Quadratic Functions: Key GCSE Topics | 二次函数考点精讲

📚 Quadratic Functions: Key GCSE Topics | 二次函数考点精讲

A quadratic function is one of the most important topics in GCSE Mathematics. It forms the foundation for understanding parabolas, solving equations, and tackling real-world problems. In this revision guide, we cover everything you need to know – from the basic shape of a quadratic graph to using the quadratic formula and discriminant. Let’s break down the key concepts step by step.

二次函数是 GCSE 数学中最关键的主题之一。它为理解抛物线、解方程和处理现实问题奠定了基础。在这份复习指南中，我们将涵盖你需要掌握的所有内容——从二次函数图像的基本形状到求根公式和判别式的使用。让我们一步步拆解核心考点。

1. Introduction to Quadratic Functions | 二次函数简介

A quadratic function is any function that can be written in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The highest power of x is 2, which gives the function its characteristic ‘U-shaped’ curve called a parabola.

二次函数是可以写成 f(x) = ax² + bx + c 形式的函数，其中 a、b、c 为常数且 a ≠ 0。x 的最高次幂是 2，这使得函数图像呈现出特有的 ‘U 形’ 曲线，即抛物线。

In GCSE questions, you will often see expressions like y = x² – 5x + 6 or f(x) = -2x² + 8x – 3. The graph of a quadratic function always has a single turning point (the vertex) and is symmetric about a vertical line known as the axis of symmetry.

在 GCSE 考题中，你会经常看到诸如 y = x² – 5x + 6 或 f(x) = -2x² + 8x – 3 的表达式。二次函数的图像总是有一个拐点（顶点），并且关于一条称为对称轴的竖直线对称。

2. Standard Form and Key Terminology | 标准形式与关键术语

The standard form of a quadratic is y = ax² + bx + c. Each coefficient has a role: ‘a’ controls the shape and direction; ‘b’ influences the position of the axis of symmetry; ‘c’ is the y-intercept. The graph crosses the y-axis at the point (0, c).

二次函数的标准形式是 y = ax² + bx + c。每个系数的作用如下：a 控制形状和开口方向；b 影响对称轴的位置；c 是图像与 y 轴的交点纵坐标。图像总在点 (0, c) 处穿过 y 轴。

The roots (or solutions) of a quadratic equation ax² + bx + c = 0 are the x-values where the graph crosses the x-axis. These are also called x-intercepts or zeros. A quadratic can have 0, 1, or 2 real roots.

二次方程 ax² + bx + c = 0 的根（或解）是图像与 x 轴交点的横坐标。它们也被称为 x 轴截距或零点。二次函数可以有 0 个、1 个或 2 个实根。

Tip: Always write a quadratic in standard form before you start working with it.

技巧：在处理二次函数之前，始终先将其写成标准形式。

3. Shape and Orientation of Parabola | 抛物线的形状与开口方向

If a > 0, the parabola opens upwards (smiley face). The vertex is a minimum point. If a < 0, the parabola opens downwards (sad face) and the vertex is a maximum point. The magnitude of a determines how 'wide' or 'narrow' the graph appears – a smaller |a| makes the parabola wider.

如果 a > 0，抛物线开口向上（像笑脸）；顶点为最小值点。如果 a < 0，抛物线开口向下（像哭脸）；顶点为最大值点。a 的绝对值大小决定了图像的“宽窄”——|a| 越小，抛物线越宽。

For example, y = 2x² is narrower than y = x², while y = ½x² is wider. Understanding the effect of ‘a’ helps you quickly visualise the graph before plotting points.

例如，y = 2x² 的图像比 y = x² 窄，而 y = ½x² 的图像更宽。理解 a 的作用有助于在描点之前快速想象出图像的大致形状。

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

The vertex is the turning point of the parabola. For a quadratic in standard form, the x-coordinate of the vertex is given by x = -b / (2a). The axis of symmetry is the vertical line x = -b / (2a). Once you have x, substitute back into the equation to find the y-coordinate of the vertex.

顶点是抛物线的拐点。对于标准形式的二次函数，顶点的 x 坐标由 x = -b / (2a) 给出。对称轴是垂直线 x = -b / (2a)。求得 x 后，代入原方程即可得到顶点的 y 坐标。

If the quadratic is given in completed-square form y = a(x – h)² + k, then the vertex is simply (h, k) and the axis of symmetry is x = h. Converting to this form is very useful.

如果二次函数以配方式 y = a(x – h)² + k 给出，那么顶点就是 (h, k)，对称轴为 x = h。转化为这种形式非常有用。

Example: For y = x² – 4x + 3, a=1, b=-4 so x = -(-4)/(2×1) = 2. The axis is x = 2, and the vertex is (2, -1).

例子：对于 y = x² – 4x + 3，a=1，b=-4，所以 x = -(-4)/(2×1) = 2。对称轴为 x = 2，顶点为 (2, -1)。

5. Finding Roots by Factorising | 因式分解求根

One of the most common methods to solve a quadratic equation is factorising. You rewrite ax² + bx + c as a product of two linear brackets (px + q)(rx + s) = 0. Then set each bracket to zero to find the roots.

解二次方程最常用的方法之一就是因式分解。将 ax² + bx + c 改写为两个一次括号的乘积 (px + q)(rx + s) = 0。然后令每个括号分别等于零即可求出根。

For simple quadratics where a = 1, you look for two numbers that multiply to give c and add to give b. For example, x² – 5x + 6 = 0 → (x – 2)(x – 3) = 0, so x = 2 or x = 3.

对于 a = 1 的简单二次式，寻找两个数，使得它们的乘积等于 c，和等于 b。例如，x² – 5x + 6 = 0 → (x – 2)(x – 3) = 0，因此 x = 2 或 x = 3。

When a ≠ 1, you may need to use the ‘split the middle term’ method or trial and error. Always check by expanding back.

当 a ≠ 1 时，你可能需要使用“拆中项”方法或试探法。务必通过展开括号来检验。

6. Completing the Square | 配方法

Completing the square rewrites a quadratic as a(x – h)² + k. This form is essential for finding the vertex, solving when factorising is difficult, and for deriving the quadratic formula. The procedure: take half of the coefficient of x, square it, and adjust the constant term.

配方法将二次式改写为 a(x – h)² + k 的形式。这种形式对于求顶点、在难以因式分解时解方程以及推导求根公式都至关重要。步骤为：取 x 系数的一半，将其平方，并调整常数项。

For example, to complete the square for x² + 6x + 7: halve 6 to get 3, square to get 9. Write (x + 3)² – 9 + 7 = (x + 3)² – 2. So vertex is (-3, -2). If a ≠ 1, factor it out from the x-terms first.

例如，对 x² + 6x + 7 进行配方：取 6 的一半得 3，平方得 9。写成 (x + 3)² – 9 + 7 = (x + 3)² – 2。因此顶点为 (-3, -2)。如果 a ≠ 1，先让 a 从含 x 的项中提出来。

Note: Many GCSE papers require you to express a quadratic in completed square form and then read off the vertex coordinates or solve the equation.

注意：许多 GCSE 试卷要求你将二次式写成配方式，然后读出顶点坐标或解方程。

7. The Quadratic Formula | 求根公式

When factorising is messy or impossible, the quadratic formula works for any quadratic ax² + bx + c = 0. The formula is:

当因式分解困难或无法进行时，求根公式适用于任何二次方程 ax² + bx + c = 0。公式为：

x = (-b ± √(b² – 4ac)) / 2a

This formula gives two solutions, one using the ‘+’ sign and the other using the ‘-‘ sign. The term under the square root, (b² – 4ac), is the discriminant. Be careful with signs when substituting negative b.

该公式给出两个解，一个使用 ‘+’ 号，另一个使用 ‘-‘ 号。平方根下的表达式 (b² – 4ac) 是判别式。在代入负的 b 时要小心符号。

Example: Solve 2x² – 4x – 3 = 0. Here a=2, b=-4, c=-3. Substituting:

例子：解方程 2x² – 4x – 3 = 0。这里 a=2，b=-4，c=-3。代入：

x = (4 ± √((-4)² – 4×2×(-3))) / (2×2) = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4

Simplify √40 = 2√10, so x = (4 ± 2√10)/4 = 1 ± ½√10. Leave in surd form unless told otherwise.

化简 √40 = 2√10，所以 x = (4 ± 2√10)/4 = 1 ± ½√10。除非另有要求，通常保留根式形式。

8. Discriminant and Nature of Roots | 判别式与根的性质

The discriminant Δ = b² – 4ac tells you how many real roots the quadratic has without solving it completely:

判别式 Δ = b² – 4ac 能让你在不完全解方程的情况下判断二次函数有多少个实根：

Δ > 0: Two distinct real roots (graph crosses x-axis twice).
Δ > 0：有两个不同的实根（图像与 x 轴相交于两点）。
Δ = 0: One repeated real root (graph touches x-axis at vertex).
Δ = 0：有一个重根（图像在顶点处与 x 轴相切）。
Δ < 0: No real roots (graph does not cross x-axis).
Δ < 0：没有实根（图像与 x 轴不相交）。

This is frequently tested: “Find the value of k for which the equation has equal roots.” Set Δ = 0 and solve for k.

这一点经常被考查：“求使方程有等根的 k 值。”令 Δ = 0 并解出 k。

For example, x² + 2x + k = 0 has equal roots when b² – 4ac = 4 – 4k = 0 → k = 1.

例如，x² + 2x + k = 0 有等根时，需要 b² – 4ac = 4 – 4k = 0 → k = 1。

9. Sketching Quadratic Graphs | 二次函数图像绘制

To sketch a quadratic graph, you need to identify key features: shape (opens up or down), y-intercept (0, c), roots (if any), and vertex. Always label these points on your sketch. A sketch does not need a perfect scale but must show the correct orientation and intercepts.

要绘制二次函数图像，你需要标出关键特征：形状（开口向上还是向下）、y 轴截距 (0, c)、根（如果有的话）以及顶点。务必在草图上标注这些点。草图不需要完美的比例，但必须展现正确的开口方向和截距。

Step-by-step: (1) Note the value of a to decide direction. (2) Plot the y-intercept c. (3) Mark the vertex using x = -b/(2a) and compute y. (4) Find the roots by factorising or the formula, and mark them. (5) Draw a smooth parabola through the points.

分步操作：(1) 观察 a 的值以决定开口方向。(2) 标出 y 轴截距 c。(3) 用 x = -b/(2a) 计算顶点坐标并标出。(4) 通过因式分解或公式求出根并标出。(5) 用平滑的曲线穿过这些点画出抛物线。

Example: Sketch y = -x² + 2x + 3. a < 0 → opens down; y-intercept 3; vertex at x = -2/(2×(-1)) = 1, y = -(1)²+2+3=4 → (1,4); roots: -x²+2x+3=0 → x²-2x-3=0 → (x-3)(x+1)=0 → x=3, x=-1.

例子：绘制 y = -x² + 2x + 3 的草图。a < 0 → 开口向下；y 轴截距 3；顶点 x = -2/(2×(-1)) = 1，y = -(1)²+2+3=4 → (1,4)；根：-x²+2x+3=0 → x²-2x-3=0 → (x-3)(x+1)=0 → x=3, x=-1。

10. Solving Problems and Applications | 应用题解题

Quadratic equations often appear in real-world contexts: area of a rectangle, projectile motion, profit maximisation, and more. Read the problem carefully to form an equation, then solve using any suitable method. Don’t forget to check if the solution makes sense (e.g. a length cannot be negative).

二次方程经常出现在现实情境中：矩形的面积、抛体运动、利润最大化等等。仔细读题，列出方程，然后用任何合适的方法求解。别忘了检查解是否合理（例如长度不能为负）。

Typical GCSE problem: “A rectangular garden has length (x+3) m and width (x-1) m. Its area is 12 m². Find x.” Set up (x+3)(x-1)=12, expand to x²+2x-3=12 → x²+2x-15=0 → (x+5)(x-3)=0 → x=3 (since x must be positive).

典型 GCSE 题目：“一个矩形花园长为 (x+3) 米，宽为 (x-1) 米，面积为 12 平方米。求 x。” 建立方程 (x+3)(x-1)=12，展开得 x²+2x-3=12 → x²+2x-15=0 → (x+5)(x-3)=0 → x=3（因为 x 必须为正）。

Always give the final answer in the units requested and to the required accuracy if using the quadratic formula.

如果使用求根公式，最终答案务必以题目要求的单位表示，并达到要求的精度。

11. Intersections with Lines | 与直线的交点

To find where a quadratic curve y = ax² + bx + c meets a straight line y = mx + d, set them equal: ax² + bx + c = mx + d. Then rearrange to form a new quadratic equation and solve. The solutions give the x-coordinates of intersection points; substitute back to find y.

要找到二次曲线 y = ax² + bx + c 与直线 y = mx + d 的交点，令两式相等：ax² + bx + c = mx + d。然后重新整理成一个新的二次方程并求解。得到的解就是交点横坐标；代回原式求出纵坐标。

The discriminant of the resulting quadratic tells you the number of intersection points: Δ > 0 → two intersections; Δ = 0 → line is tangent; Δ < 0 → no intersection.

所得二次方程的判别式能告诉你交点的个数：Δ > 0 → 两个交点；Δ = 0 → 直线为切线；Δ < 0 → 没有交点。

Example: Find where y = x² – 3x + 1 meets y = 2x – 5. Set x² – 3x + 1 = 2x – 5 → x² – 5x + 6 = 0 → (x-2)(x-3)=0 → x=2, x=3. Points: (2, -1) and (3, 1).

例子：求 y = x² – 3x + 1 与 y = 2x – 5 的交点。令 x² – 3x + 1 = 2x – 5 → x² – 5x + 6 = 0 → (x-2)(x-3)=0 → x=2, x=3。交点：(2, -1) 和 (3, 1)。

12. Summary and Tips | 总结与技巧

Mastering quadratics at GCSE means being comfortable switching between standard form, factorised form, and completed-square form. Practise all three solving methods: factorising, quadratic formula, and completing the square. Remember to always check the sign of a before sketching.

在 GCSE 中掌握二次函数意味着能够熟练地在标准形式、因式分解形式和配方式之间转换。练习所有三种求解方法：因式分解、求根公式和配方法。在绘图前务必检查 a 的符号。

Method	When to Use
Factorising	When a, b, and c are small and factors are easy to spot
因式分解	当 a、b、c 较小且容易找到因式时
Quadratic Formula	Always works, best for non-factorable equations or surds
求根公式	始终有效，最适合无法分解或带有根式的方程
Completing the Square	Great for finding vertex, solving when a=1, and proving the formula
配方法	适合求顶点、当 a=1 时解方程以及推导求根公式

Finally, show all working steps clearly. Even if you make a sign error, you can earn method marks. Practise past paper questions regularly, especially discriminated and intersection problems.

最后，请清晰展示所有解题步骤。即便符号出错，你仍然可能得到过程分。定期练习往年真题，尤其要重视判别式和交点问题。

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Quadratic Functions: Key GCSE Topics | 二次函数考点精讲