📚 IGCSE Mathematics: Quadratic Functions and Their Graphs | IGCSE 数学：二次函数及其图像
Quadratic functions are central to the IGCSE Mathematics syllabus, linking algebra, graphs, and real-life problem solving. This revision guide walks through the key ideas, methods, and common pitfalls you need for the exam.
二次函数是 IGCSE 数学课程的核心内容，它将代数、图像与现实问题求解紧密联系起来。本复习指南将带你梳理考试所需的关键概念、方法和常见易错点。
1. What is a Quadratic Function? | 什么是二次函数？
≠ 0 is essential because if a = 0, the x² term disappears and the function becomes linear.
二次函数的一般形式为 y = ax² + bx + c，其中 a、b、c 为常数且 a ≠ 0。a ≠ 0 这一条件至关重要，因为如果 a = 0，x² 项就会消失，函数就变成一次函数。
ic curved shape called a parabola. Examples include y = x² - 4x + 3 and y = -2x² + 5x - 1.
x 的最高次数为 2，正是这一点使图像具有特有的曲线形状，称为抛物线。例如 y = x² - 4x + 3 和 y = -2x² + 5x - 1 都是二次函数。
y = ax² + bx + c, a ≠ 0
2. Standard Form and Coefficients | 标准式与系数
h coefficient has a clear role. The coefficient a controls the direction and width of the parabola: a positive a gives an upward opening curve, while a negative a gives a downward opening curve.
在标准式 y = ax² + bx + c 中，每个系数都有明确的作用。系数 a 控制抛物线的开口方向和宽窄：a 为正时曲线开口向上，a 为负时曲线开口向下。
ifts the curve left or right. The constant c is the y-intercept because when x = 0, y = c.
系数 b 影响对称轴的位置和顶点的横坐标。它不会改变图像形状，但会使曲线左右移动。常数 c 是 y 轴截距，因为当 x = 0 时，y = c。
It is useful to identify a, b and c quickly before using any formula, as most calculation mistakes come from copying the wrong coefficients.
在使用任何公式之前，先快速确定 a、b、c 的值会很有帮助，因为大多数计算错误都来自系数抄写错误。
3. The Shape of a Quadratic Graph | 二次函数图像的形状
The graph of every quadratic function is a parabola. If a > 0, the parabola opens upwards and has a minimum point at the vertex. If a < 0, the parabola opens downwards and has a maximum point at the vertex.
每个二次函数的图像都是一条抛物线。如果 a > 0，抛物线开口向上，顶点为最低点；如果 a < 0，抛物线开口向下，顶点为最高点。
The curve is symmetric about a vertical line called the axis of symmetry. This line passes through the vertex, so once you know the vertex, you also know the axis of symmetry.
曲线关于一条称为对称轴的竖直直线对称。这条直线经过顶点，因此一旦知道了顶点，也就知道了对称轴。
wo points. These crossing points are the roots of the quadratic equation ax² + bx + c = 0.
抛物线可能与 x 轴有零个、一个或两个交点。这些交点就是二次方程 ax² + bx + c = 0 的根。
4. Finding Key Features: Vertex and Axis of Symmetry | 求关键特征：顶点与对称轴
÷ (2a). To find the y-coordinate, substitute this x value back into the original function.
对于标准式二次函数，顶点的横坐标可用公式 x = -b ÷ (2a) 求出。要得到纵坐标，只需将这个 x 值代回原函数即可。
x = -b ÷ (2a)
the vertex is exactly (h, k). This form is very efficient when you need to state the vertex directly.
另一种方法是通过配方法得到形式 y = a(x - h)² + k，此时顶点恰好为 (h, k)。当你需要直接写出顶点坐标时，这种形式非常高效。
y = a(x - h)² + k, vertex = (h, k)
+ 3 becomes y = (x - 2)² - 1, so the vertex is (2, -1) and the axis of symmetry is x = 2.
例如，通过配方，y = x² - 4x + 3 变为 y = (x - 2)² - 1，所以顶点为 (2, -1)，对称轴为 x = 2。
5. Roots and x-intercepts | 根与 x 轴截距
ns are the roots of the equation and the x-coordinates where the curve crosses the x-axis.
要求二次图像与 x 轴的交点，可令 y = 0 并解方程 ax² + bx + c = 0。这些解就是方程的根，也是曲线与 x 轴交点的横坐标。
ample, x² - 5x + 6 = 0 can be written as (x - 2)(x - 3) = 0, giving roots x = 2 and x = 3.
如果二次式可以进行因式分解，就很容易找到根。例如，x² - 5x + 6 = 0 可写为 (x - 2)(x - 3) = 0，从而得到根 x = 2 和 x = 3。
(x - 2)(x - 3) = 0, so x = 2 or x = 3
h factorises to (x - 3)² = 0. In this case the graph touches the x-axis at one point only.
有时二次方程有重根，例如 x² - 6x + 9 = 0，可因式分解为 (x - 3)² = 0。此时图像只在一点与 x 轴相切。
6. The Discriminant and Nature of Roots | 判别式与根的性质
= b² - 4ac. It tells you how many real roots the equation has without actually solving it.
二次方程 ax² + bx + c = 0 的判别式为 Δ = b² - 4ac。它可以在不实际求解的情况下告诉你方程有多少个实根。
Δ = b² - 4ac
epeated real root and the graph touches the x-axis at one point. If Δ < 0, there are no real roots and the graph does not cross the x-axis.
如果 Δ > 0，方程有两个不同的实根，图像与 x 轴相交两次。如果 Δ = 0，有一个重根，图像与 x 轴相切于一点。如果 Δ < 0，则没有实根，图像不与 x 轴相交。
Examiners often ask you to use the discriminant to show that a curve is always above the x-axis, so you should be confident interpreting the three cases.
考官经常要求利用判别式证明曲线始终位于 x 轴上方，因此你应当熟练掌握这三种情况的解释。
7. Sketching Quadratic Graphs | 绘制二次图像
find the vertex using x = -b ÷ (2a). Finally, draw a smooth parabola through these points.
要准确地绘制二次图像，请按照清晰的步骤进行。首先，代入 x = 0 求出 y 轴截距。其次，如果可能，解方程 ax² + bx + c = 0 求出根。第三，用 x = -b ÷ (2a) 求顶点。最后，通过这些点画出一条平滑的抛物线。
1 and x = 3, and the vertex is (2, -1). The curve opens upwards because a = 1 is positive.
例如，对于 y = x² - 4x + 3，y 轴截距为 3，根为 x = 1 和 x = 3，顶点为 (2, -1)。由于 a = 1 为正，曲线开口向上。
Always label the key points on your sketch and use a symmetrical shape. A sketch does not need to be perfectly to scale, but it must show the correct shape and positions of intercepts and vertex.
在草图上始终标出关键点，并使用对称形状。草图不要求完全按比例，但必须正确表示截距和顶点的位置及形状。
8. Solving Quadratic Equations | 解二次方程
There are three main methods for solving quadratic equations: factorising, using the quadratic formula, and completing the square. Choose the method that is quickest for the given equation.
解二次方程主要有三种方法：因式分解法、使用二次公式法和配方法。应选择对给定方程最快捷的方法。
The quadratic formula works for any quadratic equation and is especially useful when the equation does not factorise easily. The formula is:
二次公式适用于任何二次方程，尤其是当方程不容易因式分解时。公式为：
x = (-b ± √b² - 4ac) ÷ (2a)
When using the formula, write down the values of a, b and c first, then substitute them carefully. Pay attention to negative signs, because b is often negative in IGCSE questions.
使用公式时，先写下 a、b、c 的值，然后仔细代入。要特别注意负号，因为在 IGCSE 题目中 b 常常为负数。
entify a = 2, b = 3, c = -5, then substitute into the formula to obtain the two solutions.
例如，解方程 2x² + 3x - 5 = 0 时，确定 a = 2、b = 3、c = -5，然后代入公式得到两个解。
9. Quadratic Inequalities | 二次不等式
e roots of the corresponding equation x² - 4x + 3 = 0. Here the roots are x = 1 and x = 3.
要解二次不等式，例如 x² - 4x + 3 < 0，先求相应方程 x² - 4x + 3 = 0 的根。此处根为 x = 1 和 x = 3。
is positive, the graph opens upwards and lies below the x-axis between the roots. Therefore the solution to x² - 4x + 3 < 0 is 1 < x < 3.
接下来，画出抛物线的草图。由于 a = 1 为正，图像开口向上，且在两根之间位于 x 轴下方。因此 x² - 4x + 3 < 0 的解为 1 < x < 3。
1 < x < 3
If the inequality were x² - 4x + 3 > 0, the solution would be x < 1 or x > 3, because the graph is above the x-axis outside the roots.
如果不等式是 x² - 4x + 3 > 0，解则为 x < 1 或 x > 3，因为图像在两根之外位于 x 轴上方。
10. Applications and Common Mistakes | 应用与常见错误
Quadratic functions model many real-life situations, including projectile motion, area problems, and maximum profit calculations. In these questions, the vertex often gives the maximum or minimum value required.
二次函数可以模拟许多现实情境，包括抛体运动、面积问题和最大利润计算。在这些题目中，顶点往往给出所需的最大值或最小值。
A common mistake is forgetting that a must not equal zero. Another is copying b or c with the wrong sign when substituting into the quadratic formula. Always check the discriminant before trying to take a square root.
一个常见错误是忘记 a 不能为零。另一个是在代入二次公式时把 b 或 c 的符号抄错。在求平方根之前，一定要先检查判别式。
Students also lose marks by rounding too early in calculator questions. Keep exact values until the final step, and state the answer to the required degree of accuracy.
学生在计算器题目中还常因过早四舍五入而失分。应保留精确值到最后一步，并按照要求的精确度写出答案。
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