📚 Quadratic Functions at KS3 | KS3 数学：二次函数 考点精讲
A quadratic function is one of the first non‑linear functions you meet at Key Stage 3. Understanding its shape, features and transformations will give you a solid foundation for GCSE and beyond. This article breaks down everything you need to know about quadratic functions, from plotting their graphs to solving real‑world problems.
二次函数是KS3阶段会接触到的第一种非线性函数。理解它的图像形状、关键特征以及变换方式，能为你进入GCSE及更高阶段的学习打下坚实基础。本文详细梳理了二次函数的核心内容，从绘制图像到解决实际问题，一网打尽。
1. What Is a Quadratic Function? | 什么是二次函数？
A quadratic function is any function that can be written in the form y = ax² + bx + c, where a, b and c are constants and a ≠ 0. The x² term is what makes it ‘quadratic’ – it comes from the Latin word ‘quadratus’ meaning ‘square’. If a = 0, the function would become linear, so the coefficient of x² must not be zero.
二次函数是能写成 y = ax² + bx + c 形式的函数，其中 a、b、c 是常数且 a ≠ 0。x² 这一项决定了它是“二次”的——这个词源于拉丁语“quadratus”，意为“平方”。如果 a = 0，函数就退化成了线性函数，因此 x² 的系数必须不为零。
In KS3, we usually focus on simpler quadratics where b and c may be zero, for example y = x², y = 2x² or y = x² + 3. These simple forms make it easier to see how the graph changes when you alter a, b or c.
在KS3阶段，我们通常关注 b 和 c 可能为零的简单二次函数，例如 y = x²、y = 2x² 或 y = x² + 3。这些简单形式有助于你更直观地理解改变 a、b、c 时图像的变化。
2. The Standard Form y = ax² + bx + c | 标准形式 y = ax² + bx + c
The graph of any quadratic function is a smooth curve called a parabola. The sign of a tells you whether the parabola opens upwards (a > 0) like a U‑shape, or downwards (a < 0) like an inverted U, often called a 'sad face' curve.
任何二次函数的图像都是一条光滑的曲线，叫做抛物线。系数 a 的正负决定抛物线的开口方向：a > 0 时开口朝上，呈U形；a < 0 时开口朝下，像一个倒过来的U，常被形容为“难过的脸”形状。
The value of a also affects the width of the parabola: the larger |a|, the narrower the graph; the smaller |a| (closer to 0), the wider the graph. In KS3, you will mainly work with positive a values and curves that open upwards.
a 的绝对值大小还影响抛物线的宽窄：|a| 越大，图像越窄；|a| 越小（越接近0），图像越宽。在KS3中，你主要会接触到 a 为正数、开口向上的抛物线。
3. The Parent Function: y = x² | 基础函数：y = x²
The simplest quadratic is y = x². Its graph is a U‑shaped parabola that opens upwards and passes through the origin (0,0). This curve is symmetrical about the y‑axis, which means if you fold the graph along the y‑axis, the two halves match perfectly.
最简单的二次函数是 y = x²。它的图像是一条经过原点 (0,0) 且开口向上的U形抛物线。这条曲线关于y轴对称，也就是说，如果你沿着y轴对折图像，左右两半会完全重合。
To plot y = x², you can create a table of values:
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | 9 | 4 | 1 | 0 | 1 | 4 | 9 |
You can see the y‑values are the squares of the x‑values, and the graph gets steeper as you move away from the origin.
绘制 y = x² 的图像时，你可以先列出数值表：
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | 9 | 4 | 1 | 0 | 1 | 4 | 9 |
可以看到，y值是x值的平方，图像在远离原点时变得越来越陡。
4. Vertex and Axis of Symmetry | 顶点与对称轴
Every parabola has a lowest point (for a > 0) or a highest point (for a < 0). This turning point is called the vertex. For y = x², the vertex is at (0,0). The vertical line that passes through the vertex is called the axis of symmetry. For y = x², the axis of symmetry is the line x = 0 (the y‑axis).
每条抛物线都有一个最低点（当 a > 0 时）或最高点（当 a < 0 时）。这个转折点叫做顶点。对于 y = x²，顶点在 (0,0)。穿过顶点的竖直线称为对称轴。y = x² 的对称轴是直线 x = 0（即y轴）。
When you change the equation by adding or subtracting a constant, the vertex moves but the axis of symmetry remains vertical. Knowing the vertex helps you sketch the graph quickly and accurately.
当你通过加减常数来改变方程时，顶点会移动，但对称轴始终保持竖直。掌握顶点位置能帮助你快速准确地画出示意图。
5. Vertical Translations: y = x² + k | 垂直平移：y = x² + k
If you add a positive number k to x², the whole graph shifts upwards by k units. For example, the vertex of y = x² + 3 is (0,3), and the parabola still opens upwards. If you subtract a number, like y = x² − 4, the graph shifts downwards so the vertex becomes (0,−4).
如果你给 x² 加上一个正数 k，整个图像会向上平移 k 个单位。比如 y = x² + 3 的顶点是 (0,3)，抛物线仍然开口向上。如果减去一个数，如 y = x² − 4，图像就会向下平移，顶点变为 (0,−4)。
These are called vertical translations. The shape of the parabola does not change – only its position along the y‑axis changes. This is a key concept at KS3 and is often tested with graph‑sketching questions.
这类操作叫做垂直平移。抛物线的形状没有改变——改变的只是它在y轴方向的位置。这是KS3阶段的重要概念，常出现在画图题中。
6. Horizontal Translations: y = (x − h)² | 水平平移：y = (x − h)²
Replacing x by (x − h) inside the square moves the graph horizontally. For y = (x − 2)², the entire graph shifts 2 units to the right, so the vertex moves to (2,0). For y = (x + 3)², think of it as y = (x − (−3))², so it shifts 3 units to the left, giving a vertex at (−3,0).
在平方内用 (x − h) 替代 x 会让图像水平移动。对于 y = (x − 2)²，整个图像向右平移 2 单位，顶点变为 (2,0)。对于 y = (x + 3)²，可以看成 y = (x − (−3))²，因此向左平移 3 单位，顶点变为 (−3,0)。
Notice that the sign inside the bracket is opposite to the direction of the shift: (x − 2)² moves right, (x + 3)² moves left. This is a common confusion, so always double‑check with a quick table of values.
注意，括号内的符号与平移方向相反：(x − 2)² 向右移，(x + 3)² 向左移。这一点很容易混淆，建议画图时用数值表快速验证。
7. Combined Translations: y = (x − h)² + k | 组合平移：y = (x − h)² + k
When you have both a horizontal and a vertical shift, the vertex is simply (h, k). For example, y = (x − 1)² + 2 has its vertex at (1, 2). The axis of symmetry is x = h, or x = 1 in this case.
当你同时进行水平和垂直平移时，顶点就位于 (h, k)。例如，y = (x − 1)² + 2 的顶点在 (1, 2)，对称轴为 x = h，在这个例子中即 x = 1。
The graph still opens upwards because the coefficient of (x − 1)² is positive 1. This form is sometimes called the vertex form of a quadratic, and it is extremely useful for quickly sketching parabolas without needing a full table of values.
由于 (x − 1)² 的系数为正1，图像依然开口向上。这种形式有时被称为二次函数的顶点式，它非常实用，可以让你无需列出完整数值表就能快速画出抛物线。
8. Stretching and Reflecting: y = ax² | 拉伸与反射：y = ax²
When the coefficient a is not 1, the graph becomes steeper or flatter. For instance, y = 2x² is narrower than y = x² because the y‑values grow more quickly. On the other hand, y = ½x² is wider because the y‑values increase more slowly.
当系数 a 不为1时，图像会变得更陡或更平。例如，y = 2x² 比 y = x² 窄，因为y值增长速度更快。而 y = ½x² 较宽，因为y值增长更慢。
If a is negative, the parabola reflects over the x‑axis and opens downwards. For example, y = −x² is an upside‑down U with vertex at (0,0). At KS3, you might also explore simple combinations like y = −2x² + 1 to see how the reflection and translation work together.
如果 a 为负数，抛物线会关于x轴反射，开口朝下。例如，y = −x² 就是一条顶点在 (0,0) 的倒U形曲线。在KS3，你可能还会接触到 y = −2x² + 1 这样的简单组合，来观察反射和平移如何共同起作用。
9. Plotting Quadratics Using a Table | 利用表格绘制二次函数图像
To sketch any quadratic, you can follow a reliable method:
- Choose a range of x‑values (usually from −3 to 3 or −4 to 4).
- Substitute each x into the equation to find y.
- Record the points in a table.
- Plot the points on a coordinate grid and join them with a smooth, U‑shaped curve.
Even when the vertex is not at (0,0), this method works and helps you see the symmetry in the y‑values.
想要画出任一二次函数的图像，你可以遵循一个可靠的方法：
- 选择x的取值范围（通常从 −3 到 3 或 −4 到 4）。
- 将每个x值代入方程求出y值。
- 在表格中记录对应点。
- 在坐标网格上描点，并用光滑的U形曲线连接。
即使顶点不在 (0,0)，这个方法也同样适用，并能帮助你从y值中观察到对称性。
10. Introduction to Solving Quadratic Equations | 解二次方程入门
A quadratic equation is when a quadratic expression is set equal to zero, like x² − 5x + 6 = 0. At KS3, you learn to solve these by factorising the expression into two brackets, such as (x − 2)(x − 3) = 0. If the product of two terms is zero, then at least one of the terms must be zero.
二次方程是指将二次表达式设为零，如 x² − 5x + 6 = 0。在KS3，你会学习通过将表达式因式分解成两个括号来解方程，比如 (x − 2)(x − 3) = 0。如果两个因式的乘积为零，那么至少有一个因式为零。
Setting each bracket equal to zero gives the solutions x = 2 and x = 3. Graphically, these are the x‑intercepts of the parabola y = x² − 5x + 6, where the curve crosses the x‑axis. This visual link between algebra and graphs is a powerful idea.
令每个括号等于零，就得到解 x = 2 和 x = 3。从图像上看，这两个值正是抛物线 y = x² − 5x + 6 与x轴交点的横坐标。代数与图像之间的这种视觉联系，是一个很强大的概念。
11. Factorising Simple Quadratics | 因式分解简单的二次式
To factorise x² + bx + c, look for two numbers that multiply to give c and add to give b. For example, with x² + 7x + 10, the numbers 2 and 5 multiply to 10 and add to 7. So the factorised form is (x + 2)(x + 5).
要对 x² + bx + c 进行因式分解，需要寻找两个数，它们的乘积等于 c，和等于 b。例如，对于 x² + 7x + 10，2和5 的乘积为10，和为7，因此因式分解的结果是 (x + 2)(x + 5)。
Always expand back to check your answer: (x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10. At KS3, you mainly work with positive coefficients, but the same logic applies when negative numbers are involved.
一定要展开回去来检查答案：(x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10。在KS3，你主要处理正系数，但当涉及负数时，同样的逻辑也适用。
12. Real‑World Applications and Summary | 实际应用与总结
Quadratic functions model many real‑life situations, such as the area of a rectangle with a fixed perimeter, the height of a thrown ball over time, or the shape of a satellite dish. For instance, if a rectangle has length x + 2 and width x, the area is x(x + 2) = x² + 2x, a quadratic expression. Solving x² + 2x = 15 helps you find possible dimensions.
二次函数可以模拟许多现实情境，例如固定周长的矩形面积、抛出的球随时间变化的高度，或者卫星天线的形状。比如，如果一个矩形的长为 x + 2、宽为 x，那么面积就是 x(x + 2) = x² + 2x，一个二次表达式。求解 x² + 2x = 15 就能帮助你找出可能的尺寸。
By mastering the shape, vertex, axis of symmetry, translations and basic solution methods, you build a strong base for more advanced algebra. Keep practising with tables of values and sketching graphs – these skills will serve you well throughout your maths journey.
通过掌握图像形状、顶点、对称轴、平移以及基本的求解方法，你就能为更高阶的代数学习打下坚实基础。坚持练习数值表和画图技巧，这些能力将在你的数学学习中持续发挥作用。
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