某区间上 f'(x) > 0，则 f 在该区间递增；若 f'(x) < 0，则递减。
At a local extremum, f'(x) = 0 or f' undefined. Use the first derivative test or second derivative test to classify the point.
在局部极值处，f'(x) = 0 或 f' 不存在。用一阶导数测试或二阶导数测试判断该点的类型。
For example, f(x) = x³ − 3x has f' = 3x² − 3 = 3(x − 1)(x + 1). Thus x = −1 is a local maximum and x = 1 is a local minimum.
例如，f(x) = x³ − 3x 的导数为 f'(x) = 3x² − 3 = 3(x − 1)(x + 1)。因此 x = −1 是局部极大值，x = 1 是局部极小值。
Stationary points: f'(x) = 0 | f increasing: f'(x) > 0
9. Concavity and Points of Inflection | 凹凸性与拐点
The second derivative describes the curvature of a graph. A function that opens upward is concave up, while one that opens downward is concave down.
二阶导数描述图像的弯曲方向。向上开口的函数为凹向上，向下开口则为凹向下。
If f''(x) > 0 on an interval, the graph is concave up (like y = x²). If f''(x) < 0, it is concave down (like y = −x²).
若在某区间上 f''(x) > 0，则图像凹向上（如 y = x²）；若 f''(x) < 0，则凹向下（如 y = −x²）。
A point of inflection occurs where f''(x) = 0 or is undefined, and the concavity changes sign across that point.
当 f''(x) = 0 或不存在，且凹凸性在该点两侧发生改变时，该点为拐点。
Combine concavity with monotonicity to produce a smooth, accurate curve rather than straight-line segments between points.
将凹凸性与单调性结合，绘制出平滑准确的曲线，而不是在点之间用直线连接。
Concave up: f''(x) > 0 | Concave down: f''(x) < 0 | Inflection: f''(x) changes sign
10. A Systematic Approach to Sketching | 绘制综合图像的系统步骤
Put all the tools together in a reliable order. This prevents missing key features and produces a graph that is easy to mark and verify.
按可靠顺序综合运用所有工具，这样可以避免遗漏关键特征，并绘制出易于评分和验证的图像。
Determine the domain and range, and mark any restrictions.
确定定义域和值域，并标出限制条件。
Find intercepts with both axes.
求与两坐标轴的交点。
Locate any vertical, horizontal or oblique asymptotes using limits.
利用极限确定垂直、水平或斜渐近线。
Compute f'(x) and f''(x) to identify intervals of increase/decrease, extrema, concavity and inflection points.
计算 f'(x) 和 f''(x)，确定增减区间、极值、凹凸性和拐点。
Plot all key points, including stationary points and inflection points, then sketch the curve through them while respecting asymptotes and symmetry.
描出所有关键点，包括驻点和拐点，然后按照渐近线和对称性要求通过这些点绘制曲线。
Check the graph with a quick test: does it behave correctly as x → ±∞? Does it match the function's parity?
快速检验图像：当 x → ±∞ 时行为是否正确？是否符合函数的奇偶性？
Key steps: domain → intercepts → asymptotes → derivatives → key points → final curve
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