📚 Quadratic Functions and Graphs | 二次函数与图像
Quadratic functions are among the most important topics in the IGCSE Mathematics syllabus. They appear in algebra, coordinate geometry, and problem-solving questions. Mastering the standard form, factorisation, completing the square, and graph sketching will give you a strong foundation for both Paper 2 and Paper 4.
二次函数是 IGCSE 数学大纲中最重要的内容之一。它出现在代数、坐标几何和应用题中。掌握标准形式、因式分解、配方法以及图像绘制，将为你应对 Paper 2 和 Paper 4 打下坚实基础。
1. The General Form of a Quadratic Function | 二次函数的一般形式
x² + bx + c, where a, b and c are constants and a ≠ 0. The value of a determines whether the parabola opens upward or downward.
二次函数是指可以写成 f(x) = ax² + bx + c 形式的函数，其中 a、b、c 为常数，且 a ≠ 0。a 的值决定了抛物线的开口方向。
f(x) = ax² + bx + c, a ≠ 0
If a > 0, the parabola opens upward and has a minimum point.
如果 a > 0，抛物线开口向上，有最小值点。
If a < 0, the parabola opens downward and has a maximum point.
如果 a < 0，抛物线开口向下，有最大值点。
The curve is always symmetric about a vertical line called the axis of symmetry.
曲线始终关于一条称为对称轴的竖直线对称。
2. Roots and the Discriminant | 根与判别式
The roots of a quadratic equation are the x-values where the graph crosses the x-axis. They can be found by factorising, using the quadratic formula, or completing the square.
二次方程的根是图像与 x 轴交点处的 x 值。可以通过因式分解、求根公式或配方法来求解。
x = (−b ± √b² − 4ac) / 2a
The expression b² − 4ac is called the discriminant. It tells us how many real roots the equation has:
表达式 b² − 4ac 称为判别式。它告诉我们方程有多少个实数根：
| Discriminant b² − 4ac | Number of Real Roots | Graph Intersection with x-axis |
| Positive ( > 0 ) | Two distinct roots | Two intersection points |
| Zero ( = 0 ) | One repeated root | One touching point |
| Negative ( < 0 ) | No real roots | No intersection |
If discriminant > 0: two distinct real roots.
若判别式 > 0：两个不相等的实数根。
If discriminant = 0: one repeated root (graph touches the axis).
若判别式 = 0：一个重根（图像与轴相切）。
If discriminant < 0: no real roots.
若判别式 < 0：无实数根。
3. The Vertex (Turning Point) | 顶点（极值点）
Every parabola has exactly one vertex. This is the point where the curve changes direction. If the parabola opens upward, the vertex is the minimum point; if it opens downward, the vertex is the maximum point.
每条抛物线都只有一个顶点。这是曲线改变方向的点。如果抛物线开口向上，顶点为最小值点；如果开口向下，顶点为最大值点。
For f(x) = ax² + bx + c, the x-coordinate of the vertex is given by:
对于 f(x) = ax² + bx + c，顶点的 x 坐标为：
x = −b / 2a
Once you have this x-value, substitute it back into the original function to find the corresponding y-coordinate. The vertex is then written as (x, y).
求出 x 值后，将其代入原函数即可得到对应的 y 坐标。顶点坐标即为 (x, y)。
4. Axis of Symmetry | 对称轴
The axis of symmetry is a vertical line that passes through the vertex. It divides the parabola into two mirror-image halves.
对称轴是经过顶点的一条竖直线。它将抛物线分成互为镜像的两半。
x = −b / 2a
Notice that the axis of symmetry has the same equation as the x-coordinate of the vertex. This makes sense because the vertex always lies on the axis of symmetry.
注意，对称轴的方程与顶点 x 坐标的公式完全相同。这很合理，因为顶点始终位于对称轴上。
The axis of symmetry is always vertical.
对称轴始终是竖直线。
Every point on the parabola has a mirror point on the opposite side of this axis.
抛物线上的每一个点在对称轴的另一侧都有一个镜像点。
5. Completing the Square | 配方法
Completing the square is a powerful technique for rewriting a quadratic function in vertex form:
配方法是一种强大的技巧，可以将二次函数改写为顶点形式：
f(x) = a(x − h)² + k
In this form, the vertex is simply (h, k). The value of h is the x-coordinate of the vertex, and k is the y-coordinate (the minimum or maximum value of the function).
在这种形式下，顶点坐标就是 (h, k)。其中 h 是顶点的 x 坐标，k 是 y 坐标（函数的最小值或最大值）。
For example, to complete the square for f(x) = x² + 6x + 5:
例如，对 f(x) = x² + 6x + 5 配方：
x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4
Therefore, the vertex is at (−3, −4), and the axis of symmetry is x = −3.
因此，顶点坐标为 (−3, −4)，对称轴为 x = −3。
6. Sketching a Quadratic Graph | 绘制二次函数图像
To sketch a quadratic graph accurately, you need to identify four key features:
要准确地绘制二次函数图像，你需要确定四个关键特征：
| Feature | How to Find It |
| y-intercept | Set x = 0, read c from f(x) = ax² + bx + c |
| x-intercepts (roots) | Set f(x) = 0 and solve |
| Vertex | Use x = −b/2a, then substitute |
| Direction of opening | Look at the sign of a |
Plot the y-intercept first, then the x-intercepts if they exist, then the vertex.
先标出 y 轴截距，然后是 x 轴截距（如果存在），最后标出顶点。
Draw a smooth U-shaped or ∩-shaped curve through these points.
用平滑的 U 形或 ∩ 形曲线连接这些点。
Always label the vertex, axis of symmetry, and intercepts clearly.
始终清晰地标注顶点、对称轴和截距。
7. Solving Quadratic Equations by Factorisation | 用因式分解解二次方程
Factorisation is the fastest method when the quadratic expression can be factorised. For a quadratic in the form x² + bx + c, look for two numbers that multiply to give c and add to give b.
当二次表达式可以因式分解时，因式分解是最快的方法。对于 x² + bx + c 形式的二次式，找两个数，它们相乘等于 c，相加等于 b。
For example, solve x² − 5x + 6 = 0:
例如，解 x² − 5x + 6 = 0：
(x − 2)(x − 3) = 0
Therefore, x = 2 or x = 3.
因此，x = 2 或 x = 3。
Always factor out any common factor first before attempting to factorise the quadratic.
在尝试因式分解二次式之前，先提取公因式。
If the coefficient of x² is not 1, use the method of splitting the middle term or use the quadratic formula.
如果 x² 的系数不为 1，使用拆中项法或求根公式。
Check your answers by substituting back into the original equation.
通过代回原方程来检查答案。
8. The Quadratic Formula | 求根公式
When a quadratic equation cannot be factorised easily, use the quadratic formula. This formula works for all quadratic equations, including those with irrational roots.
当二次方程不易因式分解时，使用求根公式。这个公式适用于所有二次方程，包括具有无理数根的情况。
x = (−b ± √b² − 4ac) / 2a
For example, solve 2x² + 3x − 1 = 0:
例如，解 2x² + 3x − 1 = 0：
Here a = 2, b = 3, c = −1. Substituting into the formula gives:
这里 a = 2，b = 3，c = −1。代入公式得：
x = (−3 ± √(9 − 4 × 2 × (−1))) / 4 = (−3 ± √17) / 4
So x ≈ 0.281 or x ≈ −1.781 (to 3 decimal places).
因此 x ≈ 0.281 或 x ≈ −1.781（精确到三位小数）。
9. Applications in Real-World Problems | 实际应用问题
Quadratic functions model many real-world situations, such as projectile motion, area optimisation, and profit calculations. In IGCSE exams, these appear as word problems where you must form a quadratic equation and solve it.
二次函数可以建模许多实际情况，如抛体运动、面积优化和利润计算。在 IGCSE 考试中，这些问题以应用题的形式出现，你需要建立二次方程并求解。
en by h = −5t² + 20t + 1. To find when the ball hits the ground, solve −5t² + 20t + 1 = 0.
例如，球在 t 秒后的高度 h（米）由 h = −5t² + 20t + 1 给出。要求球落地的时间，即解 −5t² + 20t + 1 = 0。
Read the problem carefully and identify what variable is x and what expression is f(x).
仔细阅读题目，确定哪个变量是 x，哪个表达式是 f(x)。
Decide whether a negative root makes sense in the context of the problem.
判断负根在题目背景下是否有意义。
Always write the final answer with the correct units.
最终答案始终要写上正确的单位。
10. Graphs of y = (x − p)² + q and Transformations | y = (x − p)² + q 的图像与变换
ic graph y = x². The value of p shifts the graph horizontally, and q shifts it vertically.
顶点形式直接揭示了基本图像 y = x² 的变换。p 的值使图像水平移动，q 使图像垂直移动。
| Transformation | Equation | Effect |
| Horizontal shift right by p | y = (x − p)² | Vertex moves to (p, 0) |
| Horizontal shift left by p | y = (x + p)² | Vertex moves to (−p, 0) |
| Vertical shift up by q | y = x² + q | Vertex moves to (0, q) |
| Vertical shift down by q | y = x² − q | Vertex moves to (0, −q) |
A negative sign before the square term, y = −x², reflects the graph in the x-axis.
平方项前有负号，即 y = −x²，表示图像关于 x 轴翻转。
Combining transformations: y = (x − 3)² + 2 means shift right by 3 and up by 2.
组合变换：y = (x − 3)² + 2 表示向右平移 3 个单位，再向上平移 2 个单位。
11. Common Exam Pitfalls | 常见考试误区
Students often lose marks in quadratic function questions due to a few repeated mistakes. Being aware of these pitfalls can help you avoid them.
学生在二次函数题目中常因几个反复出现的错误而失分。意识到这些误区可以帮助你避免它们。
Forgetting that a ≠ 0 in a quadratic function.
忘记二次函数中 a ≠ 0 的条件。
Mixing up the sign of h in the vertex form y = a(x − h)² + k.
混淆顶点形式 y = a(x − h)² + k 中 h 的符号。
When factorising, forgetting to set each factor to zero separately.
因式分解后，忘记分别令每个因式为 0。
Substituting incorrectly into the quadratic formula, especially the sign of c.
代入求根公式时出错，尤其是 c 的符号。
Sketching the graph without finding the vertex.
绘制图像时没有求出顶点。
12. Exam Preparation Checklist | 考前复习清单
Use this checklist to confirm you are fully prepared for quadratic function questions in your IGCSE exam.
使用这份清单来确认你已经为 IGCSE 考试中的二次函数题目做好充分准备。
I can identify the values of a, b and c from a quadratic expression.
我能从二次表达式中识别 a、b 和 c 的值。
I can calculate the discriminant and state the number of roots.
我能计算判别式并判断根的个数。
I can find the vertex and axis of symmetry.
我能求出顶点坐标和对称轴。
I can complete the square and rewrite in vertex form.
我能用配方法将函数改写为顶点形式。
I can sketch a quadratic graph with all key features labelled.
我能绘制二次函数图像并标出所有关键特征。
I can solve quadratic equations by factorisation, formula and completing the square.
我能用因式分解、求根公式和配方法解二次方程。
I understand how transformations affect the graph of y = x².
我理解变换对 y = x² 图像的影响。
I can apply quadratic equations to solve word problems.
我能应用二次方程解决应用题。
Practice makes permanent. Make sure you attempt a wide variety of questions, including past paper questions, before your exam day.
熟能生巧。确保在考试前练习各类题目，包括历年真题。
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