📚 Mastering Quadratic Graphs for Edexcel A Level Maths | 掌握爱德思 A Level 数学中的二次函数图像
Quadratic graphs are one of the most frequently examined topics in Edexcel A Level Mathematics. They connect algebra, coordinate geometry, and modelling, so a strong visual understanding of y = ax² + bx + c will help you solve equations, inequalities, and optimisation problems with confidence.
二次函数图像是爱德思 A Level 数学中最常考查的主题之一。它们将代数、坐标几何和建模联系在一起，因此扎实掌握 y = ax² + bx + c 的图像，能帮助你自信地解决方程、不等式和优化问题。
In the Edexcel specification, quadratic graphs appear in both Pure Mathematics and applied contexts. You are expected to sketch parabolas accurately, interpret roots and turning points, and use graphs to justify solutions.
在爱德思考试大纲中，二次函数图像同时出现在纯数学与应用背景中。考官要求你准确画出抛物线，解释根和顶点，并利用图像证明解的合理性。
1. The General Quadratic and Its Graph | 一般二次函数及其图像
A quadratic function has the standard form y = ax² + bx + c, where a, b and c are constants and a ≠ 0. Its graph is a smooth, symmetric curve called a parabola.
二次函数的一般形式为 y = ax² + bx + c，其中 a、b 和 c 为常数且 a ≠ 0。它的图像是一条光滑、对称的曲线，称为抛物线。
The sign of a controls whether the parabola opens upward or downward. If a > 0, the graph has a minimum turning point; if a < 0, the graph has a maximum turning point.
a 的符号控制抛物线的开口方向。若 a > 0，图像存在最小值点；若 a < 0，图像存在最大值点。
The constant c gives the y-intercept, while the values of a and b together determine the position of the vertex along the x-axis. Understanding these roles makes sketching much easier.
常数 c 决定 y 轴截距，而 a 与 b 共同决定顶点在 x 轴方向的位置。理解这些作用能让画图变得容易得多。
2. Shape and Orientation | 开口方向与形状
The coefficient a also affects the width of the parabola. A larger value of |a| makes the graph steeper and narrower, while a smaller value of |a| makes it flatter and wider.
系数 a 还会影响抛物线的宽窄。|a| 越大，图像越陡、越窄；|a| 越小，图像越平、越宽。
For example, y = 2x² is narrower than y = x², while y = ½x² is wider. If a is negative, the graph is reflected in the x-axis compared with y = |a|x².
例如，y = 2x² 比 y = x² 更窄，而 y = ½x² 更宽。如果 a 为负数，图像相对于 y = |a|x² 会关于 x 轴反射。
The value of b causes a horizontal shift combined with a vertical shift. This is why the completed square form y = a(x − h)² + k is so useful for sketching.
b 的值会引起水平平移并伴随垂直平移。这就是为什么配方式 y = a(x − h)² + k 对画图非常有用。
3. Key Points: y-intercept and x-intercepts | 关键点：y 轴截距与 x 轴截距
To find the y-intercept, set x = 0. The equation becomes y = c, so the parabola always passes through the point (0, c).
求 y 轴截距时，令 x = 0。方程变为 y = c，因此抛物线总是经过点 (0, c)。
The x-intercepts, also called roots or zeros, are found by solving ax² + bx + c = 0. These are the values of x where the graph cuts or touches the x-axis.
x 轴截距，也称为根或零点，通过解方程 ax² + bx + c = 0 求得。这些 x 值是图像与 x 轴相交或相切的位置。
If the graph has two x-intercepts, it crosses the x-axis twice. If it has one repeated intercept, it touches the x-axis. If it has none, the entire graph lies entirely above or below the x-axis.
如果图像有两个 x 轴截距，它会穿过 x 轴两次；如果有一个重复截距，它会与 x 轴相切；如果没有实根，整个图像完全位于 x 轴上方或下方。
4. Factorising to Find Roots | 通过因式分解求根
When a quadratic factorises over integers, the roots can be read directly from the factors. For example, y = x² − 5x + 6 = (x − 2)(x − 3), so the roots are x = 2 and x = 3.
当二次式可以整数因式分解时，可以直接从因式中读出根。例如 y = x² − 5x + 6 = (x − 2)(x − 3)，所以根为 x = 2 和 x = 3。
If the quadratic is a perfect square, such as y = (x − 3)², the graph touches the x-axis at x = 3 and does not cross it. This is called a repeated root or double root.
如果二次式是完全平方，例如 y = (x − 3)²，图像在 x = 3 处与 x 轴相切但不穿过，这称为重根或二重根。
Always check whether the factorisation is possible before using the quadratic formula. In many Edexcel exam questions, factorising is quicker and reveals the roots immediately.
在使用求根公式之前，务必先判断能否因式分解。在许多爱德思考试题中，因式分解更快，并能直接显示出根。
5. Discriminant and Number of Real Roots | 判别式与实根个数
The discriminant is the expression under the square root in the quadratic formula. It tells you how many real roots the equation ax² + bx + c = 0 has.
判别式是求根公式中平方根号下的表达式。它告诉你方程 ax² + bx + c = 0 有多少个实根。
Δ = b² − 4ac
判别式公式：Δ = b² − 4ac。
The discriminant directly describes the graph: if Δ > 0, the parabola crosses the x-axis twice; if Δ = 0, it touches the x-axis once; if Δ < 0, it does not meet the x-axis at all.
判别式直接描述图像：若 Δ > 0，抛物线与 x 轴相交两次；若 Δ = 0，它与 x 轴相切一次；若 Δ < 0，它与 x 轴完全不相交。
- Δ > 0: two distinct real roots, the graph crosses the x-axis twice.
- Δ > 0：两个不同实根，图像与 x 轴相交两次。
- Δ = 0: one repeated real root, the graph touches the x-axis at one point.
- Δ = 0：一个重根，图像与 x 轴相切于一点。
- Δ < 0: no real roots, the graph does not meet the x-axis.
- Δ < 0：没有实根，图像不与 x 轴相交。
In Edexcel exam questions, always state the discriminant condition clearly before giving your conclusion about the number of roots.
在爱德思考试题中，给出关于根的个数的结论之前，务必先清楚地写出判别式条件。
6. Completing the Square and the Turning Point | 配方法与顶点
Completing the square rewrites y = ax² + bx + c in vertex form: y = a(x − h)² + k. The turning point, or vertex, is then (h, k).
配方法将 y = ax² + bx + c 改写为顶点式：y = a(x − h)² + k。那么顶点（转折点）为 (h, k)。
For example, y = x² − 6x + 5 can be written as y = (x − 3)² − 4, so the vertex is (3, −4). Since a > 0, this vertex is a minimum point.
例如，y = x² − 6x + 5 可以写成 y = (x − 3)² − 4，所以顶点为 (3, −4)。由于 a > 0，该顶点为最小值点。
If a < 0, the vertex is a maximum. The completed square form is essential when a question asks for the maximum or minimum value of a quadratic expression.
如果 a < 0，顶点为最大值点。当题目要求求二次表达式的最大值或最小值时，配方式至关重要。
7. Axis of Symmetry | 对称轴
Every quadratic graph is symmetric about a vertical line passing through its vertex. If the vertex is (h, k), the axis of symmetry has equation x = h.
每个二次函数图像都关于经过顶点的竖直线对称。如果顶点为 (h, k)，则对称轴方程为 x = h。
Using the general form, the axis of symmetry is x = −b / 2a. This value is also the x-coordinate of the turning point.
由一般式可知，对称轴为 x = −b / 2a。这个值同时也是顶点（转折点）的 x 坐标。
x = −b ÷ 2a
对称轴公式：x = −b ÷ 2a。
The axis of symmetry helps you sketch the graph accurately: once you know one side of the parabola, the other side is its mirror image.
对称轴有助于准确画图：一旦知道抛物线的一侧，另一侧就是它的镜像。
8. Step-by-Step Sketching | 二次函数图像绘制步骤
To sketch a quadratic accurately in an exam, follow these key steps. They will help you show all the essential features that examiners look for.
在考试中要准确画出二次函数图像，请遵循以下关键步骤。它们能帮助你展示考官希望看到的所有重要特征。
- Step 1: Write y = ax² + bx + c and check the sign of a.
- 第一步：写出 y = ax² + bx + c 并检查 a 的符号。
- Step 2: Put x = 0 to find the y-intercept (0, c).
- 第二步：令 x =
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