📚 Quadratic Functions and Equations | 二次函数与方程
In this article, we will review the key concepts of quadratic functions and equations in the IGCSE Mathematics syllabus. We will cover standard forms, graphs, solving methods, the discriminant, and common applications.
本文将复习IGCSE数学大纲中二次函数与方程的核心概念，包括标准形式、图像、求解方法、判别式以及常见应用。
1. Standard Form of Quadratic Functions | 二次函数的标准形式
A quadratic function can be written in the general form f(x) = ax² + bx + c, where a, b and c are constants, and a ≠ 0. The highest power of the variable is 2.
二次函数可以写成一般形式 f(x) = ax² + bx + c，其中 a、b、c 是常数，且 a ≠ 0。变量的最高次数是2。
A quadratic equation is formed when the function is set to zero: ax² + bx + c = 0. Solving this equation means finding the values of x that make the function equal to zero.
当函数值为零时，就得到二次方程：ax² + bx + c = 0。解这个方程就是找到使函数值为零的 x 的值。
2. Graph Shape and Key Features | 图像的形状与关键特征
The graph of a quadratic function is called a parabola. If a > 0, the parabola opens upwards, and the vertex is a minimum point. If a < 0, it opens downwards, and the vertex is a maximum point.
二次函数的图像称为抛物线。如果 a > 0，抛物线开口向上，顶点是最小值点；如果 a < 0，开口向下，顶点是最大值点。
The axis of symmetry is the vertical line x = −b/(2a). The vertex lies on this line. The y-intercept of the graph is at (0, c).
对称轴是垂直直线 x = −b/(2a)，顶点位于这条直线上。图像与 y 轴的交点为 (0, c)。
3. Vertex Form and Completing the Square | 顶点式与配方法
A quadratic function can also be written in vertex form: f(x) = a(x − h)² + k, where (h, k) is the vertex. This form is very useful for sketching graphs.
二次函数还可以写成顶点式：f(x) = a(x − h)² + k，其中 (h, k) 是顶点。这种形式非常便于画图。
To convert from general form to vertex form, we use the method of completing the square. For example, for x² + 6x + 5, we write (x + 3)² − 4.
要将一般式化为顶点式，我们使用配方法。例如，对于 x² + 6x + 5，可以写成 (x + 3)² − 4。
x² + 6x + 5 = (x + 3)² − 4
4. Solving by Factorisation | 用因式分解求解
If a quadratic expression can be factorised, we can solve the equation by setting each factor to zero. For example, x² − 5x + 6 = 0 becomes (x − 2)(x − 3) = 0.
如果一个二次表达式可以因式分解，我们可以通过令每个因式等于零来解方程。例如，x² − 5x + 6 = 0 变为 (x − 2)(x − 3) = 0。
Therefore, x = 2 or x = 3. Always check whether the equation is in the form ax² + bx + c = 0 before factorising.
因此，x = 2 或 x = 3。在因式分解前，一定要确保方程化为 ax² + bx + c = 0 的形式。
(x − 2)(x − 3) = 0 → x = 2, x = 3
5. Solving by the Quadratic Formula | 用求根公式求解
When factorisation is difficult or impossible, we use the quadratic formula. For ax² + bx + c = 0, the solutions are:
当因式分解困难或不可能时，我们使用求根公式。对于 ax² + bx + c = 0，其解为：
x = (−b ± √b² − 4ac) / (2a)
This formula works for all quadratic equations, including those with irrational or complex roots (although IGCSE usually focuses on real roots).
这个公式适用于所有二次方程，包括有无理数根或复数根的情况（不过IGCSE通常只关注实数根）。
6. The Discriminant and the Nature of Roots | 判别式与根的性质
The expression Δ = b² − 4ac is called the discriminant. It tells us how many real roots a quadratic equation has.
表达式 Δ = b² − 4ac 称为判别式。它告诉我们二次方程有多少个实数根。
- If Δ > 0: two distinct real roots.
- If Δ = 0: one repeated real root (a double root).
- If Δ < 0: no real roots.
- 如果 Δ > 0：有两个不同的实数根。
- 如果 Δ = 0：有一个重根（二重根）。
- 如果 Δ < 0：没有实数根。
This is a very common exam topic. Remember the relationship between the discriminant and the shape of the graph.
这是非常常见的考点。请记住判别式与图像形状之间的关系。
7. Graph and Roots | 图像与根的关系
The real roots of a quadratic equation are the x-coordinates of the points where the graph crosses the x-axis. If the discriminant is positive, there are two crossing points; if zero, the vertex touches the x-axis; if negative, the graph never touches the x-axis.
二次方程的实数根是图像与 x 轴交点的横坐标。如果判别式为正，有两个交点；如果为零，顶点与 x 轴相切；如果为负，图像与 x 轴没有交点。
For example, the graph of y = x² − 4 crosses the x-axis at x = −2 and x = 2.
例如，y = x² − 4 的图像在 x = −2 和 x = 2 处与 x 轴相交。
8. Solving Quadratic Inequalities | 二次不等式简介
Quadratic inequalities can be solved by sketching the graph and reading the intervals that satisfy the inequality. For example, x² − 5x + 6 > 0 is satisfied when x < 2 or x > 3.
二次不等式可以通过画图并找出满足不等式的区间来求解。例如，x² − 5x + 6 > 0 在 x < 2 或 x > 3 时成立。
For x² − 5x + 6 < 0, the solution is the interval between the roots: 2 < x < 3.
对于 x² − 5x + 6 < 0，解是两根之间的区间：2 < x < 3。
9. Applications: Maximum and Minimum Problems | 应用：最大值与最小值问题
Many real-world problems involve finding the maximum or minimum value of a quadratic function. The vertex gives this value. For a function f(x) = ax² + bx + c, the x-coordinate of the vertex is −b/(2a), and the maximum or minimum value is f(−b/(2a)).
许多实际问题涉及求二次函数的最大值或最小值。顶点给出这个值。对于函数 f(x) = ax² + bx + c，顶点的横坐标是 −b/(2a)，最大值或最小值就是 f(−b/(2a))。
For example, if the height of a projectile is given by h(t) = −5t² + 20t + 1, the maximum height occurs at t = 2 and equals 21.
例如，如果抛射体的高度由 h(t) = −5t² + 20t + 1 给出，则最大高度在 t = 2 时取得，其值为 21。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Here are some common pitfalls to avoid in the exam:
以下是一些考试中需要避免的常见错误：
- Forgetting to set the equation to zero before factorising or using the formula.
- Mixing up the signs in the quadratic formula, especially −b and b² − 4ac.
- When completing the square, forgetting to adjust the constant term correctly.
- 在因式分解或使用公式前忘记将方程化为零的形式。
- 在求根公式中搞错符号，尤其是 −b 和 b² − 4ac。
- 配方法中忘记正确调整常数项。
Always check your solutions by substituting them back into the original equation. Practice drawing graphs to verify the number of roots.
始终通过代回原方程来检验解。练习画图以验证根的个数。
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