Sketching Gradient Functions | 绘制导函数图像

📚 Sketching Gradient Functions | 绘制导函数图像

Sketching the gradient function f'(x) is a key Edexcel A-Level Maths skill. You are often given the graph of y = f(x) and asked to draw the graph of its derivative without knowing the equation of f(x). This topic tests your understanding of how the slope of a curve changes as x changes, and how stationary points, intervals of increase and decrease, and concavity are transferred to the derivative graph.

绘制导函数 f'(x) 的图像是 Edexcel A-Level 数学的一项关键技能。题目常常给出 y = f(x) 的图像，要求你在不知道 f(x) 解析式的情况下画出其导函数的图像。这一主题考察你是否理解曲线斜率随 x 变化的规律，以及驻点、递增递减区间和凹凸性如何体现在导函数图像上。

1. What Is a Gradient Function? | 什么是导函数？

The gradient function of a curve y = f(x) is the derivative, usually written as f'(x) or dy/dx. It gives the gradient of the tangent to the curve at each value of x.

曲线 y = f(x) 的导函数就是导数，通常写作 f'(x) 或 dy/dx。它表示曲线在每一个 x 值处切线的斜率。

When you sketch f'(x), you are drawing a new graph that shows how the gradient of f(x) changes along the x-axis. You do not always need the equation of f(x); you can use the shape and stationary points of the original curve.

绘制 f'(x) 的图像，就是画出原函数 f(x) 的斜率如何随 x 变化的新图像。你并不总需要 f(x) 的解析式，也可以利用原曲线的形状和驻点来作图。

2. The Core Sign Link: Increasing, Decreasing and f'(x) | 核心符号联系：递增、递减与 f'(x)

The most important rule is: where f(x) is increasing, f'(x) > 0; where f(x) is decreasing, f'(x) < 0; where f(x) has a horizontal tangent, f'(x) = 0.

最重要的规则是：当 f(x) 递增时，f'(x) > 0；当 f(x) 递减时，f'(x) < 0；当 f(x) 有水平切线时，f'(x) = 0。

This sign link allows you to place the graph of f'(x) above or below the x-axis based on the slope of f(x). The actual y-values of f'(x) are less important than the sign and the location of its roots in a sketch.

这种符号联系可以让你根据 f(x) 的斜率，把 f'(x) 的图像放在 x 轴上方或下方。在草图中，f'(x) 的具体 y 值不如符号和根的位置重要。

3. Stationary Points Become x-Intercepts | 驻点变成 x 轴截距

At any stationary point of f(x), the derivative is zero. Therefore, each stationary point of f(x) corresponds to an x-intercept of the graph y = f'(x).

在 f(x) 的任意驻点处，导数都为零。因此，f(x) 的每一个驻点都对应 y = f'(x) 图像上的一个 x 轴截距。

If f(x) has a local maximum or local minimum, f'(x) changes sign at that point, so the graph of f'(x) crosses the x-axis. If f(x) has a stationary point of inflection, f'(x) touches the x-axis but does not cross it.

如果 f(x) 有局部极大值或局部极小值，f'(x) 在该点变号，所以 f'(x) 的图像会穿过 x 轴。如果 f(x) 有驻点拐点，f'(x) 会与 x 轴相切但不穿过。

The table below summarises the difference.

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