Curve Sketching for Edexcel A-Level Maths | Edexcel A-Level 数学：曲线草图

📚 Curve Sketching for Edexcel A-Level Maths | Edexcel A-Level 数学：曲线草图

Curve sketching is the process of producing a quick but accurate graph by identifying key algebraic and calculus features. In Edexcel A-Level Mathematics, this topic brings together intercepts, symmetry, asymptotes, stationary points, transformations, and standard graphs such as rational, modulus, exponential, logarithmic, and trigonometric functions. Instead of plotting many points, you use limits, derivatives, and known shapes to build the graph efficiently.

曲线草图是通过确定关键的代数与微积分特征来快速准确地画出函数图像的过程。在 Edexcel A-Level 数学中，这一主题综合了截距、对称性、渐近线、驻点、图像变换以及有理函数、绝对值函数、指数函数、对数函数和三角函数等标准图像。与其描出大量点，不如利用极限、导数和已知形状高效地构建图像。

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1. What Is Curve Sketching? | 什么是曲线草图？

Curve sketching means drawing a graph that shows the most important features without needing a calculator or a large table of values. A good sketch includes intercepts, turning points, asymptotes, symmetry, and the correct behaviour as x tends to positive or negative infinity. You should always label key points and asymptotes clearly.

曲线草图是指在不需要计算器或大量数值表的情况下，画出显示最重要特征的图像。一幅好的草图应包括截距、极值点、渐近线、对称性以及当 x 趋向正无穷或负无穷时的正确趋势。你应始终清楚标注关键点和渐近线。

In an Edexcel exam, curve sketching often appears as part of a larger question on differentiation or functions. You may be asked to find stationary points and asymptotes first, then use them to sketch the curve. Marks are usually awarded for the method, the key features, and the overall shape.

在 Edexcel 考试中，曲线草图常作为微分或函数综合题的一部分出现。题目可能要求先求驻点和渐近线，再利用它们画出曲线。分数通常按方法、关键特征和整体形状给出。

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2. Domain, Zeros and Intercepts | 定义域、零点与截距

Before sketching any function y = f(x), determine its domain. Exclude x-values that make a denominator zero, make the expression inside a square root negative, or make the argument of a logarithm less than or equal to zero. The domain tells you where the graph can exist.

在绘制任何函数 y = f(x) 之前，首先要确定其定义域。需要排除使分母为零、使根号内表达式为负或使对数真数小于等于零的 x 值。定义域告诉你图像可能存在的区域。

To find x-intercepts, solve f(x) = 0. These are the points where the curve crosses the x-axis. To find the y-intercept, evaluate f(0), provided x = 0 is in the domain. A curve can have several x-intercepts but at most one y-intercept if it is a function.

求 x 轴截距时，解方程 f(x) = 0。这些点是曲线与 x 轴的交点。求 y 轴截距时，如果 x = 0 在定义域内，则计算 f(0)。一个函数曲线可能有多个 x 轴截距，但最多只有一个 y 轴截距。

For example, y = (x – 1)(x + 2) has x-intercepts at x = 1 and x = -2, and y-intercept at y = -2. For y = ln(x – 3), the domain is x > 3, so there is no y-intercept because x = 0 is not allowed.

例如，y = (x – 1)(x + 2) 的 x 轴截距为 x = 1 和 x = -2，y 轴截距为 y = -2。对于 y = ln(x – 3)，定义域为 x > 3，因此不存在 y 轴截距，因为 x = 0 不在定义域内。

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3. Symmetry: Even and Odd Functions | 对称性：偶函数与奇函数

Checking symmetry can reduce the amount of work needed for a sketch. A function is even if f(-x) = f(x) for all x in its domain. Even functions are symmetric about the y-axis, so you only need to sketch the right side and reflect it.

检查对称性可以减少画图所需的工作量。如果对定义域内所有 x 都有 f(-x) = f(x)，则函数为偶函数。偶函数关于 y 轴对称，因此只需画出右侧图像再反射即可。

A function is odd if f(-x) = -f(x). Odd functions have rotational symmetry of 180° about the origin. You can sketch the graph for positive x and rotate it about the origin to obtain the negative side.

如果 f(-x) = -f(x)，则函数为奇函数。奇函数具有关于原点的 180° 旋转对称性。你可以先画出 x 为正时的图像，再绕原点旋转得到负半轴图像。

Examples: y = x² + 3 is even, y = x³ – 4x is odd, and y = x² + x is neither even nor odd. Polynomials with only even powers are even, and those with only odd powers are odd.

例如：y = x² + 3 是偶函数，y = x³ – 4x 是奇函数，而 y = x² + x 既不是偶函数也不是奇函数。只含有偶次幂的多项式为偶函数，只含有奇次幂的多项式为奇函数。

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4. Vertical and Horizontal Asymptotes | 垂直渐近线与水平渐近线

Vertical asymptotes occur where the function tends to positive or negative infinity as x approaches a finite value. For rational functions, set the denominator equal to zero after cancelling common factors. If x = a makes the denominator zero and the factor does not cancel, the line x = a is a vertical asymptote.

垂直渐近线出现在函数随 x 趋近某个有限值时趋向正无穷或负无穷的位置。对于有理函数，先约去公因式，再令分母等于零。如果 x = a 使分母为零且该因式未被约去，则直线 x = a 是一条垂直渐近线。

Horizontal asymptotes are found by taking the limit of f(x) as x → +∞ and as x → -∞. For a rational function, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degrees are equal, the horizontal asymptote is y = leading coefficient of numerator ÷ leading coefficient of denominator.

水平渐近线通过求 x → +∞ 和 x → -∞ 时 f(x) 的极限来确定。对于有理函数，比较分子与分母的次数。如果分子次数小于分母次数，水平渐近线为 y = 0。如果次数相等，水平渐近线为 y = 分子首项系数 ÷ 分母首项系数。

If the degree of the numerator is exactly one more than the degree of the denominator, there is no horizontal asymptote. Instead, divide the numerator by the denominator to find an oblique asymptote of the form y = mx + c.

如果分子的次数恰好比分母高一次，则没有水平渐近线。此时可用长除法将分子除以分母，求得形如 y = mx + c 的斜渐近线。

Example: for y = (2x + 1)/(x – 3), the vertical asymptote is x = 3 and the horizontal asymptote is y = 2, because the leading coefficients are 2 and 1.

例如：对于 y = (2x + 1)/(x – 3)，垂直渐近线为 x = 3，水平渐近线为 y = 2，因为首项系数分别为 2 和 1。

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5. Stationary Points and Turning Points | 驻点与极值点

Stationary points occur where the first derivative is zero. Solve dy/dx = 0 to find the x-coordinates, then substitute them into y = f(x) to find the y-coordinates. These points are where the tangent to the curve is horizontal.

驻点出现在一阶导数为零的位置。解方程 dy/dx = 0 求得 x 坐标，再代入 y = f(x) 求得 y 坐标。这些点处曲线的切线是水平的。

To classify a stationary point, use the second derivative. If d²y/dx² < 0 at the point, it is a local maximum. If d²y/dx² > 0, it is a local minimum. If d²y/dx² = 0, the second derivative test is inconclusive, and you should use the sign change of dy/dx on either side.

判断驻点类型时可使用二阶导数。如果该点处 d²y/dx² < 0，则为局部极大值；如果 d²y/dx² > 0，则为局部极小值；如果 d²y/dx² = 0，则二阶导数判别法失效，应使用该点两侧 dy/dx 的符号变化来判断。

For y = x³ – 3x, dy/dx = 3x² – 3 = 3(x² – 1). Setting this equal to zero gives x = -1 and x = 1. The second derivative is d²y/dx² = 6x, so x = -1 is a local maximum and x = 1 is a local minimum.

对于 y = x³ – 3x，dy/dx = 3x² – 3 = 3(x² – 1)。令其为零得到 x = -1 和 x = 1。二阶导数为 d²y/dx² = 6x，因此 x = -1 处是局部极大值，x = 1 处是局部极小值。

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6. Concavity and Points of Inflection | 凹凸性与拐点

The sign of the second derivative tells you the concavity of the curve. If d²y/dx² > 0 on an interval, the curve is concave up and the gradient is increasing. If d²y/dx² < 0, the curve is concave down and the gradient is decreasing.

二阶导数的符号反映曲线的凹凸性。如果在某区间上 d²y/dx² > 0，则曲线凹向上，且斜率递增；如果 d²y/dx² < 0，则曲线凹向下，且斜率递减。

A point of inflection is where the concavity changes from up to down or from down to up. At a genuine point of inflection, d²y/dx² = 0 or is undefined, but you must check that the second derivative changes sign across that point.

拐点是指曲线凹凸性发生改变的位置。在真正的拐点处，d²y/dx² = 0 或不存在，但必须验证二阶导数在该点两侧是否变号。

For example, y = x³ has d²y/dx² = 6x. This is negative for x < 0 and positive for x > 0, so the concavity changes at x = 0. Therefore (0, 0) is a point of inflection, even though dy/dx = 0 there.

例如，y = x³ 的二阶导数为 d²y/dx² = 6x。当 x < 0 时为负，当 x > 0 时为正，因此凹凸性在 x = 0 处改变。所以 (0, 0) 是一个拐点，尽管该处 dy/dx = 0。

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7. Rational Function Worked Example | 有理函数实例

Consider y = (x² – 1)/(x² – 4). The domain excludes x = 2 and x = -2, so there are vertical asymptotes at x = 2 and x = -2. The degrees of numerator and denominator are both 2, so the horizontal asymptote is y = 1, the ratio of the leading coefficients.

考虑函数 y = (x² – 1)/(x² – 4)。定义域排除了 x = 2 和 x = -2，因此在 x = 2 和 x = -2 处有垂直渐近线。分子和分母的次数均为 2，因此水平渐近线为 y = 1，即首项系数之比。

For intercepts, y = 0 when x² – 1 = 0, giving x = 1 and x = -1. The y-intercept is y = (-1)/(-4) = 1/4 at x = 0. Since f(-x) = f(x), the function is even, so the graph is symmetric about the y-axis.

求截距时，当 x² – 1 = 0 时 y = 0，得到 x = 1 和 x = -1。y 轴截距为 x = 0 时 y = (-1)/(-4) = 1/4。由于 f(-x) = f(x)，该函数为偶函数，图像关于 y 轴对称。

Differentiate to find stationary points: dy/dx = -6x/(x² – 4)². Setting dy/dx = 0 gives x = 0. The corresponding y-value is 1/4. For x < 0, dy/dx > 0, and for x > 0, dy/dx < 0, so (0, 1/4) is a local maximum.

求导以确定驻点：dy/dx = -6x/(x² – 4)²。令 dy/dx = 0 得到 x = 0，对应的 y 值为 1/4。当 x < 0 时 dy/dx > 0，当 x > 0 时 dy/dx < 0，因此 (0, 1/4) 是局部极大值点。

The graph approaches y = 1 from below as x → +∞ or x → -∞. Near x = 2, the curve tends to +∞ on one side and -∞ on the other, so a sketch should show both branches rising or falling along the vertical asymptotes.

当 x → +∞ 或 x → -∞ 时，图像从下方趋近 y = 1。在 x = 2 附近，曲线一侧趋向 +∞，另一侧趋向 -∞，因此草图中应画出沿垂直渐近线两侧上升或下降的分支。

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8. Transformations of Graphs | 图像变换

Knowing basic graphs allows you to sketch transformed functions quickly. A vertical translation y = f(x) + a moves the graph up by a units if a > 0, or down if a < 0. A horizontal translation y = f(x + a) moves the graph left by a units if a > 0, or right if a < 0.

掌握基本图像后，你可以快速画出经过变换的函数图像。纵向平移 y = f(x) + a 在 a > 0 时将图像上移 a 个单位，在 a < 0 时下移 |a| 个单位。横向平移 y = f(x + a) 在 a > 0 时将图像左移 a 个单位，在 a < 0 时右移 |a| 个单位。

Vertical stretches are given by y = af(x). If |a| > 1, the graph is stretched away from the x-axis; if 0 < |a| < 1, it is compressed towards the x-axis. A negative a also reflects the graph in the x-axis.

纵向伸缩由 y = af(x) 表示。如果 |a| > 1，图像沿纵向远离 x 轴拉伸；如果 0 < |a| < 1，则向 x 轴压缩。a 为负时图像还会关于 x 轴反射。

Horizontal stretches are written as y = f(ax). If |a| > 1, the graph is compressed towards the y-axis by factor 1/|a|. If 0 < |a| < 1, it is stretched away from the y-axis. The transformation y = -f(x) reflects the graph in the x-axis, while y = f(-x) reflects it in the y-axis.

横向伸缩写作 y = f(ax)。如果 |a| > 1，图像以 1/|a| 的倍数向 y 轴压缩；如果 0 < |a| < 1，则远离 y 轴拉伸。变换 y = -f(x) 使图像关于 x 轴反射，而 y = f(-x) 使图像关于 y 轴反射。

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9. Modulus Graphs | 绝对值函数图像

To sketch y = |f(x)|, first sketch y = f(x). Keep any part that lies on or above the x-axis unchanged. Take any part below the x-axis and reflect it in the x-axis to make it positive. The resulting graph never goes below the x-axis.

画 y = |f(x)| 时，先画出 y = f(x)。保留位于 x 轴

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