📚 Quadratic Functions & Graphs | 二次函数与图像
Quadratic functions are one of the most important topics in the IGCSE Mathematics syllabus. A quadratic function is a polynomial of degree 2, typically written in the form f(x) = ax² + bx + c, where a ≠ 0. Its graph is a smooth, symmetric curve called a parabola.
二次函数是 IGCSE 数学大纲中最重要的考点之一。二次函数是最高次数为 2 的多项式，通常写作 f(x) = ax² + bx + c 的形式，其中 a ≠ 0。它的图像是一条光滑且对称的曲线，称为抛物线。
1. The Standard Form | 标准形式
The standard form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants. The coefficient a determines the direction of the parabola: if a > 0, the parabola opens upward; if a < 0, it opens downward. The constant c represents the y-intercept, where the graph crosses the y-axis.
二次函数的标准形式为 f(x) = ax² + bx + c，其中 a、b、c 为常数。系数 a 决定抛物线的开口方向：当 a > 0 时，抛物线开口向上；当 a < 0 时，抛物线开口向下。常数 c 表示 y 轴截距，即图像与 y 轴的交点。
- If a > 0: minimum point exists | 若 a > 0：存在最小值点
- If a < 0: maximum point exists | 若 a < 0：存在最大值点
- The larger |a| is, the narrower the parabola | |a| 越大，抛物线越窄
2. The Vertex Form | 顶点式
The vertex form of a quadratic function is f(x) = a(x – h)² + k. In this form, the vertex (turning point) of the parabola is located at the point (h, k). This form is particularly useful because it allows us to read the coordinates of the vertex directly without further calculation.
二次函数的顶点式为 f(x) = a(x – h)² + k。在这种形式下，抛物线的顶点（转向点）位于点 (h, k)。这种形式非常实用，因为我们可以直接读出顶点的坐标，无需进一步计算。
Vertex = (h, k) | 顶点 = (h, k)
To convert from standard form to vertex form, we use the method of completing the square. This involves rewriting ax² + bx + c as a(x – h)² + k.
要将标准形式转换为顶点式，我们使用配方法。即将 ax² + bx + c 改写为 a(x – h)² + k 的形式。
3. Completing the Square | 配方法
Completing the square is a systematic technique used to rewrite a quadratic expression. Given f(x) = ax² + bx + c, we first factor out a from the first two terms, then add and subtract (b/2a)² inside the bracket.
配方法是一种系统化的技巧，用于改写二次表达式。给定 f(x) = ax² + bx + c，我们首先从前两项中提取 a，然后在括号内加上并减去 (b/2a)²。
x² + bx = (x + b/2)² − (b/2)²
For example, to complete the square for x² + 6x + 5, we write x² + 6x = (x + 3)² − 9, so the original expression becomes (x + 3)² − 4.
例如，对 x² + 6x + 5 配方时，我们写 x² + 6x = (x + 3)² − 9，因此原表达式变为 (x + 3)² − 4。
4. The Axis of Symmetry | 对称轴
Every parabola is symmetric about a vertical line called the axis of symmetry. For a quadratic function in standard form f(x) = ax² + bx + c, the axis of symmetry is given by the equation x = −b/(2a). This line passes through the vertex, dividing the parabola into two mirror-image halves.
每条抛物线都关于一条称为对称轴的垂直线对称。对于标准形式的二次函数 f(x) = ax² + bx + c，对称轴的方程为 x = −b/(2a)。这条线穿过顶点，将抛物线分成两个镜像对称的部分。
x = −b/(2a)
In vertex form, the axis of symmetry is simply x = h. Understanding the axis of symmetry helps us sketch the graph accurately and locate the vertex.
在顶点式中，对称轴就是 x = h。理解对称轴有助于我们准确绘制图像并找到顶点。
5. The Discriminant and Roots | 判别式与根
The discriminant, denoted Δ = b² − 4ac, determines the nature of the roots of the quadratic equation ax² + bx + c = 0. This value tells us whether the equation has real or complex roots, and how many distinct real roots exist.
判别式，记作 Δ = b² − 4ac，决定了二次方程 ax² + bx + c = 0 的根的性质。这个值告诉我们方程是否有实根，以及存在多少个不同的实根。
| Discriminant | 判别式 | Nature of Roots | 根的性质 |
| Δ > 0 | Two distinct real roots | 两个不同的实根 |
| Δ = 0 | One repeated real root | 一个重根（两个相等实根） |
| Δ < 0 | No real roots | 无实根 |
The roots can be found using the quadratic formula x = (−b ± √Δ)/(2a). This formula works for all quadratic equations and is derived from completing the square.
根可以通过求根公式 x = (−b ± √Δ)/(2a) 求得。这个公式适用于所有二次方程，由配方法推导得出。
6. Solving Quadratic Equations | 解二次方程
There are three main methods for solving quadratic equations: factorisation, using the quadratic formula, and completing the square. Factorisation is often the quickest method, but it only works when the quadratic can be factorised over the rational numbers.
解二次方程有三种主要方法：因式分解、使用求根公式和配方法。因式分解通常是最快的方法，但仅在二次式能够在有理数范围内因式分解时才适用。
- Factorisation: find two numbers whose product equals ac and sum equals b | 因式分解：寻找两个数，其积等于 ac，和等于 b
- Quadratic formula: x = (−b ± √(b² − 4ac))/(2a) | 求根公式：x = (−b ± √(b² − 4ac))/(2a)
- Completing the square: rewrite and solve for x | 配方法：改写后求解 x
x = (−b ± √(b² − 4ac)) / (2a)
When solving by factorisation, we use the zero product property: if AB = 0, then A = 0 or B = 0. This is a fundamental principle in algebra.
用因式分解求解时，我们用零积性质：若 AB = 0，则 A = 0 或 B = 0。这是代数中的基本原理。
7. Sketching Quadratic Graphs | 绘制二次函数图像
To sketch a quadratic graph accurately, we need to identify four key features: the y-intercept (when x = 0), the roots (when y = 0), the vertex, and the axis of symmetry. The y-intercept is simply the constant term c in standard form.
要准确绘制二次函数图像，我们需要确定四个关键特征：y 轴截距（x = 0 时）、根（y = 0 时）、顶点以及对称轴。y 轴截距就是标准形式中的常数项 c。
A typical sketching procedure is as follows: first, find the y-intercept by evaluating f(0). Second, find the roots by solving f(x) = 0. Third, find the vertex using the formula x = −b/(2a), then substitute back to find y. Finally, plot these points and draw a smooth curve through them.
典型的作图步骤如下：首先，通过计算 f(0) 求 y 轴截距。其次，通过解 f(x) = 0 求根。第三，用公式 x = −b/(2a) 求顶点的 x 坐标，再代回求 y。最后，标出这些点并画一条光滑曲线连接它们。
8. Word Problems Involving Quadratics | 二次函数应用问题
Quadratics are widely used in real-world contexts, such as projectile motion, area optimisation, and profit maximisation. In these problems, we typically form a quadratic equation from the given information and then solve it to answer the question.
二次函数广泛用于现实情境，如抛体运动、面积优化和利润最大化。在这些问题中，我们通常根据已知信息建立一个二次方程，然后解方程来回答问题。
For example, if a rectangle has a perimeter of 20 m and an area of 21 m², we can set up the equation x(10 − x) = 21, which simplifies to x² − 10x + 21 = 0. Solving this gives x = 3 or x = 7, meaning the rectangle is 3 m by 7 m.
例如，若一个矩形的周长为 20 m，面积为 21 m²，我们可以建立方程 x(10 − x) = 21，化简为 x² − 10x + 21 = 0。解方程得 x = 3 或 x = 7，即矩形为 3 m × 7 m。
When solving word problems, always check whether both solutions make sense in the context of the problem. Sometimes only one solution is valid; for instance, a length cannot be negative.
在解决应用问题时，务必检查两个解在问题情境下是否都有意义。有时只有一个解有效；例如，长度不能为负数。
9. The Sum and Product of Roots | 根的和与积
For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum of the roots is α + β = −b/a and the product of the roots is α × β = c/a. These relationships allow us to write a quadratic equation given its roots.
对于二次方程 ax² + bx + c = 0，若其根为 α 和 β，则两根之和为 α + β = −b/a，两根之积为 α × β = c/a。这些关系让我们能够在已知根的情况下写出二次方程。
α + β = −b/a, α × β = c/a
A quadratic equation with roots α and β can be written as x² − (α + β)x + α × β = 0. This is a useful technique for constructing equations in reverse engineering problems.
以 α 和 β 为根的二次方程可以写作 x² − (α + β)x + α × β = 0。这是一个在逆向问题中构造方程的有用技巧。
10. Quadratic Inequalities | 二次不等式
Quadratic inequalities involve expressions like ax² + bx + c > 0 or ax² + bx + c ≤ 0. To solve them, we first find the roots of the corresponding quadratic equation, then use a sign diagram or test points to determine which intervals satisfy the inequality.
二次不等式涉及形如 ax² + bx + c > 0 或 ax² + bx + c ≤ 0 的表达式。解这类不等式时，我们首先求出对应二次方程的根，然后使用符号图或测试点来确定哪些区间满足不等式。
For example, to solve x² − 5x + 6 > 0, we factor to get (x − 2)(x − 3) > 0. The roots are x = 2 and x = 3. Testing intervals: x < 2 gives positive, 2 < x < 3 gives negative, x > 3 gives positive. Therefore, the solution is x < 2 or x > 3.
例如，解 x² − 5x + 6 > 0，因式分解得 (x − 2)(x − 3) > 0。根为 x = 2 和 x = 3。测试区间：x < 2 时为正，2 < x < 3 时为负，x > 3 时为正。因此，解为 x < 2 或 x > 3。
When the coefficient a is negative, the inequality sign reverses in the solution. Always sketch a quick graph if you are unsure.
当系数 a 为负数时，解中的不等号方向会反转。如果不确定，建议快速画一个草图辅助判断。
11. Graphs of Quadratic Functions and Transformations | 二次函数图像的变换
Quadratic graphs can be transformed through translations, reflections, and stretches. Replacing x with (x − h) shifts the graph h units to the right; adding k to the function shifts it k units upward. A negative sign before the whole function reflects the graph across the x-axis.
二次函数图像可以通过平移、翻折和伸缩进行变换。用 (x − h) 替换 x 使图像向右平移 h 个单位；在函数后加上 k 使其向上平移 k 个单位。在函数前加负号则使其关于 x 轴翻折。
- y = (x − h)²: shift right by h | y = (x − h)²：向右平移 h
- y = x² + k: shift up by k | y = x² + k：向上平移 k
- y = −f(x): reflection in x-axis | y = −f(x)：关于 x 轴翻折
- y = f(−x): reflection in y-axis | y = f(−x)：关于 y 轴翻折
These transformations help us sketch more complex quadratic curves and understand the relationships between different forms of quadratic functions.
这些变换帮助我们绘制更复杂的二次曲线，并理解不同形式二次函数之间的关系。
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