📚 Quadratic Functions: Equations and Graphs | 二次函数：方程与图像
Quadratic functions are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, problem solving, and even calculus foundations. This guide covers everything you need to master quadratics, from solving equations to sketching graphs.
二次函数是 IGCSE 数学中最重要的内容之一。它出现在代数、图形、应用题乃至微积分基础中。本指南涵盖掌握二次函数所需的全部内容，从解方程到画图。
1. Introduction to Quadratic Functions | 二次函数入门
A quadratic function is a polynomial of degree 2. Its standard form is f(x) = ax² + bx + c, where a ≠ 0. The graph of a quadratic function is a parabola, which opens upward if a > 0 and downward if a < 0.
二次函数是次数为 2 的多项式。其标准形式为 f(x) = ax² + bx + c，其中 a ≠ 0。二次函数的图像是抛物线，若 a > 0 则开口向上，若 a < 0 则开口向下。
- The highest power of x is 2 (x²). | x 的最高次数为 2（x²）。
- The coefficient a determines the width and direction. | 系数 a 决定抛物线的开口方向和宽度。
- The constant term c is the y-intercept. | 常数项 c 是 y 轴截距。
2. Standard Form and Key Features | 标准形式与关键特征
The standard form ax² + bx + c allows us to extract useful information. The axis of symmetry is given by x = -b/(2a), and the vertex lies on this line.
标准形式 ax² + bx + c 可以帮助我们提取有用信息。对称轴为 x = -b/(2a)，顶点位于这条直线上。
x = -b/(2a)
To find the vertex, substitute x = -b/(2a) back into the original equation. The y-coordinate gives the maximum or minimum value of the function.
将 x = -b/(2a) 代回原方程即可求出顶点。y 坐标就是函数的最大值或最小值。
| Feature | 特征 | Formula | 公式 |
|---|---|
| Axis of symmetry | 对称轴 | x = -b/(2a) |
| Vertex x-coordinate | 顶点 x 坐标 | x = -b/(2a) |
| Vertex y-coordinate | 顶点 y 坐标 | f(-b/(2a)) |
| y-intercept | y 轴截距 | c |
3. Solving Quadratic Equations by Factoring | 因式分解法解二次方程
Factoring is the fastest method when the expression can be written as a product of two binomials. For example, x² + 5x + 6 = (x + 2)(x + 3). Setting each factor to zero gives roots x = -2 and x = -3.
因式分解是当表达式能写成两个二项式乘积时最快的方法。例如，x² + 5x + 6 = (x + 2)(x + 3)。令每个因子为零，可得根 x = -2 和 x = -3。
- Rewrite the equation in the form ax² + bx + c = 0. | 将方程改写为 ax² + bx + c = 0 的形式。
- Factor the left-hand side. | 将左侧因式分解。
- Set each factor equal to zero and solve. | 令每个因式为 0，然后求解。
If ab = 0, then a = 0 or b = 0 | 若 ab = 0，则 a = 0 或 b = 0
This method requires the discriminant to be a perfect square for rational roots. Otherwise, factoring becomes difficult.
这种方法要求判别式是完全平方数，才能得到有理数根。否则，因式分解会变得困难。
4. Solving by Completing the Square | 配方法
Completing the square transforms a quadratic into the form a(x – h)² + k. This is useful for deriving the vertex form and solving equations.
配方法将二次方程转化为 a(x – h)² + k 的形式。这有助于得到顶点形式并解方程。
x² + bx = (x + b/2)² – (b/2)²
For the general equation ax² + bx + c = 0, divide by a first, then complete the square.
对于一般方程 ax² + bx + c = 0，先除以 a，然后配方。
- Divide both sides by a. | 两边同时除以 a。
- Move the constant to the right side. | 将常数项移到右边。
- Add (b/2a)² to both sides. | 两边加上 (b/2a)²。
- Write the left side as a binomial square. | 将左边写成二项式的平方。
5. The Quadratic Formula | 二次公式
The quadratic formula works for all quadratic equations. It is derived by completing the square on the general form. The formula is:
二次公式适用于所有二次方程。它通过对一般形式进行配方推导得出。公式为：
x = [-b ± √(b² – 4ac)] / (2a)
This formula gives the exact solutions, whether the roots are rational, irrational, or complex. On the IGCSE syllabus, we only consider real roots.
此公式给出精确解，无论根是有理数、无理数还是复数。在 IGCSE 考纲中，我们只考虑实根。
| Example | 示例 | Values | 数值 | Solution | 解 |
|---|---|---|
| 2x² + 3x – 5 = 0 | a=2, b=3, c=-5 | x = 1 or x = -2.5 |
6. The Discriminant | 判别式
The discriminant is the part under the square root in the quadratic formula: Δ = b² – 4ac. It determines the nature of the roots.
判别式是二次公式中平方根下的部分：Δ = b² – 4ac。它决定根的性质。
- If Δ > 0: two distinct real roots. | 若 Δ > 0：两个不同实根。
- If Δ = 0: one repeated real root (a double root). | 若 Δ = 0：一个重根。
- If Δ < 0: no real roots (the graph does not cross the x-axis). | 若 Δ < 0：无实根（图像不与 x 轴相交）。
Δ = b² – 4ac
The discriminant also helps when solving problems involving tangents to parabolas or intersections with lines.
判别式还用于解决与抛物线切线或直线交点相关的问题。
7. Graphing Quadratics | 画二次函数图像
To sketch a quadratic graph, follow these steps: find the y-intercept, find the x-intercepts by solving f(x) = 0, find the vertex, and then plot the parabola.
画二次函数图像时，按以下步骤：求 y 截距；通过解 f(x) = 0 求 x 截距；求顶点；然后画出抛物线。
- y-intercept: set x = 0. | y 截距：令 x = 0。
- x-intercepts: solve ax² + bx + c = 0. | x 截距：解 ax² + bx + c = 0。
- Vertex: use x = -b/(2a). | 顶点：使用 x = -b/(2a)。
- Direction: check the sign of a. | 方向：检查 a 的符号。
If the discriminant is negative, the parabola does not touch the x-axis, and the graph lies entirely above or below the axis.
若判别式为负，抛物线不与 x 轴相交，图像完全位于 x 轴上方或下方。
8. Transformations of Graphs | 图像变换
Quadratic graphs can be transformed by changing the values of a, h, and k in the vertex form f(x) = a(x – h)² + k.
二次函数图像可以通过改变顶点形式 f(x) = a(x – h)² + k 中的 a、h、k 进行变换。
| Change | 变化 | Effect | 效果 |
|---|---|
| a > 1 | a > 1 | Narrower graph | 图像变窄 |
| 0 < a < 1 | 0 < a < 1 | Wider graph | 图像变宽 |
| a < 0 | a < 0 | Reflects across x-axis | 关于 x 轴对称翻转 |
| h changes | h 变化 | Horizontal shift | 水平平移 |
| k changes | k 变化 | Vertical shift | 垂直平移 |
For example, f(x) = (x – 3)² + 2 shifts the graph of x² three units right and two units up.
例如，f(x) = (x – 3)² + 2 将 x² 的图像向右平移 3 个单位，向上平移 2 个单位。
9. Applications in Problem Solving | 应用题中的应用
Quadratic functions model real-world situations such as projectile motion, area problems, and profit optimization. For example, the height of a ball after t seconds is given by h(t) = -5t² + 20t + 1.
二次函数可以模拟现实情境，如抛体运动、面积问题和利润优化。例如，球在 t 秒后的高度为 h(t) = -5t² + 20t + 1。
To solve such problems:
解决此类问题的步骤：
- Identify the unknown and assign a variable. | 确定未知量并设变量。
- Write a quadratic equation based on the given conditions. | 根据已知条件写出二次方程。
- Solve the equation using an appropriate method. | 用合适的方法解方程。
- Check whether the solution makes sense in the context. | 检查解在情境中是否合理。
For a rectangle with area 24 m² and length 2 m longer than the width, we can write x(x + 2) = 24, leading to x² + 2x – 24 = 0.
对于一个面积为 24 m²、长比宽多 2 m 的长方形，可写为 x(x + 2) = 24，得到 x² + 2x – 24 = 0。
10. Common Exam Pitfalls and Tips | 常见考试陷阱与技巧
Students often lose marks on sign errors, incorrect factorisation, and forgetting to check the discriminant. Here are essential tips.
学生常因符号错误、因式分解错误和忘记检查判别式而失分。以下是一些重要建议。
- Always rearrange into ax² + bx + c = 0 before solving. | 求解前务必将方程化为 ax² + bx + c = 0。
- Check your factors by expanding. | 通过展开检查因式是否正确。
- When using the quadratic formula, be careful with negative b and c. | 使用二次公式时，注意 b 和 c 为负数的情况。
- For word problems, reject negative or impossible answers if they do not fit the context. | 在应用题中，排除不符合情境的负数解或不可能的解。
- Use a graphing calculator to verify your sketches in the exam if allowed. | 若考试允许，用图形计算器验证草图。
Mastering quadratics takes practice. Work through past papers and check every step carefully.
掌握二次函数需要多加练习。认真做历年真题，并逐步检查。
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