📚 Quadratic Functions and Equations | 二次函数与方程
The quadratic function and its associated equation are among the most important topics in IGCSE Mathematics. Mastering the methods of solving quadratic equations and sketching their graphs is essential for success in the Edexcel examinations.
二次函数及其相关方程是 IGCSE 数学中最重要的主题之一。掌握解二次方程的方法并会画出它们的图像，对于在 Edexcel 考试中取得好成绩至关重要。
1. The General Form of a Quadratic | 二次函数的一般形式
A quadratic expression is written in the form ax² + bx + c, where a, b and c are constants, and a ≠ 0. The highest power of the variable x is 2, which gives the function its characteristic curved shape.
二次表达式写成 ax² + bx + c 的形式，其中 a、b、c 是常数，且 a ≠ 0。变量 x 的最高次数是 2，这赋予了函数特有的弯曲形状。
For example, y = 2x² − 3x + 5 is a quadratic, while y = x³ + 2 is not.
例如，y = 2x² − 3x + 5 是二次函数，而 y = x³ + 2 不是。
2. Solving by Factorisation | 因式分解法
be solved easily. Set each factor equal to zero and solve the resulting linear equations.
当二次表达式可以写成两个一次因式的乘积时，方程 f(x) = 0 就可以轻松求解。令每个因式等于零，然后解所得的一次方程。
Example: Solve x² − 5x + 6 = 0. Factorise: (x − 2)(x − 3) = 0. Therefore x = 2 or x = 3.
例：解 x² − 5x + 6 = 0。因式分解得 (x − 2)(x − 3) = 0，所以 x = 2 或 x = 3。
If ab = 0, then a = 0 or b = 0.
若 ab = 0，则 a = 0 或 b = 0。
3. Solving by Completing the Square | 配方法
Completing the square rewrites the quadratic in the form a(x + p)² + q. This is useful for finding turning points and solving equations when factorisation is not straightforward.
配方法将二次式改写成 a(x + p)² + q 的形式。这在求顶点以及因式分解不方便时解方程非常有用。
Solve x² + 6x + 1 = 0. First write (x + 3)² − 9 + 1 = 0, so (x + 3)² = 8. Hence x + 3 = ±√8, giving x = −3 ± 2√2.
例：解 x² + 6x + 1 = 0。先写 (x + 3)² − 9 + 1 = 0，所以 (x + 3)² = 8。因此 x + 3 = ±√8，解得 x = −3 ± 2√2。
Remember that the vertex of a quadratic in the form a(x + p)² + q is at (−p, q).
注意，a(x + p)² + q 形式的二次函数的顶点在 (−p, q)。
4. The Quadratic Formula | 二次公式
dratic formula. This method works for all quadratics, including those that cannot be factorised.
对于任何二次方程 ax² + bx + c = 0，其解由二次公式给出。此方法适用于所有二次方程，包括无法因式分解的方程。
x = (−b ± √b² − 4ac) / (2a)
You must memorise this formula for the Edexcel IGCSE examination, as it is not always provided on the formula sheet.
你必须牢记这个公式，因为在 Edexcel IGCSE 考试中，公式表不一定会给出它。
5. The Discriminant and Nature of Roots | 判别式与根的性质
The expression b² − 4ac is called the discriminant. It tells us how many real roots the quadratic equation has, without solving it fully.
表达式 b² − 4ac 称为判别式。它告诉我们二次方程有多少个实数根，而无需完整求解。
If b² − 4ac > 0, there are two distinct real roots.
若 b² − 4ac > 0，则有两个不同的实根。
If b² − 4ac = 0, there is exactly one repeated real root.
若 b² − 4ac = 0，则有一个二重实根。
If b² − 4ac < 0, there are no real roots.
若 b² − 4ac < 0，则没有实根。
This is closely connected to how many times the graph of the quadratic crosses the x-axis.
这一点与二次函数图像与 x 轴的交点个数密切相关。
6. Sum and Product of Roots | 根的和与积
= 0 with roots α and β, the sum of the roots is α + β = −b/a, and the product is αβ = c/a.
对于根为 α 和 β 的二次方程 ax² + bx + c = 0，根的和为 α + β = −b/a，根的积为 αβ = c/a。
to form a quadratic equation from its roots. The equation is x² − (sum)x + (product) = 0.
利用这些关系，我们可以从根构造一个二次方程：x² − (和)x + (积) = 0。
7. Sketching Quadratic Graphs | 绘制二次函数图像
ax² + bx + c is a parabola. When a > 0, it opens upwards; when a < 0, it opens downwards.
y = ax² + bx + c 的图像是抛物线。当 a > 0 时开口向上；当 a < 0 时开口向下。
ots), and the turning point. The axis of symmetry is the vertical line through the turning point.
画图像时，你需要 y 轴截距（x = 0）、x 轴截距（根）以及顶点。对称轴是经过顶点的竖直直线。
the roots are 1 and 3. The axis of symmetry is x = 2, and the turning point is at (2, −1).
例如，对于 y = x² − 4x + 3，因式分解得 (x − 1)(x − 3)，因此根为 1 和 3。对称轴是 x = 2，顶点在 (2, −1)。
8. Key Features of the Graph | 图像的关键特征
A complete sketch of a quadratic should clearly show the following features:
完整的二次函数草图应清晰地显示以下特征：
| Feature | How to find it |
| y-intercept | Set x = 0, read c in ax² + bx + c |
| x-intercepts (real roots) | Solve ax² + bx + c = 0 |
| Turning point | Complete the square to get (p, q) |
| Axis of symmetry | x = −b/(2a) or the midline of roots |
These features are essential for solving graphical problems and for interpreting worded questions.
这些特征对于解决图像问题和理解应用题至关重要。
9. Finding a Quadratic from Its Graph | 由图像求二次函数
can write the function as y = a(x − r)(x − s). Substitute the known point to determine a.
如果已知二次函数的两个根 r 和 s 以及另一个点，你可以将函数写成 y = a(x − r)(x − s)。代入已知点即可确定 a 的值。
nt (h, k), use the form y = a(x − h)² + k. Then find a using any other point on the curve.
或者，如果已知顶点 (h, k)，可使用 y = a(x − h)² + k 的形式，然后用曲线上的另一个点求出 a。
10. Quadratic Inequalities | 二次不等式
Solving a quadratic inequality such as x² − 5x + 6 < 0 is closely linked to the graph of the quadratic. First solve the corresponding equation to find the critical values.
解二次不等式（如 x² − 5x + 6 < 0）与二次函数图像密切相关。先解对应的方程以找到临界值。
0. The graph lies below the x-axis between x = 2 and x = 3, so the solution is 2 < x < 3.
对于 x² − 5x + 6 < 0，因式分解得 (x − 2)(x − 3) < 0。图像在 x = 2 与 x = 3 之间位于 x 轴下方，所以解为 2 < x < 3。
For inequalities with >, the solution is usually x < r or x > s, where r and s are the roots.
对于含 > 的不等式，解通常是 x < r 或 x > s，其中 r 和 s 是根。
11. Applications of Quadratics | 二次函数的实际应用
n object thrown upwards can be modelled by h = −5t² + 20t + 2, where t is time in seconds.
二次方程出现在涉及面积、抛体运动和优化的问题中。例如，向上抛出的物体高度可以用 h = −5t² + 20t + 2 建模，其中 t 是时间（秒）。
atic formula. To find the maximum height, complete the square to locate the turning point.
要求物体何时落地，可用二次公式解 h = 0。要求最大高度，可通过配方法求顶点。
These real-world connections are frequently tested in the Edexcel IGCSE papers.
这些实际应用联系在 Edexcel IGCSE 试卷中经常出现。
12. Common Pitfalls and Exam Tips | 常见错误与考试技巧
the quadratic formula. Always rearrange the equation into the form ax² + bx + c = 0 first.
学生在因式分解或使用二次公式前，经常忘记将方程化为等于零的形式。务必先将方程整理为 ax² + bx + c = 0。
When sketching, remember to label the turning point and all intercepts. Use a smooth curve, not a set of straight lines.
画草图时，记得标出顶点和所有截距。用平滑曲线画出，而不是连成直线。
Always check whether the discriminant is positive, zero, or negative before writing your final answer about the number of roots.
在写出关于根个数的最终答案前，务必先判断判别式是正、零还是负。
Practice using the quadratic formula on non-factorisable quadratics, as this is a favourite exam question.
练习用二次公式解不可因式分解的二次方程，这是考试中非常喜欢考查的题型。
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