Mastering Quadratic Functions and Graphs | 掌握二次函数与图像

📚 Mastering Quadratic Functions and Graphs | 掌握二次函数与图像

Quadratic functions and their graphs are among the most frequently tested topics in IGCSE Mathematics. A solid grasp of the standard form, completing the square, and graph transformations not only secures marks in exams but also builds a strong foundation for advanced studies. This article provides a structured revision guide for teachers and students, covering key concepts, worked examples, and common pitfalls.

二次函数及其图像是 IGCSE 数学中最常考查的知识点之一。熟练掌握标准形式、配方法以及图像变换，不仅能在考试中稳拿分数，还能为高阶学习奠定坚实基础。本文将作为教师与学生的系统复习指南，涵盖关键概念、典型例题与常见易错点。

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

A quadratic function is any function of the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of such a function is a parabola, which opens upward when a > 0 and downward when a < 0. The value of a also determines the "width" of the parabola: larger absolute values of a produce a narrower graph, while smaller absolute values produce a wider one.

二次函数是指形如 f(x) = ax² + bx + c 的函数，其中 a、b、c 为常数，且 a ≠ 0。这类函数的图像称为抛物线：当 a > 0 时开口向上，当 a < 0 时开口向下。a 的取值还决定抛物线的"宽窄"：|a| 越大，图像越窄；|a| 越小，图像越宽。

The y-intercept of the graph is simply c, the constant term, because setting x = 0 yields f(0) = c. This single point is often the easiest feature to identify and serves as a useful anchor when sketching the curve.

抛物线与 y 轴的交点就是常数项 c，因为令 x = 0 时 f(0) = c。这个点是图像上最容易确定的特征，也是画草图时最有效的起点。

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

The vertex is the turning point of the parabola. Its x-coordinate can be found using the formula x = -b / (2a). Substituting this value back into the function gives the y-coordinate of the vertex. The vertical line x = -b / (2a) is called the axis of symmetry; the parabola is perfectly mirrored across this line.

顶点是抛物线的转折点。其 x 坐标可通过公式 x = -b / (2a) 求出。将该值代回函数即可得到顶点的 y 坐标。垂直线 x = -b / (2a) 被称为对称轴，抛物线关于这条线完全对称。

For example, given f(x) = 2x² – 4x + 1, the x-coordinate of the vertex is x = -(-4) / (2 × 2) = 4 / 4 = 1. Substituting x = 1 gives f(1) = 2(1)² – 4(1) + 1 = -1. Therefore, the vertex is at (1, -1) and the axis of symmetry is x = 1.

例如，对于 f(x) = 2x² – 4x + 1，顶点的 x 坐标为 x = -(-4) / (2 × 2) = 4 / 4 = 1。代入 x = 1 得 f(1) = 2(1)² – 4(1) + 1 = -1。因此顶点位于 (1, -1)，对称轴为 x = 1。

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3. Completing the Square | 配方法

Completing the square rewrites f(x) = ax² + bx + c in the form f(x) = a(x – h)² + k. In this form, the vertex is immediately visible as (h, k). To complete the square, halve the coefficient of x, square it, and adjust the constant term accordingly. This technique is essential for solving equations, finding maximum or minimum values, and sketching graphs.

配方法将 f(x) = ax² + bx + c 改写为 f(x) = a(x – h)² + k 的形式。在这种形式下，顶点一目了然，即为 (h, k)。配方的步骤是：将 x 的系数取半、平方，并相应地调整常数项。这一技巧在解方程、求最大值或最小值以及画图时至关重要。

Worked example: Write f(x) = x² + 6x + 5 in the form (x – h)² + k. Halving 6 gives 3; squaring gives 9. Thus f(x) = (x² + 6x + 9) – 9 + 5 = (x + 3)² – 4. The vertex is therefore (-3, -4).

示例：将 f(x) = x² + 6x + 5 改写为 (x – h)² + k 的形式。6 的一半是 3，平方得 9。因此 f(x) = (x² + 6x + 9) – 9 + 5 = (x + 3)² – 4。故顶点为 (-3, -4)。

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

When the quadratic expression factorises, solving ax² + bx + c = 0 becomes straightforward. Factorising involves finding two numbers whose product is ac and whose sum is b, then using the zero product property. This method is fast and reliable for equations with integer solutions.

当二次式可以因式分解时，解 ax² + bx + c = 0 就变得非常直接。因式分解的关键是找到两个数，使其乘积等于 ac，和等于 b，再利用”零积性质”。对于整数解的情况，这种方法快速且可靠。

For instance, x² – 5x + 6 = 0 factorises to (x – 2)(x – 3) = 0. Thus x = 2 or x = 3. It is important to remember to set each factor to zero separately. A common error is to forget the second solution, especially when students rush after finding one root.

例如，x² – 5x + 6 = 0 可分解为 (x – 2)(x – 3) = 0，因此 x = 2 或 x = 3。注意要分别令每个因式为零。一个常见错误是在找到一个解后就草率收工，漏掉了第二个解。

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5. Solving Quadratic Equations by the Formula | 用公式法解二次方程

For quadratics that do not factorise easily, the quadratic formula provides a universal method. The solutions to ax² + bx + c = 0 are given by:

对于不易因式分解的二次方程，求根公式提供了一种通用解法。方程 ax² + bx + c = 0 的解为：

x = (-b ± √(b² – 4ac)) / (2a)

The discriminant, Δ = b² – 4ac, determines the nature of the roots. If Δ > 0, there are two distinct real roots. If Δ = 0, there is exactly one real root (a repeated root). If Δ < 0, there are no real roots. Students should be able to compute the discriminant first and use it to predict the number of solutions before applying the full formula.

判别式 Δ = b² – 4ac 决定根的性质。若 Δ > 0，方程有两个不相等的实根；若 Δ = 0，有且仅有一个实根（重根）；若 Δ < 0，则没有实根。学生应先计算判别式来预判解的个数，再套用完整公式，这样能有效避免盲目计算。

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

When sketching a quadratic graph, students should identify four key features in the following order: (1) the y-intercept, (2) the roots, (3) the axis of symmetry and vertex, and (4) whether the parabola opens upward or downward. This systematic approach ensures accuracy and clarity.

画二次函数图像时，学生应按照以下顺序确定四个关键特征：（1）y 轴截距；（2）根（与 x 轴的交点）；（3）对称轴与顶点；（4）抛物线开口方向。按部就班地操作能保证图像的准确与清晰。

If there are no real roots, the parabola does not cross the x-axis at all, so students should still plot the y-intercept and the vertex to produce a correct sketch. A table of additional points can help when the graph is not symmetric enough for a confident sketch.

如果方程无实根，抛物线就不会与 x 轴相交。此时学生仍需标出 y 轴截距和顶点来绘制正确的草图。当仅凭对称性不足以自信作图时，可以额外列一张数值表取点辅助。

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7. Transformations of Quadratic Graphs | 二次函数图像的变换

Understanding graph transformations is essential for IGCSE. Adding a constant outside the function, f(x) + k, shifts the graph vertically. Adding a constant inside the brackets, f(x + h), shifts the graph horizontally. Replacing x by -x reflects the graph in the y-axis, and replacing f(x) by -f(x) reflects it in the x-axis.

理解图像变换是 IGCSE 的必修内容。在函数外加常数 f(x) + k 使图像上下平移；在括号内加常数 f(x + h) 使图像水平平移。用 -x 替换 x 得到关于 y 轴的对称图像，用 -f(x) 替换 f(x) 则得到关于 x 轴的对称图像。

Vertical stretch and compression are achieved by multiplying the whole function, a·f(x), while horizontal stretch and compression are achieved by f(b·x). Students often confuse the direction of horizontal shifts: remember that f(x + 2) shifts the graph to the left, not the right.

纵向拉伸与压缩通过将整个函数乘以 a·f(x) 实现，横向拉伸与压缩则通过 f(b·x) 实现。学生常弄反水平平移的方向：请记住 f(x + 2) 使图像向左移动，而不是向右。

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8. The Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0, the sum of the two roots is -b/a and the product is c/a. This relationship is extremely useful for checking answers and for constructing quadratics with given roots. For example, if a quadratic has roots 2 and -3, then the sum is -1 and the product is -6, leading to the equation x² + x – 6 = 0.

对于二次方程 ax² + bx + c = 0，两根之和为 -b/a，两根之积为 c/a。这一关系在验算答案和由已知根构造二次方程时极为实用。例如，若一个二次方程的两个根为 2 和 -3，则根和为 -1、根积为 -6，因此方程为 x² + x – 6 = 0。

This is an elegant shortcut that students can mention in the exam working to demonstrate deeper understanding. Judges and examiners reward concise, mathematically fluent solutions.

这是一个非常优雅的捷径，在考试步骤中适当引用能展现出更深层的理解。阅卷者通常会对简洁而数学逻辑流畅的解法给予高分。

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9. Quadratic Inequalities | 二次不等式

Solving quadratic inequalities involves first solving the corresponding equation and then determining the range of x that satisfies the inequality using a graph or a sign table. For instance, x² – x – 6 < 0 factorises to (x - 3)(x + 2) < 0. The critical points are x = -2 and x = 3. Testing intervals shows the solution is -2 < x < 3.

解二次不等式时，先解对应方程，再利用图像或符号表确定满足不等式的 x 范围。例如，x² – x – 6 < 0 分解为 (x - 3)(x + 2) < 0。关键点 x = -2 和 x = 3，检验各区间的符号可得解为 -2 < x < 3。

Students should make the distinction between strict (<) and non-strict (≤) inequalities when writing final answers. Open circles on the number line indicate strict inequalities, while filled circles indicate non-strict ones.

学生需在写出最终答案时区分严格不等式 (<) 与非严格不等式 (≤)。在数轴上，空心圆圈表示严格不等，实心圆圈表示非严格不等。

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10. Common Pitfalls and Exam Tips | 常见易错点与考试建议

One of the most frequent errors is sign errors when substituting negative values into the quadratic formula. Always double-check each sign carefully. Another common mistake is forgetting to divide every term by a when factoring out a common factor. Also, when sketching graphs, many students lose marks for failing to label the vertex coordinates explicitly.

最常见的错误之一是将负数代入求根公式时出现符号错误，务必仔细核对每一步的符号。另一个常见误区是提取公因数时忘记将每一项都除以 a。此外，在画图时，许多学生因未明确标注顶点坐标而失分。

Exam tips: always write down the equation before substituting numbers; use a ruler for axes and clear labels; when using a calculator, work through the process step by step rather than jumping straight to the answer. Remember that calculators can verify solutions but should not replace a thorough understanding of the method.

考试建议：先完整写下原始方程再代入数值；用直尺画坐标轴并清晰标记；使用计算器时按步骤运算，切忌直接跳到答案。计算器可用于验证解，但不应替代对方法的透彻理解。

Additionally, do not round intermediate values when using the quadratic formula. Keep all values exact until the final step, then round as instructed by the question.

此外，使用求根公式时不要在中间步骤四舍五入。保持所有数值精确到最后一步，再按照题目要求进行舍入。

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