E – Finding a Quadratic from its Graph | 由图像求二次函数解析式

📚 E – Finding a Quadratic from its Graph | 由图像求二次函数解析式

In IB Mathematics, being able to derive the equation of a quadratic function from its graph is a fundamental skill. Whether you are given the vertex, the intercepts, or simply a set of points, there are several reliable methods to reconstruct the function. This article will guide you through each approach, illustrate the reasoning with worked examples, and highlight common mistakes.

在IB数学中，由图像推导二次函数解析式是一项基本技能。无论题目给出的是顶点、截距还是几个点，都有多种可靠的方法来还原函数。本文将逐一介绍这些方法，结合实例展示推理过程，并指出常见错误。

1. Introduction to Quadratic Graphs | 二次函数图像导论

A quadratic function has the general form y = ax² + bx + c, where a ≠ 0. Its graph is a parabola that opens upwards if a > 0, or downwards if a < 0. The shape is symmetric about a vertical line called the axis of symmetry.

二次函数的一般形式为 y = ax² + bx + c，其中 a ≠ 0。它的图像是一条抛物线，若 a > 0 则开口向上，若 a < 0 则开口向下。图像关于一条称为对称轴的竖直线对称。

There are two other useful forms: the vertex form y = a(x − h)² + k, which immediately shows the vertex (h, k), and the intercept form y = a(x − p)(x − q), which reveals the x‑intercepts at x = p and x = q. Depending on the information visible on the graph, you can choose the most efficient algebraic approach.

另外两种常用形式是：顶点式 y = a(x − h)² + k，能直接显示顶点 (h, k)；交点式 y = a(x − p)(x − q)，能直接读出 x 截距 x = p 和 x = q。根据图上能获取的信息，你可以选择最高效的代数方法。

2. Key Features of a Parabola | 抛物线的关键特征

When you look at the graph of a quadratic, identify these features: (1) The vertex, the maximum or minimum point. (2) The axis of symmetry, the vertical line x = h through the vertex. (3) The x‑intercepts (roots), where the graph crosses the x‑axis. (4) The y‑intercept, where x = 0. (5) The direction of opening.

观察二次函数图像时，请识别以下特征：(1) 顶点

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