📚 Quadratic Functions and Equations | 二次函数与方程
Quadratic functions and equations are a central topic in IGCSE Mathematics. This article provides a clear, exam-focused guide to understanding, solving, and graphing quadratic expressions.
二次函数与方程是IGCSE数学的核心主题。本文将提供一份清晰、紧扣考点的指南，帮助你理解、求解并绘制二次表达式。
1. Standard Form of a Quadratic Function | 二次函数的标准形式
A quadratic function can be written in the standard form:
二次函数可以写成标准形式：
f(x) = ax² + bx + c
where a, b and c are constants, and a ≠ 0. The graph of a quadratic function is called a parabola.
其中a、b、c为常数，且a ≠ 0。二次函数的图像称为抛物线。
The coefficient a determines the direction of the parabola. If a > 0, it opens upwards; if a < 0, it opens downwards.
系数a决定抛物线的开口方向。若a > 0，开口向上；若a < 0，开口向下。
If a > 0, the parabola has a minimum turning point.
若a > 0，抛物线有最小值点。
If a < 0, the parabola has a maximum turning point.
若a < 0，抛物线有最大值点。
Understanding the roles of a, b and c is essential for sketching and transforming quadratic graphs.
理解a、b、c的作用对于画图和图形变换至关重要。
2. Solving Quadratic Equations by Factorisation | 因式分解法解二次方程
way to solve a quadratic equation of the form ax² + bx + c = 0, when simple factors exist.
因式分解通常是解形如ax² + bx + c = 0的二次方程最快的方法，前提是存在简单因式。
To factorise, look for two numbers that multiply to give ac and add to give b, then split the middle term.
因式分解时，寻找两个数，它们的乘积等于ac，和等于b，然后拆分中间项。
For example, solve x² + 5x + 6 = 0:
例如，解x² + 5x + 6 = 0：
(x + 2)(x + 3) = 0
So x = −2 or x = −3.
因此x = −2 或 x = −3。
Set each factor equal to zero: x + 2 = 0 or x + 3 = 0.
令每个因式等于零：x + 2 = 0 或 x + 3 = 0。
Solve each linear equation to find the roots.
解每个一次方程即可得到根。
Check your answers by substituting back into the original equation.
将答案代回原方程进行检验。
3. Completing the Square | 配方法
Completing the square rewrites a quadratic expression in the form a(x − h)² + k. This form reveals the vertex of the parabola directly.
配方法将二次表达式改写为a(x − h)² + k的形式。这种形式直接揭示抛物线的顶点。
For x² + bx + c, add and subtract (b/2)²:
对于x² + bx + c，加上并减去(b/2)²：
x² + bx + c = (x + b/2)² − (b/2)² + c
Example: Complete the square for x² + 6x + 1.
示例：对x² + 6x + 1配方。
x² + 6x + 1 = (x + 3)² − 9 + 1 = (x + 3)² − 8
Hence the vertex is (−3, −8).
因此顶点为(−3, −8)。
This method is also used to solve equations when factorisation is difficult.
当因式分解困难时，这种方法也用于解方程。
4. The Quadratic Formula | 求根公式
Any quadratic equation ax² + bx + c = 0 can be solved using the quadratic formula:
任何二次方程ax² + bx + c = 0都可以用求根公式求解：
x = (−b ± √b² − 4ac) / (2a)
This formula works for all quadratic equations, including those with irrational or imaginary roots.
这个公式适用于所有二次方程，包括含无理根或虚根的方程。
To use it, identify a, b and c, then substitute them into the formula carefully.
使用它时，先确定a、b、c，然后小心地代入公式。
First calculate the discriminant b² − 4ac.
先计算判别式b² − 4ac。
Then find the square root and simplify.
然后求平方根并化简。
Remember to show your working clearly in the exam.
考试中务必清晰展示计算步骤。
5. The Discriminant | 判别式
noted by Δ = b² − 4ac, tells us the number and type of roots without solving the equation.
判别式表示为Δ = b² − 4ac，无需解方程即可判断根的数量和类型。
Δ = b² − 4ac
If Δ > 0, the equation has two distinct real roots.
若Δ > 0，方程有两个不同的实数根。
If Δ = 0, the equation has exactly one real root (a repeated root).
若Δ = 0，方程恰有一个实数根（重根）。
If Δ < 0, the equation has no real roots (two complex roots).
若Δ < 0，方程没有实数根（有两个复数根）。
Graphically, the discriminant tells us whether the parabola crosses the x-axis, touches it, or never meets it.
在图像上，判别式告诉我们抛物线与x轴相交、相切还是永不相交。
6. Graphing Quadratic Functions | 二次函数图像
s a parabola. To sketch it, you need the vertex, the y-intercept, and the roots (if real).
y = ax² + bx + c的图像是抛物线。为了画草图，需要顶点、y轴截距以及根（如果存在）。
The y-intercept is simply the value of y when x = 0, which is c.
y轴截距是x = 0时y的值，即c。
The vertex can be found using x = −b/(2a). Substitute this x into the equation to find y.
顶点可由x = −b/(2a)求得。将这个x代入方程求y。
Plot the roots, the y-intercept and the vertex, then draw a smooth curve through these points.
标出根、y轴截距和顶点，然后通过它们画光滑曲线。
If a > 0, the curve opens upward.
若a > 0，曲线开口向上。
If a < 0, the curve opens downward.
若a < 0，曲线开口向下。
Always label the coordinates and the axis of symmetry.
始终标出坐标和对称轴。
7. Vertex Form and Transformations | 顶点式与图形变换
The vertex form of a quadratic is f(x) = a(x − h)² + k, where (h, k) is the vertex.
二次函数的顶点式为f(x) = a(x − h)² + k，其中(h, k)是顶点。
This form makes it easy to apply transformations to the graph of y = x².
这种形式使得对y = x²的图像进行变换变得容易。
The graph shifts right by h units if h > 0, and left by |h| units if h < 0.
若h > 0，图像向右平移h个单位；若h < 0，向左平移|h|个单位。
The graph shifts up by k units if k > 0, and down by |k| units if k < 0.
若k > 0，图像向上平移k个单位；若k < 0，向下平移|k|个单位。
The coefficient a stretches or compresses the graph vertically.
系数a使图像垂直拉伸或压缩。
3)² + 1 is y = x² shifted right 3 units, stretched vertically by 2, and shifted up 1 unit.
例如，f(x) = 2(x − 3)² + 1是把y = x²向右平移3个单位、垂直拉伸2倍、再向上平移1个单位。
8. Solving by Using the Graph | 利用图像求解
The roots of a quadratic equation correspond to the x-intercepts of its graph.
二次方程的根对应其图像的x轴截距。
To solve an equation graphically, draw the parabola and read off the coordinates where it crosses the x-axis.
要用图像法解方程，画出抛物线并读取它与x轴交点的坐标。
the equation is f(x) = k, draw the horizontal line y = k and find the intersection points.
如果方程是f(x) = k，则画水平线y = k，找出交点。
This method is useful for checking algebraic solutions, especially when roots are not integers.
这种方法对于验证代数解特别有用，尤其是当根不是整数时。
Example: y = x² − 2x − 3 has roots at x = 3 and x = −1.
例如：y = x² − 2x − 3的根为x = 3和x = −1。
9. Quadratic Inequalities | 二次不等式
Quadratic inequalities such as ax² + bx + c > 0 are solved by first finding the roots and then testing intervals.
二次不等式如ax² + bx + c > 0，先求根，再在各区间内进行测试。
Sketch the parabola or use a sign diagram to determine where the expression is positive or negative.
画出抛物线或使用符号图来确定表达式在何处为正或为负。
For example, solve x² − x − 6 > 0.
例如，解x² − x − 6 > 0。
(x − 3)(x + 2) > 0
The critical points are x = 3 and x = −2. Testing intervals gives x < −2 or x > 3.
临界点为x = 3和x = −2。对各区间测试得到x < −2 或 x > 3。
Remember to use open circles for strict inequalities and closed circles for ≤ or ≥.
记住：严格不等号用空心圆，≤或≥用实心圆。
10. Applications in Word Problems | 应用题
Quadratic equations frequently appear in real-life problems involving areas, projectile motion, and profit maximisation.
二次方程经常出现在涉及面积、抛体运动和利润最大化的实际生活问题中。
When solving word problems, define the unknown variable first, then set up the quadratic equation.
解应用题时，先定义未知变量，再建立二次方程。
Example: The area of a rectangle is 24 cm². Its length is 2 cm longer than its width. Find the width.
示例：矩形的面积是24 cm²。长比宽长2 cm。求宽。
Let x be the width, then the length is x + 2. So x(x + 2) = 24.
设宽为x，则长为x + 2。因此x(x + 2) = 24。
x² + 2x − 24 = 0
Factorise to get x = 4 (discard the negative solution).
因式分解得到x = 4（舍弃负数解）。
Always interpret the solutions in the context of the problem.
始终根据问题背景解释解的意义。
11. Sum and Product of Roots | 根的和与积
If α and β are the roots of ax² + bx + c = 0, then:
若α和β是二次方程ax² + bx + c = 0的根，则：
α + β = −b/a
αβ = c/a
These relationships allow us to form a quadratic equation from its roots without solving.
这些关系使得我们可以不需要求解就能从根构造二次方程。
For example, if the roots are 2 and 5, then the equation is:
例如，若根是2和5，则方程为：
x² − (2 + 5)x + (2 × 5) = 0
x² − 7x + 10 = 0
This is a quick check for factorisation problems.
这是因式分解题的一个快速检验方法。
12. Common Mistakes and Exam Tips | 常见错误与考试技巧
Below are common errors students make when dealing with quadratics, and how to avoid them.
以下是学生在处理二次方程时常见的错误以及如何避免它们。
Forgetting that a ≠ 0. If a = 0, the equation is not quadratic.
忘记a ≠ 0。若a = 0，方程就不是二次的。
Misplacing signs when using the quadratic formula. Always write down a, b, c first.
使用求根公式时弄错符号。务必先写出a、b、c。
Dropping the ± when taking square roots. x² = 9 gives x = ±3.
开平方时漏掉±。x² = 9得到x = ±3。
Not simplifying surds like √12 into 2√3.
没有将√12化简为2√3。
Sketching graphs without finding the vertex or intercepts.
画图时没有求顶点或截距。
Practice past-paper questions and always check your roots by substitution.
多做历年真题，并通过代入检验你的根。
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