Quadratic Functions: From Factoring to Applications | 二次函数：从因式分解到应用

📚 Quadratic Functions: From Factoring to Applications | 二次函数：从因式分解到应用

Quadratic functions form one of the most important topics in IGCSE Mathematics. They appear in algebra, coordinate geometry, and real-world problem solving. This article provides a complete revision guide covering standard form, factorisation, the quadratic formula, the discriminant, graph sketching, and applied problems.

二次函数是 IGCSE 数学中最重要的内容之一。它出现在代数、坐标几何以及真实世界的问题求解中。本文将提供一份完整的复习指南，涵盖标准形式、因式分解、求根公式、判别式、图像草图以及应用题。

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

A quadratic expression in one variable has the general form ax² + bx + c, where a, b, and c are constants and a ≠ 0. The highest power of x is 2, which is what distinguishes a quadratic from a linear equation.

一元二次表达式的一般形式为 ax² + bx + c，其中 a、b、c 是常数，且 a ≠ 0。x 的最高次数为 2，这正是二次式与一次式的区别所在。

In standard form, the coefficient a controls the width and direction of the parabola, b affects the position of the axis of symmetry, and c represents the y-intercept. If a > 0, the parabola opens upward; if a < 0, it opens downward.

在标准形式中，系数 a 控制抛物线的宽度和开口方向，b 影响对称轴的位置，c 表示 y 轴截距。若 a > 0，抛物线开口向上；若 a < 0，抛物线开口向下。

f(x) = ax² + bx + c, where a ≠ 0

f(x) = ax² + bx + c，其中 a ≠ 0

2. Solving by Factorisation | 因式分解法求解

Factorisation is a technique used to express a quadratic expression as the product of two linear factors. To solve ax² + bx + c = 0 by factorisation, we first factorise the left-hand side and then set each factor equal to zero.

因式分解是将二次表达式表示为两个一次因式乘积的方法。要用因式分解法解形如 ax² + bx + c = 0 的方程，我们先将左边分解因式，再令每个因式分别等于零。

For simple cases where a = 1, we look for two numbers whose sum is b and whose product is c. For example, x² − 5x + 6 = 0 can be written as (x − 2)(x − 3) = 0, giving x = 2 or x = 3.

对于 a = 1 的简单情况，我们寻找两个数的和为 b、积为 c。例如，x² − 5x + 6 = 0 可写成 (x − 2)(x − 3) = 0，得到 x = 2 或 x = 3。

When a > 1, an effective method is the “cross method” or “grouping method”. Multiply a by c, factorise this product, and rewrite the middle term as the sum of two terms before grouping.

当 a > 1 时，用十字相乘法或分组分解法比较有效。将 a 乘以 c，分解这个乘积，把中间项写成两项之和，再分组分解。

Factorisation only works when the equation has rational roots. If the roots are irrational or complex, another method is required.

因式分解法仅适用于方程有有理数根的情况。如果根是无理数或复数，就需要使用其他方法。

3. Completing the Square | 配方法

Completing the square is a powerful algebraic technique that rewrites a quadratic in the form a(x + p)² + q. This form explicitly shows the vertex of the parabola and is essential for solving equations that do not factorise easily.

配方法是将二次式改写为 a(x + p)² + q 的代数技巧。该形式直接显示抛物线的顶点，也用于求解不易因式分解的方程。

To complete the square, take half the coefficient of x, square it, and adjust the constant term. For x² + 6x + 5 = 0, we write (x + 3)² − 4 = 0, since (x + 3)² = x² + 6x + 9.

配方的步骤是：取 x 系数的一半、平方，然后调整常数项。对于 x² + 6x + 5 = 0，可写成 (x + 3)² − 4 = 0，因为 (x + 3)² = x² + 6x + 9。

x² + bx + c = (x + b⁄2)² − (b⁄2)² + c

Solving (x + 3)² = 4 gives x + 3 = ±2, so x = −1 or x = −5. Note that when completing the square, do not forget to include the ± sign when taking the square root.

解 (x + 3)² = 4，得 x + 3 = ±2，因此 x = −1 或 x = −5。注意在使用配方法时，开平方根不要漏掉 ± 号。

4. The Quadratic Formula | 求根公式

The quadratic formula is a universal method for solving any quadratic equation of the form ax² + bx + c = 0. It works for rational, irrational, and even complex roots.

求根公式是解任意形式 ax² + bx + c = 0 的通用方法。它适用于有理数根、无理数根，甚至复数根。

x = (−b ± √(b² − 4ac)) ÷ 2a

To use this formula, substitute the values of a, b, and c directly. Before substituting, ensure the equation has been rearranged into standard form with all terms on one side equal to zero.

使用该公式时，直接将 a、b、c 的值代入。代入前，请先将方程化为标准形式——所有项移到一边且等于零。

For example, solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, and c = −3. Substitution yields x = (4 ± √(16 + 24)) ÷ 4 = (4 ± √40) ÷ 4, which simplifies to x = (2 ± √10) ÷ 2.

例如，解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。代入得 x = (4 ± √(16 + 24)) ÷ 4 = (4 ± √40) ÷ 4，化简为 x = (2 ± √10) ÷ 2。

Candidates often forget to divide both terms in the numerator by 2a. Write out each step carefully to avoid algebraic slips.

考生经常忘记将分子中的两项同时除以 2a。写出每一步过程可避免代数错误。

5. The Discriminant | 判别式

The discriminant, denoted by Δ = b² − 4ac, determines the nature of the roots of a quadratic equation without solving it completely.

判别式用 Δ = b² − 4ac 表示，可以在不完全求解的情况下判断二次方程根的性质。

If Δ > 0, the equation has two distinct real roots. If Δ = 0, there is exactly one repeated real root. If Δ < 0, there are no real roots, but two complex conjugate roots.

若 Δ > 0，方程有两个不相等的实数根；若 Δ = 0，有一个二重实数根；若 Δ < 0，没有实数根，但有两个共轭复数根。

Geometrically, Δ > 0 means the parabola intersects the x-axis at two points; Δ = 0 means it touches the x-axis at one point; and Δ < 0 means it does not meet the x-axis at all.

从几何上看，Δ > 0 表示抛物线与 x 轴有两个交点；Δ = 0 表示与 x 轴相切于一点；Δ < 0 表示与 x 轴没有交点。

Consider the equation x² − kx + 4 = 0. For the equation to have two equal roots, the discriminant must be zero: k² − 16 = 0, so k = ±4.

考虑方程 x² − kx + 4 = 0。若方程有等根，则判别式必须为零：k² − 16 = 0，因此 k = ±4。

6. Vertex and Axis of Symmetry | 顶点与对称轴

The vertex is the turning point of the parabola. For a quadratic y = ax² + bx + c, the x-coordinate of the vertex is given by x = −b⁄2a. The y-coordinate is found by substituting this x-value back into the original equation.

顶点是抛物线的转折点。对于二次函数 y = ax² + bx + c，顶点的 x 坐标为 x = −b⁄2a。将得到的 x 值代回原方程，即可求出 y 坐标。

The axis of symmetry is the vertical line x = −b⁄2a, which passes through the vertex and divides the parabola into two mirror-image halves.

对称轴是经过顶点的竖直直线 x = −b⁄2a，它将抛物线分成左右对称的两半。

For example, in y = x² + 4x + 1, the axis of symmetry is x = −4 ÷ 2 = −2. Substituting x = −2 gives y = 4 − 8 + 1 = −3, so the vertex is (−2, −3).

例如，在 y = x² + 4x + 1 中，对称轴为 x = −4 ÷ 2 = −2。代入 x = −2 得 y = 4 − 8 + 1 = −3，因此顶点为 (−2, −3)。

When the quadratic is already in the form y = a(x + p)² + q, the vertex is simply (−p, q). When a > 0, the vertex is the minimum point; when a < 0, it is the maximum point.

当二次函数已写成 y = a(x + p)² + q 的形式时，顶点坐标直接为 (−p, q)。当 a > 0 时，顶点是最低点；当 a < 0 时，顶点是最高点。

7. Sketching Quadratic Graphs | 画二次函数图像

To sketch the graph of y = ax² + bx + c, you need three key features: the y-intercept, the x-intercepts (if any), and the vertex. Start by setting x = 0 to find the y-intercept, then solve the equation ax² + bx + c = 0 to find the x-intercepts.

画 y = ax² + bx + c 的图像需要三个关键特征：y 轴截距、x 轴截距（如果有的话）和顶点。先令 x = 0 求 y 轴截距，再解方程 ax² + bx + c = 0 求 x 轴截距。

If the discriminant is negative, the curve does not cross the x-axis, so instead use a table of values to plot additional points around the vertex.

如果判别式为负，曲线不与 x 轴相交，这时需要用数值表在顶点附近取点来绘制图像。

Mark the axis of symmetry, then reflect points to obtain the other half of the curve. Join the points with a smooth curve, never straight lines.

标出对称轴，然后利用对称性取另一半的点。用平滑曲线连接这些点，切勿使用直线段连接。

Always check the sign of a to verify that the parabola opens in the correct direction. A curve opening the wrong way is a common mistake even for strong students.

始终检查 a 的符号以确认抛物线开口方向正确。开口方向画反是即使优秀学生也会犯的常见错误。

8. Transformations of Graphs | 图像变换

Understanding graph transformations helps you sketch quadratics quickly and interpret equations in context. The graph of y = x² can be shifted or stretched to form any quadratic.

理解图像变换有助于快速绘制二次函数图像，并在具体情境中理解方程的含义。y = x² 的图像通过平移或伸缩，可以形成任何二次函数。

Replacing x with (x − h) shifts the graph h units to the right if h > 0, and to the left if h < 0. Adding a constant k to the whole expression shifts the graph k units in the vertical direction.

将 x 替换为 (x − h)，当 h > 0 时图像右移 h 个单位，h < 0 时左移。在整个表达式后加常数 k，图像在竖直方向移动 k 个单位。

Multiplying the whole expression by a factor a stretches or compresses the graph vertically. If a is negative, the graph is reflected in the x-axis.

将整个表达式乘以因子 a，图像在竖直方向拉伸或压缩。若 a 为负数，图像关于 x 轴对称翻转。

For example, y = (x − 2)² − 3 is the graph of y = x² shifted 2 units right and 3 units down. Its vertex is (2, −3).

例如，y = (x − 2)² − 3 是 y = x² 向右平移 2 个单位、向下平移 3 个单位得到的图像，其顶点为 (2, −3)。

9. Applications Involving Quadratics | 二次函数的实际应用

Quadratic equations are widely used in real-life problems. The most common applications include area problems, projectile motion, and profit optimisation.

二次方程在现实生活问题中应用广泛。最常见的包括面积问题、抛体运动和利润优化问题。

Consider a rectangular field with perimeter 40 m. If the length is x m, the width is (20 − x) m. The area A = x(20 − x) = 20x − x². To maximise the area, find the vertex of this quadratic: x = −20 ÷ (2 × (−1)) = 10 m. The maximum area is 20 × 10 − 100 = 100 m².

考虑一个周长为 40 m 的矩形场地。若长为 x m，则宽为 (20 − x) m。面积 A = x(20 − x) = 20x − x²。要求最大面积，找这个二次函数的顶点：x = −20 ÷ (2 × (−1)) = 10 m。最大面积为 20 × 10 − 100 = 100 m²。

For projectile motion, the height h of an object after t seconds often follows h(t) = −5t² + 20t + 2. The maximum effect occurs at t = −b⁄2a = −20 ÷ (−10) = 2 seconds, with maximum height h(2) = −20 + 40 + 2 = 22 units.

在抛体运动中，物体在 t 秒后的高度通常满足 h(t) = −5t² + 20t + 2。物体到达最高点的时间为 t = −b⁄2a = −20 ÷ (−10) = 2 秒，最大高度为 h(2) = −20 + 40 + 2 = 22 单位。

When setting up quadratic models, always define the variable clearly and check whether the answers make sense in the given context. Reject solutions that are physically impossible, such as negative lengths or times.

建立二次模型时，要清楚定义变量，并检查答案在给定情境中是否有意义。要舍弃在物理上不可能的解答，例如负的长度或时间。

10. Quadratic Inequalities | 二次不等式

Solving quadratic inequalities extends the techniques used for quadratic equations. The key idea is to find the roots first, then test intervals on a number line.

解二次不等式是二次方程解法的延伸。关键步骤是：先求根，再在数轴上测试区间符号。

For the inequality (x − 1)(x + 3) < 0, the roots are x = 1 and x = −3. Since the leading coefficient is positive, the parabola opens upward. The graph lies below the x-axis between the roots, so the solution is −3 < x < 1.

对于不等式 (x − 1)(x + 3) < 0，根为 x = 1 和 x = −3。由于二次项系数为正，抛物线开口向上。图像在两个根之间的部分位于 x 轴下方，因此解集为 −3 < x < 1。

For (x − 1)(x + 3) ≥ 0, the solution is x ≤ −3 or x ≥ 1. Remember that “less than” corresponds to the output being negative, while “greater than” corresponds to positive output.

对于 (x − 1)(x + 3) ≥ 0，解集为 x ≤ −3 或 x ≥ 1。记住，小于对应函数值为负，大于对应函数值为正。

When the inequality sign is strict (< or >), use open circles on the number line. When the sign includes equality (≤ or ≥), use closed circles.

当不等式符号为严格不等号（< 或 >）时，数轴上用空心点。当包含等号（≤ 或 ≥）时，用实心点。

11. Common Mistakes to Avoid | 常见错误与陷阱

In exams, several recurring mistakes in quadratic problems cost students marks. Recognising them in advance will improve your accuracy.

在考试中，二次函数问题中反复出现的几个错误会让学生丢分。提前识别这些错误有助于提高准确率。

Forgetting to rearrange the equation into standard form before applying the quadratic formula or completing the square. Set ax² + bx + c = 0 first.

在使用求根公式或配方前忘记将方程整理为标准形式。应先将等式化为 ax² + bx + c = 0。
Missing both solutions when solving x² = q. Always write x = ±√q, not just √q.

解 x² = q 时漏掉一个解。务必写成 x = ±√q，而不能只写 √q。
Dropping the factor a when completing the square. If a ≠ 1, factor it out from the first two terms before completing the square.

配方时忽略系数 a。若 a ≠ 1，应先将 a 从前两项中提取出来，再进行配方。
Misidentifying the vertex sign when the equation is in the form y = (x + p)² + q. The vertex is (−p, q), not (p, q).

当方程为 y = (x + p)² + q 形式时，顶点写反。顶点应为 (−p, q)，不是 (p, q)。
Misreading the discriminant result: b² − 4ac = 0 does not mean no solutions; it means exactly one repeated root.

误读判别式结果：b² − 4ac = 0 并不意味着无解，而是恰好有一个二重根。

12. Exam Tips and Final Checklist | 考试技巧与最终清单

A well-planned exam approach for quadratic questions can significantly reduce careless errors. Always check the question command word: solve, sketch, find the value of k, or factorise.

在考场上，针对二次函数问题做好规划可以显著减少粗心错误。始终注意题目的指令词：解方程、画图、求 k 的值，或因式分解。

When writing final answers, use exact values unless the question asks for decimal approximations. Avoid rounding intermediate results to prevent error accumulation.

写最终答案时，除非题目要求近似值，否则应使用精确值。避免在中间步骤四舍五入，防止误差累积。

Feature	How to find it
y-intercept	Substitute x = 0; read c directly
x-intercepts	Solve ax² + bx + c = 0
Vertex	x = −b⁄2a; substitute to get y
Axis of symmetry	x = −b⁄2a
Nature of roots	Use Δ = b² − 4ac

Use the table above as a quick reference when studying. Practise at least five problems from each section until the methods become automatic.

将上表作为复习时的快速参考。每个部分至少练习五道题，直到方法熟练为止。

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Quadratic Functions: From Factoring to Applications | 二次函数：从因式分解到应用