📚 Finding a Quadratic from its Graph | 从图像中求二次函数
A quadratic function produces a parabola when graphed. Given the graph of a parabola, how can we determine the equation of the quadratic function? By identifying key features of the graph — such as the vertex, the x‑intercepts, the y‑intercept, and the direction of opening — we can reconstruct the function in one of its standard forms. This skill is essential in IB Mathematics, as it combines algebraic manipulation with graphical interpretation.
二次函数的图像是一条抛物线。给定一条抛物线的图形，我们如何确定对应的二次函数表达式？通过识别图形的关键特征——例如顶点、x 截距、y 截距以及开口方向——我们可以以某一种标准形式还原函数。这项技能在 IB 数学中至关重要，因为它将代数运算与图形解读结合在一起。
1. The Key Features of a Parabola | 抛物线的主要特征
. Key points include the vertex (the turning point), the axis of symmetry (a vertical line through the vertex), the y‑intercept (where x = 0), and the x‑intercepts (roots or zeros, where y = 0).
二次函数 y = ax² + bx + c (a ≠ 0) 的图像是一条抛物线。a 的符号决定了抛物线开口向上（a > 0）还是向下（a < 0）。关键点包括顶点（转折点）、对称轴（经过顶点的垂直线）、y 截距（x = 0 时）和 x 截距（根或零点，y = 0 时）。
When finding the equation from a graph, the first step is to read off the coordinates of the vertex and the intercepts as accurately as possible. Sometimes the graph will clearly show these values, but you may also need to estimate them from a scale.
在从图像求方程时，第一步是尽可能准确地读出顶点和截距的坐标。有时图形会清晰地显示这些数值，但也可能需要根据比例进行估算。
2. The Three Standard Forms of a Quadratic | 二次函数的三种标准形式
Each form highlights different information and is chosen based on what the graph reveals.
要写出抛物线的方程，我们通常在三种形式中选择：一般式（y = ax² + bx + c）、顶点式（y = a(x − h)² + k）和交点式或因式分解形式（y = a(x − x₁)(x − x₂)）。每种形式突出不同的信息，选择哪一种取决于图形所揭示的特征。
| Form | Key Information (EN) | 关键信息 (ZH) |
|---|---|---|
| Standard form y = ax² + bx + c | y‑intercept is c; can be used with any three points. | y 截距为 c；可与任意三点联用。 |
| Vertex form y = a(x − h)² + k | Vertex is (h, k); easy to see the turning point. | 顶点为 (h, k)；易于看出转折点。 |
| Factored form y = a(x − x₁)(x − x₂) | Roots are x₁ and x₂; ideal when intercepts are given. | 根为 x₁ 和 x₂；适合已知截距时使用。 |
Choosing the right form makes the algebra simpler. For example, if the vertex is clearly shown, start with the vertex form. If the graph crosses the x‑axis at two nice numbers, use the factored form.
选择合适的形式可以简化代数运算。例如，如果顶点清晰显示，就从顶点式开始。如果图形在 x 轴上的两个交点是整齐的数字，就使用交点式。
3. Identifying the Vertex from the Graph | 从图像中确定顶点
The vertex is the highest point (for a < 0) or lowest point (for a > 0). You can find its x‑coordinate by drawing the axis of symmetry — the vertical line that splits the parabola into two mirror images. Once you have the x‑coordinate h, read the corresponding y‑coordinate k.
顶点是最高点（a < 0）或最低点（a > 0）。通过画出对称轴——将抛物线分成两个镜像的垂直线——可以找到其 x 坐标。一旦得到 x 坐标 h，就读取对应的 y 坐标 k。
Be careful when the vertex lies between grid lines; you may need to interpolate. In IB exam questions, graphs are usually drawn with clear scales, but always double‑check.
当顶点位于网格线之间时要小心；你可能需要进行插值。在 IB 考试题中，图形通常有清晰的比例尺，但要始终仔细核对。
4. Using Vertex Form: y = a(x − h)² + k | 使用顶点式：y = a(x − h)² + k
only unknown is the stretch factor a. Pick another point (x, y) that lies on the parabola — for instance, the y‑intercept or any clear grid intersection — and substitute its coordinates to solve for a.
一旦有了顶点 (h, k)，将这些值代入顶点式：y = a(x − h)² + k。唯一未知的是伸缩因子 a。选取抛物线上另一个点 (x, y)（例如 y 截距或任何清晰的网格交点），代入其坐标求解 a。
pose the vertex is (3, −2) and the point (5, 4) is on the graph. Then 4 = a(5 − 3)² + (−2) ⇒ 4 = 4a − 2 ⇒ 4a = 6 ⇒ a = 1.5. So the equation is y = 1.5(x − 3)² − 2.
示例：设顶点为 (3, −2)，点 (5, 4) 在图像上。则 4 = a(5 − 3)² + (−2) ⇒ 4 = 4a − 2 ⇒ 4a = 6 ⇒ a = 1.5。因此方程为 y = 1.5(x − 3)² − 2。
5. Finding the St
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