Quadratic Functions and Equations | 二次函数与方程

📚 Quadratic Functions and Equations | 二次函数与方程

Quadratic functions and equations are a cornerstone of IGCSE Mathematics. They appear in algebra, graphs, problem-solving, and even geometry. Mastering this topic not only boosts exam scores but also builds a strong foundation for A-Level mathematics and beyond.

二次函数与方程是 IGCSE 数学的基石。它们贯穿代数、图像、问题解决乃至几何领域。掌握这一主题不仅能提高考试成绩，更为进一步学习 A-Level 数学打下坚实基础。

1. Standard Form of a Quadratic Function | 二次函数的标准形式

A quadratic function is any function of the form f(x) = ax² + bx + c, where a, b, and c are constants, and a ≠ 0. The graph of such a function is a parabola, which opens upwards if a > 0 and downwards if a < 0.

二次函数是形如 f(x) = ax² + bx + c 的函数，其中 a、b、c 为常数，且 a ≠ 0。该函数的图像是抛物线：当 a > 0 时开口向上，当 a < 0 时开口向下。

The coefficient a determines the width and direction of the parabola. A larger absolute value of a makes the parabola narrower, while a smaller absolute value makes it wider. The constant c represents the y-intercept, where the graph crosses the y-axis.

系数 a 决定抛物线的宽度和方向。|a| 越大，抛物线越窄；|a| 越小，抛物线越宽。常数 c 表示 y 轴截距，即图像与 y 轴交点的纵坐标。

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

The vertex is the highest or lowest point of a parabola. For the standard form, the x-coordinate of the vertex is given by x = -b / (2a). The y-coordinate is found by substituting this x back into the function.

顶点是抛物线的最高点或最低点。对于标准形式，顶点的 x 坐标为 x = -b / (2a)。将 x 代回函数即可求出 y 坐标。

The axis of symmetry is the vertical line x = -b/(2a). The parabola is symmetric about this line. The y-intercept is always at (0, c). The x-intercepts, or roots, are the solutions to the quadratic equation ax² + bx + c = 0.

对称轴是垂直线 x = -b/(2a)，抛物线关于该线对称。y 截距恒为 (0, c)。x 截距（即根）是方程 ax² + bx + c = 0 的解。

3. Solving Quadratic Equations by Factorisation | 用因式分解解二次方程

Factorisation is often the quickest method when simple integer solutions exist. The idea is to rewrite the quadratic expression as a product of two linear factors.

当存在简单的整数解时，因式分解通常是最快捷的方法。其核心思想是将二次表达式改写为两个一次因式的乘积。

Example: Solve x² – 5x + 6 = 0. We look for two numbers that multiply to 6 and add to -5. These are -2 and -3, so (x – 2)(x – 3) = 0. Hence x = 2 or x = 3.

示例：解方程 x² – 5x + 6 = 0。寻找两个和为 -5、积为 6 的数，即 -2 和 -3，因此 (x – 2)(x – 3) = 0，解得 x = 2 或 x = 3。

Always expand your answer to check. If the coefficient of x² is not 1, use the method of grouping or the ‘ac’ method. Practise recognising common patterns such as difference of squares and perfect squares.

务必展开答案进行验算。当 x² 的系数不为 1 时，可使用分组法或 ac 法。注意识别常见的平方差和完全平方模式。

4. Solving by Completing the Square | 配方法求解

Completing the square rewrites the quadratic in the form a(x + p)² + q or (x + p)² + q. This is especially useful for finding the vertex and solving equations that cannot be factorised.

配方法将二次式改写为 a(x + p)² + q 或 (x + p)² + q 的形式。这在求顶点和解无法因式分解的方程时特别有用。

Steps for x² + bx: Take half of b, square it, and add/subtract it inside the expression. For example, x² + 6x + 2 = (x + 3)² – 9 + 2 = (x + 3)² – 7.

对于 x² + bx 的步骤：取 b 的一半，平方，然后在表达式中加上并减去该值。例如 x² + 6x + 2 = (x + 3)² – 9 + 2 = (x + 3)² – 7。

To solve an equation by completing the square, set the completed-square expression to zero and rearrange. This method also reveals the vertex directly: the vertex is (-p, q) when written as a(x + p)² + q.

通过配方法解方程时，将配方后的表达式设为 0 并重新整理。此法还能直接揭示顶点：当写为 a(x + p)² + q 时，顶点为 (-p, q)。

5. The Quadratic Formula | 求根公式

The quadratic formula x = (-b ± √(b² – 4ac)) / (2a) is a universal method that works for any quadratic equation ax² + bx + c = 0. It is derived from completing the square.

求根公式 x = (-b ± √(b² – 4ac)) / (2a) 是适用于任何二次方程 ax² + bx + c = 0 的通用方法，由配方法推导得出。

Memorise this formula exactly. In an exam, write it down first, then substitute the values of a, b, and c. Be careful with negative signs: b is the coefficient of x, including its sign.

务必准确记忆此公式。考试时先写出公式，再代入 a、b、c 的值。注意负号：b 是 x 的系数，要包含其符号。

Simplify the result as much as possible, and leave answers in surd form if required, especially when the question asks for exact values. The formula is particularly useful when the roots are irrational or complex.

尽可能化简结果。若题目要求精确值，请保留根号形式。当根为无理数或复数时，公式尤为实用。

6. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root: Δ = b² – 4ac. It determines the nature of the roots without actually solving the equation.

判别式是求根公式中根号下的部分：Δ = b² – 4ac。它能在不解方程的情况下判断根的性质。

If Δ > 0, there are two distinct real roots.
如果有 Δ > 0，则有两个不同的实数根。
If Δ = 0, there is exactly one real root (a repeated root).
如果 Δ = 0，则恰好有一个实数根（重根）。
If Δ < 0, there are no real roots.
如果 Δ < 0，则没有实数根。

The discriminant also tells about the graph: two x-intercepts, one x-intercept (touching the axis), or no x-intercept. Many IGCSE questions ask for the range of k such that a quadratic has certain root types.

判别式也反映图像特征：两个 x 截距、一个 x 截距（与轴相切）或无 x 截距。许多 IGCSE 题目要求确定 k 的取值范围，使二次方程具有特定类型的根。

7. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, we have α + β = -b/a and αβ = c/a. These relationships come directly from the factorised form a(x – α)(x – β) = 0.

对于二次方程 ax² + bx + c = 0，设两根为 α 和 β，则有 α + β = -b/a 和 αβ = c/a。这些关系直接来自因式分解形式 a(x – α)(x – β) = 0。

These formulas allow you to form a new quadratic with given roots, or to find expressions like α² + β² without solving the equation. For example, α² + β² = (α + β)² – 2αβ.

利用这些公式可以在已知根的情况下构造新的二次方程，或求诸如 α² + β² 的表达式而无需解方程。例如，α² + β² = (α + β)² – 2αβ。

Be aware of sign conventions. The sum uses -b/a, not b/a. Practise using these with fractions and negative coefficients.

注意符号约定：和是 -b/a 而非 b/a。练习时多处理系数为分数或负数的情况。

8. Graph Sketching and Transformations | 图像画法与变换

Sketching a quadratic graph involves finding the vertex, the y-intercept, and the x-intercepts (or showing there are none). Plot these key points and draw a smooth symmetric parabola.

绘制二次函数图像需要找到顶点、y 截距和 x 截距（或证明无实数解）。标出这些关键点后，画出平滑对称的抛物线。

Transformations change the graph in predictable ways. A vertical translation f(x) + k moves the graph up (k > 0) or down (k < 0). A horizontal translation f(x + h) shifts left (h > 0) or right (h < 0). Note the opposite direction for horizontal shifts.

图像的变换遵循固定规律。竖直平移 f(x) + k 使图像向上（k > 0）或向下（k < 0）移动。水平平移 f(x + h) 使图像向左（h > 0）或向右（h < 0）移动。注意水平移动的方向相反。

Reflections: -f(x) reflects over the x-axis, and f(-x) reflects over the y-axis. A stretch multiplies the y-values (vertical stretch) or x-values (horizontal stretch) by a factor. Practise sketching transformed graphs step by step.

反射：-f(x) 沿 x 轴翻转，f(-x) 沿 y 轴翻转。伸缩则是将 y 值（竖直伸缩）或 x 值（水平伸缩）乘以一个因子。练习时按步骤绘制变换后的图像。

9. Applications: Problem Solving | 应用：问题求解

Quadratic equations model many real-world situations, such as projectile motion, area problems, and profit optimisation. In IGCSE, you may be asked to translate a word problem into a quadratic equation and solve it.

二次方程可模拟许多现实场景，如抛射运动、面积问题和利润优化。在 IGCSE 中，你需要将文字问题转化为二次方程并求解。

Example: The product of two consecutive positive integers is 56. Find the integers. Let the integers be n and n + 1. Then n(n + 1) = 56, so n² + n – 56 = 0. Factorising gives (n + 8)(n – 7) = 0, so n = 7 (since n is positive). The integers are 7 and 8.

示例：两个连续正整数的乘积是 56，求这两个整数。设整数为 n 和 n + 1，则 n(n + 1) = 56，即 n² + n – 56 = 0。因式分解得 (n + 8)(n – 7) = 0，因 n 为正数，故 n = 7。两个整数为 7 和 8。

Always check whether your solutions make sense in the context of the problem. Discard any solution that violates the given conditions, such as negative lengths or times.

务必检查解在题目情境中是否合理。若解不符合条件（如长度为负或时间为负），应舍去。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Students often make errors in signs when using the quadratic formula, forget to set the equation to zero before factorising, or misread the coefficient of x². Careful systematic working helps avoid these pitfalls.

学生常在使用求根公式时弄错符号，因式分解前忘记将方程化为零，或误读 x² 的系数。细致的分步书写能避免此类错误。

Always write down the formula or method first.
首先写出公式或方法。
Rearrange the equation into standard form ax² + bx + c = 0 before applying methods.
将方程整理为标准形式 ax² + bx + c = 0 后再使用各种方法。
For factorisation, check by expanding.
因式分解后通过展开验证。
For word problems, define the variable and check the answer’s validity.
应用题中先定义变量，并检查答案的合理性。
In graph sketching, label the vertex, intercepts, and axis of symmetry.
画图时标注顶点、截距和对称轴。

Pay attention to the mark scheme: showing your working is often more important than the final answer. Round only when instructed, and leave answers in exact surd form if the question asks for it.

注意评分标准：过程往往比其他重要。只有在题目要求时才四舍五入，若要求精确值，则以根号形式保留。

11. Practice Exercises | 练习精粹

Try these questions to solidify your understanding. Work through each one fully before checking the answers.

尝试以下题目来巩固理解。请完整解答每题后再对照答案。

Question	Hint
1. Solve x² – 2x – 8 = 0 by factorisation.	Find factors of -8 summing to -2.
2. Complete the square for x² + 4x + 1.	Half of 4 is 2, so (x + 2)² – 4 + 1.
3. Determine the nature of roots of 2x² – 3x + 5 = 0.	Calculate Δ = 9 – 40 = -31.
4. If α and β are roots of x² + 6x + 2 = 0, find α² + β².	Use (α + β)² – 2αβ.
5. Sketch y = (x + 1)² – 4, labelling vertex and intercepts.	Vertex is (-1, -4).

Answers: 1) x = 4 or x = -2. 2) (x + 2)² – 3. 3) Δ = -31 < 0, no real roots. 4) 36 - 4 = 32. 5) Vertex (-1, -4), y-intercept (0, -3), x-intercepts (1, 0) and (-3, 0).

答案：1) x = 4 或 x = -2。2) (x + 2)² – 3。3) Δ = -31 < 0，无实数根。4) 36 - 4 = 32。5) 顶点 (-1, -4)，y 截距 (0, -3)，x 截距 (1, 0) 和 (-3, 0)。

12. Summary | 总结

Quadratic functions and equations are a vital part of IGCSE Maths, linking algebra, graphs, and problem solving. You have learned the standard form, key features of a parabola, and three solving methods. The discriminant and root relationships add deeper insight.

二次函数与方程是 IGCSE 数学的重要组成部分，连接着代数、图像和问题解决。你已经学习了标准形式、抛物线关键特征以及三种解法。判别式和根的关系则提供了更深层的理解。

Regular practice, careful attention to signs, and a systematic approach will help you master this topic. Use the methods flexibly, and always check your answers in the original equation or problem context.

规律练习、注意符号和系统化的方法是掌握该主题的关键。灵活运用各种解法，并始终在原始方程或问题情境中检验答案。

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Quadratic Functions and Equations | 二次函数与方程