Quadratic Functions: A Complete Guide for IGCSE Mathematics | 二次函数：IGCSE数学完整指南

📚 Quadratic Functions: A Complete Guide for IGCSE Mathematics | 二次函数：IGCSE数学完整指南

Quadratic functions form one of the most important topics in IGCSE Mathematics, appearing across algebra, graphs, and problem-solving questions. This guide covers everything you need to master quadratic expressions, equations, and their graphical representations.

二次函数是IGCSE数学中最重要的主题之一，贯穿代数、图像与解题应用。本指南将帮助你全面掌握二次表达式、二次方程及其图像表示的所有核心知识。

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

A quadratic function is any function that can be written in the standard form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a curve called a parabola.

二次函数是指可以写成标准形式 f(x) = ax² + bx + c 的函数，其中 a、b、c 为常数，且 a ≠ 0。二次函数的图像是一条称为抛物线的曲线。

Key features of a quadratic function:

二次函数的关键特征：

The coefficient a determines the direction of the parabola: if a > 0, the parabola opens upward (U-shape); if a < 0, it opens downward (∩-shape).
系数 a 决定抛物线的开口方向：当 a > 0 时，开口向上（U 形）；当 a < 0 时，开口向下（∩ 形）。
The constant c is the y-intercept, the point where the graph crosses the y-axis.
常数 c 是 y 轴截距，即图像与 y 轴的交点。
The equation y = ax² + bx + c represents a symmetrical curve with exactly one turning point (vertex).
方程 y = ax² + bx + c 表示一条具有唯一顶点（转向点）的对称曲线。

f(x) = ax² + bx + c, a ≠ 0

2. Expanding and Factorising Quadratics | 展开与因式分解二次式

Expanding involves multiplying out brackets. The most important expansions for IGCSE are the three binomial products:

展开是指将括号相乘。IGCSE 中最重要的三种二项式乘积为：

(x + p)(x + q) = x² + (p + q)x + pq
(x + p)² = x² + 2px + p²
(x − p)² = x² − 2px + p²

Factorising is the reverse process: writing a quadratic expression as a product of two binomials. To factorise x² + bx + c, find two numbers that multiply to give c and add to give b.

因式分解是相反的过程：将二次表达式写成两个二项式的乘积。对于 x² + bx + c，需要找到两个数，它们的乘积等于 c，且它们的和等于 b。

x² + bx + c = (x + m)(x + n), where m + n = b and mn = c

For quadratics where the coefficient of x² is not 1 (i.e., ax² + bx + c with a ≠ 1), use the method of decomposition: multiply a and c, find two numbers that multiply to this product and add to b, then split the middle term and factor by grouping.

对于 x² 的系数不为 1 的二次式（即 a ≠ 1 的 ax² + bx + c），使用拆项法：将 a 和 c 相乘，找到两个数使它们的乘积等于这个结果且和为 b，然后拆开中间项并分组提取公因式。

3. Solving Quadratic Equations by Factorising | 用因式分解法解二次方程

To solve a quadratic equation by factorising, follow these steps:

用因式分解法解二次方程，步骤如下：

Write the equation in the form ax² + bx + c = 0.
将方程写成 ax² + bx + c = 0 的形式。
Factorise the left-hand side completely.
将左边完全因式分解。
Set each factor equal to zero and solve for x.
令每个因式等于零，求解 x。

Example: Solve x² − 5x + 6 = 0. Factorise: (x − 2)(x − 3) = 0. Therefore x = 2 or x = 3.

示例：解方程 x² − 5x + 6 = 0。因式分解得：(x − 2)(x − 3) = 0。因此 x = 2 或 x = 3。

If (x − p)(x − q) = 0, then x = p or x = q

This method relies on the zero-product property: if the product of two factors is zero, at least one factor must be zero. Quadratic equations can yield two distinct real roots, one repeated real root, or no real roots.

该方法依赖零积性质：若两个因式的乘积为零，则至少有一个因式为零。二次方程可以有两个不同的实数根、一个重根，或者没有实数根。

4. The Quadratic Formula | 二次公式

When factorisation is not possible or difficult, use the quadratic formula:

当因式分解不可行或较困难时，使用二次公式：

x = (−b ± √(b² − 4ac)) / 2a

This formula solves any quadratic equation ax² + bx + c = 0. Simply identify a, b, and c, substitute into the formula, and simplify.

这个公式可以解决任意二次方程 ax² + bx + c = 0。只需确定 a、b、c 的值，代入公式并化简即可。

Important notes on using the formula:

使用公式时的重要注意事项：

The expression under the square root, b² − 4ac, is called the discriminant.
根号下的表达式 b² − 4ac 称为判别式。
Always simplify the final answer; check if the square root value is a perfect square.
始终化简最终答案；检查根号内的值是否为完全平方数。
When the discriminant is a perfect square, the equation could have been solved by factorising.
当判别式为完全平方数时，原方程也可以用因式分解法求解。

Example: Solve 2x² + 3x − 2 = 0 using the quadratic formula. Here a = 2, b = 3, c = −2.

示例：用二次公式解 2x² + 3x − 2 = 0。此处 a = 2，b = 3，c = −2。

x = (−3 ± √(3² − 4 × 2 × (−2))) / (2 × 2) = (−3 ± √25) / 4 = (−3 ± 5) / 4

Therefore x = 2/4 = 0.5 or x = −8/4 = −2.

因此 x = 2/4 = 0.5 或 x = −8/4 = −2。

5. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + h)² + k. This form reveals the coordinates of the vertex directly: the vertex is (−h, k).

配方法将二次式重写为 a(x + h)² + k 的形式。这种形式直接显示顶点坐标：顶点为 (−h, k)。

Steps for completing the square:

配方法的步骤：

If a ≠ 1, factor a out of the first two terms.
如果 a ≠ 1，先将 a 从前两项中提取出来。
Take half of the coefficient of x, square it, and add and subtract this value inside the bracket.
取 x 系数的一半，平方之，然后在括号内加上并减去这个值。
Simplify to the form a(x + h)² + k.
化简为 a(x + h)² + k 的形式。

Example: Complete the square for x² + 6x + 5.

示例：对 x² + 6x + 5 配方。

x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4

This shows the vertex of y = x² + 6x + 5 is (−3, −4). Completing the square is also used to solve quadratic equations and to find the maximum or minimum values of a function.

这说明 y = x² + 6x + 5 的顶点为 (−3, −4)。配方法也用于解二次方程和求函数的最大值或最小值。

6. The Discriminant and the Nature of Roots | 判别式与根的性质

The discriminant, denoted by Δ = b² − 4ac, tells us the nature of the roots without solving the equation:

判别式，记作 Δ = b² − 4ac，可判断根的性质而无需实际解方程：

Discriminant (Δ)	Nature of Roots
Δ > 0 and a perfect square	Two distinct rational roots
Δ > 0 but not a perfect square	Two distinct irrational roots
Δ = 0	Two equal (repeated) real roots
Δ < 0	No real roots

This table is a critical revision tool. In an exam, if asked to find the range of k for which an equation has real roots, set b² − 4ac ≥ 0 and solve the resulting inequality.

上表是重要的复习工具。如果考试中要求你找出使方程有实数根的 k 的取值范围，则令 b² − 4ac ≥ 0 并解所得不等式。

7. Graphs of Quadratic Functions | 二次函数的图像

The graph of a quadratic function y = ax² + bx + c is a parabola. To sketch it accurately, you need to find:

二次函数 y = ax² + bx + c 的图像是一条抛物线。为了准确画图，你需要找到：

The y-intercept: substitute x = 0; the point is (0, c).
y 轴截距：代入 x = 0；该点为 (0, c)。
The roots: solve ax² + bx + c = 0; the points where the parabola crosses the x-axis.
根（零点）：解 ax² + bx + c = 0；抛物线与 x 轴的交点。
The vertex: use the formula x = −b/(2a) for the x-coordinate, then substitute to get the y-coordinate. Alternatively, use x = (x₁ + x₂)/2 where x₁ and x₂ are the roots.
顶点：用公式 x = −b/(2a) 求顶点的 x 坐标，再代入求 y 坐标。或者用 x = (x₁ + x₂)/2，其中 x₁、x₂ 是两根。
The line of symmetry: x = −b/(2a), a vertical line passing through the vertex.
对称轴：x = −b/(2a)，一条经过顶点的垂直线。

Example: Sketch y = x² − 2x − 3. The y-intercept is (0, −3). Factorising gives (x − 3)(x + 1) = 0, so the roots are x = 3 and x = −1. The vertex occurs at x = (3 + (−1))/2 = 1, giving y = 1 − 2 − 3 = −4. Since a > 0, the parabola opens upward.

示例：画 y = x² − 2x − 3 的图像。y 截距为 (0, −3)。因式分解得 (x − 3)(x + 1) = 0，所以根为 x = 3 和 x = −1。顶点在 x = (3 + (−1))/2 = 1，代入得 y = 1 − 2 − 3 = −4。因为 a > 0，抛物线开口向上。

8. Transformations of Quadratic Graphs | 二次图像的变换

Understanding transformations helps you sketch graphs of related quadratics quickly:

理解图像变换有助于你快速画出相关二次函数的图像：

y = f(x) + k: translates the graph vertically by k units (up if k > 0, down if k < 0).
y = f(x) + k：将图像垂直平移 k 个单位（k > 0 向上，k < 0 向下）。
y = f(x + h): translates the graph horizontally by −h units (left if h > 0, right if h < 0).
y = f(x + h)：将图像水平平移 −h 个单位（h > 0 向左，h < 0 向右）。
y = −f(x): reflects the graph in the x-axis.
y = −f(x)：将图像关于 x 轴反射。
y = f(−x): reflects the graph in the y-axis.
y = f(−x)：将图像关于 y 轴反射。
y = af(x): stretches the graph vertically by a factor of a.
y = af(x)：将图像按纵向伸缩，伸缩因子为 a。

For the parabola y = (x + 2)² − 3, the graph of y = x² is shifted left by 2 units and down by 3 units. In vertex form y = (x − h)² + k, the vertex is directly read as (h, k).

对于抛物线 y = (x + 2)² − 3，y = x² 的图像向左平移 2 个单位，再向下平移 3 个单位。在顶点式 y = (x − h)² + k 中，顶点可直接读出为 (h, k)。

9. Quadratic Inequalities | 二次不等式

Solving quadratic inequalities requires sketching the parabola or identifying critical values on a number line:

解二次不等式需要画出抛物线或借助数轴确定临界值：

Step 1: Rearrange the inequality so one side is zero. Step 2: Factorise the quadratic expression. Step 3: Find the roots. Step 4: Sketch the parabola or test intervals to determine the solution set.

第一步：整理不等式，使一侧为零。第二步：因式分解二次表达式。第三步：求根。第四步：画出抛物线或检验区间来确定解集。

Example: Solve x² − 4 < 0. Factorise: (x − 2)(x + 2) < 0. The roots are −2 and 2. The parabola opens upward, so the graph is below the x-axis between the roots: −2 < x < 2.

示例：解 x² − 4 < 0。因式分解得 (x − 2)(x + 2) < 0。根为 −2 和 2。抛物线开口向上，图像在两根之间位于 x 轴下方：−2 < x < 2。

If the inequality is ≥ 0 or ≤ 0, include the endpoints by using closed inequalities. Remember: for “less than” the solution is usually between the roots; for “greater than” the solution is outside the roots.

若不等式为 ≥ 0 或 ≤ 0，则使用闭合不等式包含端点。记住：对于”小于”类不等式，解通常在两根之间；对于”大于”类不等式，解在两根之外。

10. Applications: Maximum and Minimum Problems | 应用：最大值与最小值问题

Quadratic functions are used to model real-world scenarios such as projectile motion, area optimisation, and profit calculations. The vertex represents the maximum or minimum point, which is often the key answer.

二次函数用于建模现实场景，例如抛体运动、面积优化和利润计算。顶点表示最大值或最小值点，这往往是题目的关键答案。

For a quadratic in vertex form y = a(x − h)² + k:

对于顶点式 y = a(x − h)² + k 的二次函数：

If a > 0, the minimum value of y is k, occurring at x = h.
若 a > 0，y 的最小值为 k，在 x = h 处取得。
If a < 0, the maximum value of y is k, occurring at x = h.
若 a < 0，y 的最大值为 k，在 x = h 处取得。

Example: A rectangle is to be made with a perimeter of 40 m. Find the maximum area. Let one side be x m. The other side is 20 − x m. The area is A = x(20 − x) = −x² + 20x. Completing the square: A = −(x − 10)² + 100. The maximum area is 100 m² when x = 10 m (a square).

示例：用 40 m 的周长围成一个矩形，求最大面积。设一边为 x m，则另一边为 20 − x m。面积为 A = x(20 − x) = −x² + 20x。配方得：A = −(x − 10)² + 100。最大面积为 100 m²，此时 x = 10 m（正方形）。

Maximum/Minimum value = k, achieved at x = h

11. Relationship Between Roots and Coefficients | 根与系数的关系

For a quadratic equation ax² + bx + c = 0 with roots α and β:

对于根为 α 和 β 的二次方程 ax² + bx + c = 0：

Sum of roots: α + β = −b/a

Product of roots: α × β = c/a

These relationships allow you to find new quadratics given the sum and product of their roots. If the sum is S and the product is P, the equation is x² − Sx + P = 0.

这些关系允许你在已知根的和与积时构造新的二次方程。若和为 S、积为 P，则方程为 x² − Sx + P = 0。

Example: Find a quadratic equation with roots 3 and −2. The sum is 3 + (−2) = 1, and the product is 3 × (−2) = −6. The equation is x² − x − 6 = 0.

示例：求根为 3 和 −2 的二次方程。和为 3 + (−2) = 1，积为 3 × (−2) = −6。因此方程为 x² − x − 6 = 0。

These relationships are particularly useful in questions that give one root and ask for the other, or that require symmetric expressions in α and β such as α² + β² = (α + β)² − 2αβ.

这些关系特别适用于已知一个根求另一个根的问题，或需要求 α 和 β 的对称表达式（如 α² + β² = (α + β)² − 2αβ）的题目。

12. Exam Strategies and Common Mistakes | 考试策略与常见错误

Avoid these common pitfalls in quadratic questions:

在二次函数相关题目中避免以下常见错误：

Forgetting a ≠ 0: if a = 0, the equation is linear, not quadratic.
忘记 a ≠ 0：若 a = 0，方程是线性的，而非二次方程。
Sign errors in the quadratic formula: carefully handle negative values for b and c.
二次公式中的符号错误：仔细处理 b 和 c 的负值。
Losing a root: when solving (x − 2)² = 9, remember that x − 2 = ±3, giving two solutions.
丢根：解 (x − 2)² = 9 时，记住 x − 2 = ±3，得到两个解。
Incorrect vertex coordinates: the x-coordinate of the vertex is exactly halfway between the two roots.
顶点坐标错误：顶点的 x 坐标正好位于两个根的正中间。
Misreading “no real roots”: this means the discriminant is negative, and the parabola never touches the x-axis.
误读”无实数根”：意味着判别式为负，抛物线与 x 轴不相交。

In IGCSE exams, always write down the substitution step when using the quadratic formula. Even if your final answer is wrong, the method mark is awarded for correct substitution.

在 IGCSE 考试中，使用二次公式时始终写出代入步骤。即使最终答案错误，只要代入正确，也能获得步骤分。

Practise recognising which method to use: factorise if the roots are rational, complete the square if the vertex is needed, and use the quadratic formula when factorisation is hard or the question explicitly asks for it.

练习判断应使用哪种方法：根为有理数时因式分解；需要求顶点时用配方法；因式分解困难或题目明确要求时使用二次公式。

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Quadratic Functions: A Complete Guide for IGCSE Mathematics | 二次函数：IGCSE数学完整指南