IB Mathematics: Determining Quadratic Functions from Graphs | IB数学：由图像确定二次函数解析式

📚 IB Mathematics: Determining Quadratic Functions from Graphs | IB数学：由图像确定二次函数解析式

In IB Mathematics, you are often given the graph of a quadratic function and asked to reconstruct its algebraic expression. This skill combines visual reading with algebraic solving, and it is tested in both Analysis and Approaches (AA) and Applications and Interpretation (AI).

在IB数学中，经常会给出一个二次函数的图像，要求还原其代数表达式。这一技能将图像识读与代数求解相结合，在分析与方法（AA）以及应用与解释（AI）中都会考查。

1. The Three Forms of a Quadratic Function | 二次函数的三种形式

Every quadratic function can be written in one of three standard forms, and each form is easiest to use in a different graphical situation.

每一个二次函数都可以写成三种标准形式之一，而每种形式在不同图像情境下使用最为方便。

The standard form is y = ax² + bx + c. The value of c is the y-intercept, and the axis of symmetry is x = -b/(2a).

一般式为 y = ax² + bx + c。其中 c 为 y 截距，对称轴为 x = -b/(2a)。

The vertex form is y = a(x – h)² + k, where (h, k) is the vertex of the parabola.

顶点式为 y = a(x – h)² + k，其中 (h, k) 是抛物线的顶点。

The factored form is y = a(x – p)(x – q), where p and q are the x-intercepts.

交点式为 y = a(x – p)(x – q)，其中 p 和 q 是 x 轴交点。

Form / 形式	Equation / 方程	Best used when / 适用情境
Standard / 一般式	y = ax² + bx + c	Three points are known / 已知三个点
Vertex / 顶点式	y = a(x – h)² + k	Vertex and another point are known / 已知顶点和另一个点
Factored / 交点式	y = a(x – p)(x – q)	x-intercepts are known / 已知两个 x 轴交点

2. Reading Key Features from a Graph | 从图像读取关键特征

Before choosing a method, identify the graphical features that are clearly visible. The vertex is the turning point of the parabola; it may be a maximum or a minimum.

在选择方法之前，先辨认图像上清晰可见的特征。顶点是抛物线的转折点，可能为最大值或最小值。

The x-intercepts are the points where the graph crosses the x-axis, and the y-intercept is the point where the graph crosses the y-axis.

x 轴交点是图像与 x 轴相交的点，y 截距则是图像与 y 轴相交的点。

Any other clearly labelled point on the curve can also be used to find the unknown coefficient a.

曲线上任何其他标注清晰的点也可以用来求未知系数 a。

Read the vertex (h, k) if it is labelled or clearly visible.

若顶点已标注或清晰可见，先读取 (h, k)。
Read the x-intercepts p and q, if the graph crosses the x-axis.

若图像与 x 轴相交，读取 x 轴交点 p 和 q。
Read the y-intercept c, or any other point on the curve.

读取 y 截距 c，或曲线上任意其他点。

3. Strategy 1: Vertex Form | 策略一：顶点式

If the vertex (h, k) is visible, write the equation as y = a(x – h)² + k. The only unknown is a, so one additional point gives the complete function.

如果可以直接看到顶点 (h, k)，将方程写成 y = a(x – h)² + k。此时唯一未知的是 a，因此再知道一个点即可得到完整函数。

Substitute the coordinates of the extra point into the equation and solve for a.

将额外点的坐标代入方程，解出 a 即可。

a = (y – k) / (x – h)²

Once a is found, write the final expression in the form requested by the question, which is often standard form or vertex form.

求出 a 后，按照题目要求写出最终表达式，通常为一般式或顶点式。

4. Strategy 2: Factored Form | 策略二：交点式

If the graph crosses the x-axis at x = p and x = q, the function can be written as y = a(x – p)(x – q).

若图像与 x 轴交于 x = p 和 x = q，函数可写为 y = a(x – p)(x – q)。

This form is especially useful when the roots are integers, because substituting a known point gives a directly.

当根为整数时，这种形式尤其方便，因为代入一个已知点即可直接求得 a。

If the parabola has a repeated root, the graph touches the x-axis at the vertex, so p = q.

若抛物线与 x 轴相切，则根为重根，此时 p = q，顶点位于 x 轴上。

5. Strategy 3: Standard Form with Three Points | 策略三：一般式与三个点

When no vertex or roots are obvious, use the general form y = ax² + bx + c and substitute three non-collinear points.

当顶点或根不明显时，可设一般式 y = ax² + bx + c，并代入三个不共线的点。

This creates a system of three linear equations in a, b, and c. Solve the system by elimination or substitution.

这会得到关于 a、b、c 的三元一次方程组。可通过消元法或代入法求解。

If one of the points is (0, c), then c is immediately known and only two equations are needed.

若其中一点为 (0, c)，则 c 可直接读出，只需两个方程即可求解。

6. Symmetry and the Leading Coefficient | 对称轴与二次项系数

The axis of symmetry of a parabola is the vertical line x = h. If the roots p and q are known, then h = (p + q)/2.

抛物线的对称轴是竖直直线 x = h。若已知根 p 和 q，则 h = (p + q)/2。

The sign of a determines the direction of opening: a > 0 opens upward, and a < 0 opens downward.

a 的正负决定开口方向：a > 0 时开口向上，a < 0 时开口向下。

The magnitude |a| controls the width: a larger |a| gives a narrower parabola, while a smaller |a| gives a wider one.

|a| 的大小控制开口宽窄：|a| 越大，抛物线越窄；|a| 越小，抛物线越宽。

To find a, substitute any point on the curve other than the vertex into the appropriate form. The vertex alone is never enough to determine a.

要求 a，可代入曲线上除顶点外的任意一点到适当形式中。仅凭顶点永远不足以确定 a。

7. Worked Example 1: Vertex and One Point | 例题一：已知顶点和一点

A parabola has vertex (2, 1) and passes through the point (4, 9). Find its equation in standard form.

某抛物线的顶点为 (2, 1)，且经过点 (4, 9)。求其一般式方程。

Write the vertex form: y = a(x – 2)² + 1.

先写出顶点式：y = a(x – 2)² + 1。

Substitute (4, 9): 9 = a(4 – 2)² + 1, so 9 = 4a + 1, giving a = 2.

代入 (4, 9)：9 = a(4 – 2)² + 1，即 9 = 4a + 1，得 a = 2。

y = 2(x – 2)² + 1 = 2x² – 8x + 9

Therefore the standard form is y = 2x² – 8x + 9.

因此一般式为 y = 2x² – 8x + 9。

8. Worked Example 2: Roots and One Point | 例题二：已知根和一点

A parabola crosses the x-axis at x = -2 and x = 5, and it passes through (1, -18). Find the function.

某抛物线与 x 轴交于 x = -2 和 x = 5，且经过点 (1, -18)。求该函数。

Use the factored form: y = a(x + 2)(x – 5).

使用交点式：y = a(x + 2)(x – 5)。

Substitute (1, -18): -18 = a(3)(-4), so -12a = -18, giving a = 1.5.

代入 (1, -18)：-18 = a(3)(-4)，即 -12a = -18，得 a = 1.5。

y = 1.5(x + 2)(x – 5) = 1.5x² – 4.5x – 15

The final function in standard form is y = 1.5x² – 4.5x – 15.

化简为一般式得到 y = 1.5x² – 4.5x – 15。

9. Worked Example 3: Three Points | 例题三：已知三个点

A quadratic passes through (0, -4), (2, 6), and (-1, -3). Determine its equation.

一条二次函数图像经过 (0, -4)、(2, 6) 和 (-1, -3)。求其方程。

Use standard form y = ax² + bx + c. Since (0, -4) is the y-intercept, c = -4.

设一般式 y = ax² + bx + c。由于 (0, -4) 是 y 截距，所以 c = -4。

From (2, 6): 4a + 2b – 4 = 6, so 2a + b = 5.

由 (2, 6)：4a + 2b – 4 = 6，得 2a + b = 5。

From (-1, -3): a – b – 4 = -3, so a – b = 1.

由 (-1, -3)：a – b – 4 = -3，得 a – b = 1。

Solving the system gives a = 2 and b = 1.

解方程组得 a = 2，b = 1。

y = 2x² + x – 4

Thus the required equation is y = 2x² + x – 4.

因此所求方程为 y = 2x² + x – 4。

10. Common Errors and IB Exam Advice | 常见错误与IB考试建议

Using the vertex but never solving for a. A parabola is not fully described by its vertex alone.

只写出顶点而没有求出 a。仅凭顶点无法完整描述抛物线。
Mixing up the intercepts: p and q are x-intercepts, not y-values of the vertex.

混淆截距：p 和 q 是 x 轴交点，不是顶点的 y 值。
Forgetting the sign of a when the parabola opens downward.

当抛物线开口向下时，忘记 a 为负数。
When expanding vertex form, incorrectly expanding (x – h)² as x² – h² instead of x² – 2hx + h².

展开顶点式时，将 (x – h)² 错误地展开为 x² – h²，而没有展开为 x² – 2hx + h²。
Always check the final equation by substituting the given points back into the expression.

务必把已知点代回最终表达式进行验证。

11. Practice Questions with Solutions | 练习与答案

Try these short questions without a calculator, then check your answers.

尝试用计算器完成以下

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IB Mathematics: Determining Quadratic Functions from Graphs | IB数学：由图像确定二次函数解析式