📚 Quadratic Functions and Graphs | 二次函数与图像
Quadratic functions are one of the core topics in IGCSE Mathematics. This article explains how to solve quadratic equations, sketch their graphs, and apply key techniques to exam-style problems.
二次函数是IGCSE数学的核心内容之一。本文将讲解如何求解二次方程、绘制其图像，并运用关键技巧解答考试型题目。
1. Standard Form of a Quadratic Function | 二次函数的标准形式
A quadratic function is written as f ants and a ≠ 0. The graph of such a function is a parabola. If a > 0, the parabola opens upwards; if a < 0, it opens downwards.
二次函数写作 f(x) = ax² + bx + c，其中 a、b、c 为常数且 a ≠ 0。此类函数的图像是一条抛物线。若 a > 0，抛物线开口向上；若 a < 0，开口向下。
f(x) = ax² + bx + c , a ≠ 0
- The constant c gives the y-intercept (0, c).
- 常数项 c 决定 y 轴截距 (0, c)。
2. Solving Quadratic Equations by Factorisation | 用因式分解法解二次方程
or example, solve x² − 5x + 6 = 0. We factor to (x − 2)(x − 3) = 0, giving x = 2 or x = 3.
当二次方程可写成两个一次因式相乘时，令每个因式等于零。例如解 x² − 5x + 6 = 0，因式分解为 (x − 2)(x − 3) = 0，得 x = 2 或 x = 3。
x² − 5x + 6 = (x − 2)(x − 3) = 0 → x = 2, x = 3
- If the coefficient of x² is not 1, use the product-sum method or grouping.
- 若 x² 的系数不是 1，可用乘积-求和法或分组法。
3. Completing the Square | 配方法
+ 6x + 10, half of 6 is 3, so (x + 3)² − 9 + 10 = (x + 3)² + 1. The vertex is at (−3, 1).
配方法将二次式改写为 a(x + p)² + q。对于 y = x² + 6x + 10，6 的一半是 3，所以 (x + 3)² − 9 + 10 = (x + 3)² + 1。顶点为 (−3, 1)。
x² + 6x + 10 = (x + 3)² + 1
- The turning point is (−p, q).
- 顶点坐标为 (−p, q)。
4. The Quadratic Formula | 二次公式
For ax² + bx + c = 0, the solutions are given by x = (−b ± √b² − 4ac) / (2a). This formula works for all quadratic equations, including those that cannot be factored.
对于 ax² + bx + c = 0，解为 x = (−b ± √b² − 4ac) / (2a)。该公式适用于所有二次方程，包括无法因式分解的情况。
x = (−b ± √b² − 4ac) / (2a)
For example, 2x² + 3x − 2 = 0 gives x = (−3 ± √9 + 16) / 4 = (−3 ± 5) / 4, so x = 0.5 or x = −2.
例如，2x² + 3x − 2 = 0 得 x = (−3 ± √25) / 4 = (−3 ± 5) / 4，故 x = 0.5 或 x = −2。
5. The Discriminant and Nature of Roots | 判别式与根的性质
The expression b² − 4ac is called the discriminant, written as Δ. Its sign tells us how many real roots the equation has.
表达式 b² − 4ac 称为判别式，记作 Δ。它的符号决定方程有多少实根。
| Δ = b² − 4ac | Number of real roots | 实根个数 |
| Δ > 0 | Two distinct roots | 两个不同的实根 |
| Δ = 0 | One repeated root | 一个重根 |
| Δ < 0 | No real roots | 没有实根 |
If Δ is a perfect square and a, b, c are rational, the equation can also be solved by factorisation.
若 Δ 是一个完全平方数且 a、b、c 为有理数，则方程也可用因式分解法求解。
6. Sketching the Graph of a Quadratic | 绘制二次函数图像
ing ax² + bx + c = 0, and the turning point by completing the square or using x = −b/(2a).
绘制 y = ax² + bx + c 的草图，需要找出 y 轴截距 (0, c)，通过解 ax² + bx + c = 0 找 x 轴截距，并通过配方或用 x = −b/(2a) 找顶点。
- The axis of symmetry is x = −b/(2a).
- 对称轴为 x = −b/(2a)。
- If the quadratic has no real roots, the graph does not cross the x-axis.
- 若二次方程无实根，则图像不与 x 轴相交。
Turning point x-coordinate: x = −b/(2a)
7. Transformations of Quadratic Graphs | 二次函数图像的变换
s shifted horizontally by h and vertically by k. The sign of a determines the orientation.
从配方式 y = a(x − h)² + k 可以看出，y = x² 的图像水平平移 h，垂直平移 k。a 的符号决定开口方向。
- y = (x − 1)² + 2 moves the vertex to (1, 2).
- y = (x − 1)² + 2 将顶点移到 (1, 2)。
- y = −x² reflects the graph across the x-axis.
- y = −x² 将图像关于 x 轴翻转。
8. Solving Quadratic Inequalities | 解二次不等式
To solve ax² + bx + c > 0 or < 0, first find the roots, then sketch the parabola to decide the intervals where the inequality holds.
解 ax² + bx + c > 0 或 < 0，先求根，再画抛物线草图判断满足不等式的区间。
For example, x² − x − 2 < 0 factors as (x − 2)(x + 1) < 0. The roots are 2 and −1. Since the parabola opens upwards, the solution is −1 < x < 2.
例如，x² − x − 2 < 0 因式分解为 (x − 2)(x + 1) < 0。根为 2 和 −1。因为抛物线开口向上，所以解为 −1 < x < 2。
(x − 2)(x + 1) < 0 → −1 < x < 2
9. Quadratic Functions in Real-World Problems | 二次函数在实际问题中的应用
Quadratic functions model projectile motion, area optimization, and profit maximisation. In such problems, the vertex often represents the maximum or minimum value.
二次函数常用来模拟抛体运动、面积优化和利润最大化。在这类问题中，顶点通常代表最大值或最小值。
he area is A = x(50 − x) = 50x − x². The maximum area occurs at x = 25, giving A = 625 m².
例如：一块矩形场地周长为 100 m。若一边长为 x m，则面积 A = x(50 − x) = 50x − x²。最大面积出现在 x = 25，此时 A = 625 m²。
10. Common Exam Mistakes | 常见考试错误
- Forgetting that a quadratic equation can have no real roots when solving with Δ < 0.
- 忘记当 Δ < 0 时二次方程可能没有实根。
- Mistaking the sign of the vertex in the completed square form.
- 在配方式中弄错顶点坐标的符号。
- Drawing a straight line instead of a smooth curve when sketching.
- 绘制草图时画成直线而不是平滑曲线。
11. Practice Questions | 练习题目
1. Solve x² − 4x − 5 = 0 by factorisation.
1. 用因式分解法解 x² − 4x − 5 = 0。
2. Find the coordinates of the turning point of y = 2x² − 8x + 7.
2. 求 y = 2x² − 8x + 7 的顶点坐标。
3. Determine the number of real roots of 3x² + 2x + 5 = 0.
3. 判断 3x² + 2x + 5 = 0 的实根个数。
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