📚 Sketching Gradient Functions | 绘制梯度函数图像
In Edexcel A‑Level Mathematics, you are often asked to sketch the derivative of a function given only the graph of the original function. This skill tests your understanding of how the gradient of a curve behaves and how it relates to the shape of f(x). Mastering gradient function sketching helps with interpreting graphs and tackling calculus problems with confidence.
在 Edexcel A‑Level 数学考试中，经常要求仅根据原函数的图像绘出其导数的草图。这项技能考查你对曲线斜率变化的理解，以及它与 f(x) 形状的关联。掌握梯度函数的绘制能帮助你自信地解读图形并解决微积分问题。
1. What is a Gradient Function? | 什么是梯度函数？
The gradient function f' te of change of y with respect to x, i.e. the slope of the tangent at each point. Sketching f'(x) means drawing a new graph whose y‑coordinate represents the slope of the original curve at every x‑value.
对于曲线 y = f(x) 而言，其梯度函数 f'(x) 描述了 y 关于 x 的变化率，也就是每一点处切线的斜率。绘制 f'(x) 即画出一个新图像，其纵坐标表示原曲线在每个 x 值处的斜率。
2. Key Relationship: f(x) and f'(x) | 关键关系：f(x) 与 f'(x)
The correspondence between a function and its gradient function is the foundation of curve‑sketching. Every feature on the original graph has a direct consequence on the derivative graph:
原函数与其梯度函数之间的对应关系是绘制图像的基础。原图上的每一个特征都会直接影响导函数图像：
| Feature on f(x) | Effect on f'(x) |
| Maximum or minimum point (stationary point) | f'(x) = 0, the curve crosses or touches the x‑axis |
| Increasing interval | f'(x) > 0 (graph lies above the x‑axis) |
| Decreasing interval | f'(x) < 0 (graph lies below the x‑axis) |
| Point of inflection (slope zero or non‑zero) | f'(x) may have a turning point or just touch zero |
记住这个关系可以让你在看到 f(x) 图像时，无需计算就能勾勒出 f'(x) 的大致形状。下面要点会逐一展开。
3. Stationary Points and Zeros of f'(x) | 驻点与导函数的零点
Wherever the original curve has a stationary point (a peak, a trough or a horizontal inflection), the gradient is momentarily zero. Consequently, the graph of f'(x) must touch or cross the x‑axis at those x‑coordinates.
只要原曲线存在驻点（极大值、极小值或水平拐点），该处的梯度瞬时为零。因此，f'(x) 图像必须在对应的 x 坐标处接触或穿过 x 轴。
For a local maximum, f'(x) changes sign from positive to negative as you move left to right, so the gradient graph cuts the x‑axis from above to below. For a local minimum, f'(x) changes from negative to positive, crossing from below to above. If the stationary point is a horizontal inflection, f'(x) just touches the axis (a repeated root) without changing sign.
对于局部极大值，随着从左向右移动，f'(x) 的符号由正变负，因此导数图像自上而下穿过 x 轴。对于局部极小值，f'(x) 由负变正，自下而上穿过 x 轴。若驻点为水平拐点，f'(x) 仅触碰 x 轴（重根）而不改变符号。
4. Sign of f'(x): Increasing and Decreasing Intervals | 导函数的符号：递增与递减区间
The sign of the gradient function tells you whether the original function is going up or down. Any interval where f(x) is rising (sloping upwards) corresponds to f'(x) > 0, so the derivative graph sits above the x‑axis. Conversely, where f(x) is falling, f'(x) < 0 and the graph lies below the x‑axis.
梯度函数的符号表明原函数是上升还是下降。f(x) 上升的区间对应 f'(x) > 0，即导数图像位于 x 轴上方；而 f(x) 下降的区间对应 f'(x) < 0，图像位于 x 轴下方。
f'(x) > 0 ⇔ f(x) is increasing
f'(x) > 0 ⇔ f(x) 递增
f'(x) < 0 ⇔ f(x) is decreasing
f'(x) < 0 ⇔ f(x) 递减
When sketching, mark the x‑intercepts of f'(x) at stationary points, then decide whether the curve stays above or below the axis on each side by reading the monotonicity of the original graph.
绘图时，先在驻点的 x 坐标处标出 f'(x) 的零点，随后通过观察原图的单调性来判断导数图像在各段是在轴的上方还是下方。
5. Shape of f'(x) Corresponding to f(x) Gradients | 对应原函数斜率的导函数形状
The steepness of the original curve determines the magnitude (height) of the gradient function. If f(x) becomes very steep in a certain region, |f'(x)| is large and the derivative graph moves far from the x‑axis. Where the original curve is almost flat, f'(x) is close to zero.
原曲线的陡峭程度决定了梯度函数的幅度（高度）。若 f(x) 在某段非常陡，则 |f'(x)| 很大，导数图像会远离 x 轴。原曲线接近水平的地方，f'(x) 则接近零。
Moreover, if f(x) is a smooth polynomial of degree n, its derivative will generally be of degree n−1. Therefore, when the original is a straight line, the gradient function is a horizontal line; when the original is a quadratic, the gradient function is a linear graph; a cubic gives a quadratic gradient, and so on.
此外，若 f(x) 是 n 次光滑多项式，其导数通常是 n−1 次。因此，若原函数为直线，梯度函数是一条水平线；原函数为二次，梯度为一次（直线）；三次函数则给出二次梯度函数，以此类推。
6. Inflection Points and the Gradient Function | 拐点与梯度函数
An inflection point is where the curvature of f(x) changes. Even if the slope is not zero at that point, the gradient function f'(x) will have a turning point (a local maximum or minimum). The x‑coordinate of the inflection on f(x) becomes the x‑coordinate of a stationary point on f'(x).
拐点是原曲线凹凸性改变的地方。即使该点斜率不为零，梯度函数 f'(x) 仍会出现一个转折点（局部极大或极小值）。f(x) 的拐点 x 坐标即为 f'(x) 驻点的 x 坐标。
If the original function changes from concave down to concave up at the inflection, f'(x) has a minimum. If the change is from concave up to concave down, f'(x) has a maximum. These turning points on the derivative graph must be smooth curves—do not draw sharp corners unless the original has a cusp.
如果原函数在拐点处由凹向下变为凹向上，f'(x) 取得极小值；若由凹向上变为凹向下，f'(x) 取得极大值。导数图上的这些转折点应是平滑曲线——除非原函数存在尖点，否则不要画出尖角。
7. Concavity of f(x) and the Monotonicity of f'(x) | 原函数的凹凸性与导函数的单调性
The second derivative f''(x) describes the concavity of f(x) and also tells us whether f'(x) is increasing or decreasing. Where f(x) is concave up (convex), f'(x) is increasing; where f(x) is concave down, f'(x) is decreasing.
二阶导数 f''(x) 描述了原函数的凹凸性，同时也说明 f'(x) 是递增还是递减。若 f(x) 凹向上（上凸），则 f'(x) 递增；若 f(x) 凹向下，则 f'(x) 递减。
Concave up ⇒ f'(x) is rising
凹向上 ⇒ f'(x) 递增
Concave down ⇒ f'(x) is falling
凹向下 ⇒ f'(x) 递减
This connection helps you refine the sketch: once you know the slope of f'(x) is going up or down, you can curve the derivative graph appropriately instead of simply joining points with straight lines.
这一联系能让你细化草图：一旦知道 f'(x) 本身的斜率是上升还是下降，你就能恰当地画出导数图像的弯曲方向，而不只是用直线连接关键点。
8. Step-by-Step Sketching Procedure | 分步绘制流程
Follow these steps when you are given the graph of y = f(x) and need to draw y = f'(x):
当给出 y = f(x) 的图像，需要绘制 y = f'(x) 时，可遵循以下步骤：
Identify all stationary points on f(x) – mark them as x‑intercepts on the f'(x) axes.
标出 f(x) 的所有驻点——将这些点的 x 坐标标为 f'(x) 图像上的 x 轴截距。
Decide whether each intercept is a crossing or a touch: crossing if the stationary point is a turning point, touch if it is a horizontal inflection.
判断每个截距是穿过还是相切：若驻点是极值点（转折点），则穿过；若是水平拐点，则相切。
Look at intervals between stationary points: if f(x) is increasing, draw f'(x) above the x‑axis; if decreasing, draw below.
观察驻点之间的区间：若 f(x) 递增，将 f'(x) 画在 x 轴上方；若递减，画在下方。
Estimate the steepness: the steeper f(x) is, the further f'(x) is from the x‑axis. A gentle slope keeps f'(x) near zero.
估计陡峭程度：f(x) 越陡，f'(x) 离 x 轴越远；平缓的斜率让 f'(x) 接近零。
Identify inflection points of f(x) – these give turning points on f'(x). The 'concavity' rule tells you whether f'(x) has a max or min there.
找出 f(x) 的拐点——这些点对应 f'(x) 的极值点。利用凹凸性法则判断 f'(x) 在该处是极大值还是极小值。
Connect the points with a smooth curve that obeys the sign, magnitude and monotonicity deduced. Add arrows if the function continues beyond the given window.
用光滑曲线连接各点，确保符号、幅度以及导函数的单调性都与推论一致。若函数延续到给定窗口外，加上箭头。
9. Worked Example 1: Parabola to Line | 例题一：抛物线到直线
Suppose f(x) is a concave‑up quadratic with a minimum at x = 2, crossing the y‑axis at (0,4) and going to +∞ as x → ±∞. The graph is U‑shaped.
假设 f(x) 为一个开口向上的二次函数，极小值在 x = 2，y 轴截距为 (0,4)，且当 x → ±∞ 时趋向 +∞。图像呈 U 形。
The gradient function f' of a quadratic is linear. At the minimum x=2 the slope is zero, so the line crosses the x‑axis at x=2. For x < 2 the original is decreasing, so f'(x) < 0; for x > 2 the original is increasing, so f' loping line passing through (2,0) with a positive gradient (the gradient of this line is f''(x), which is positive because the parabola is concave up).
梯度函数 f'(x) 将是一条直线，因为二次函数的导数是一次函数。在极小点 x=2 处斜率为零，故直线与 x 轴交于 x=2。当 x < 2 时原函数递减，f'(x) < 0；当 x > 2 时递增，f'(x) > 0。所得草图是一条穿过 (2,0) 且具有正斜率的直线（这条直线的斜率就是 f''(x)，因为抛物线凹向上，故为正）。
10. Worked Example 2: Cubic Curve to Quadratic | 例题二：三次曲线到二次曲线
e curve rises to the left, falls between the turning points, and rises again to the right.
考虑一个三次函数 f(x)，其在 x = −1 取得局部极大值，在 x = 3 取得局部极小值，拐点位于 x = 1。曲线左侧上升，在两个极值点之间下降，右侧再次上升。
We deduce f' ards or downwards depending on the sign of the cubic term). Using the stationary points: f'(−1)=0 and f'(3)=0 – these are the roots of the quadratic. The sign of f'(x): for x < −1, f(x) is increasing → f'(x) > 0; between −1 and 3, f(x) is decreasing → f'(x) < 0; for x > 3, increasing again → f'(x) > 0. This pattern matches a quadratic that opens upwards: its graph is below the x‑axis between the roots and above outside them.
推断 f'(x) 是一个开口向上的二次函数（取决于三次项系数的符号）。利用驻点：f'(−1)=0 且 f'(3)=0——这些是二次函数的根。f'(x) 的符号：x < −1 时原函数递增 → f'(x) > 0；在 −1 和 3 之间递减 → f'(x) < 0；x > 3 时再次递增 → f'(x) > 0。该模式正符合开口向上的抛物线：在两根之间位于 x 轴下方，两根之外位于上方。
The inflection of the cubic at x=1 gives a turning point on f'(x). Because the cubic changes from concave down to concave up, f'(x) has a minimum at x=1. This fits the vertex of the upward‑opening quadratic exactly. The sketch of f' bolic curve passing through (−1,0) and (3,0) with its lowest point at (1, negative value).
三次函数在 x=1 处的拐点给出了 f'(x) 的极值点。由于三次函数由凹向下变为凹向上，f'(x) 在 x=1 处取得极小值，正好对应开口向上的二次函数的顶点。因此 f'(x) 的草图为一条光滑抛物线，过 (−1,0) 和 (3,0)，最低点位于 (1, 负值)。
11. Common Mistakes to Avoid | 常见错误
Forgetting that a stationary point of f(x) may only touch the x‑axis on f'(x) if it is a horizontal inflection. Many pupils automatically draw a crossing.
若 f(x) 的驻点是水平拐点，f'(x) 图像仅是触碰 x 轴，这一点常被忽略，许多学生会习惯性画出穿过。
Confusing the slope of f(x) with the height of f'(x). A large positive slope gives a large positive f'(x); a gentle slope gives a value near zero – but this must be represented vertically, not by changing the slope of the f'(x) line inappropriately.
混淆原函数的斜率与梯度函数的高度。大正斜率产生大的正 f'(x) 值；平缓斜率产生接近零的值——这必须通过垂直高度体现，而不是错误地改变 f'(x) 线的斜率。
Drawing corners or cusps on f'(x) when the original is smooth. Polynomials produce smooth derivative curves; only if f
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