📚 Quadratic Functions and Their Graphs | 二次函数及其图像
A quadratic function is one of the most important topics in IGCSE Mathematics. It appears in algebra, graphs, and problem-solving questions, often worth a significant number of marks.
二次函数是 IGCSE 数学中最重要的考点之一。它贯穿代数、图像和实际问题求解，通常在考试中占较大分值。
In this revision article, we will break down the key concepts into clear sections, including standard form, the shape of the parabola, the vertex, roots, discriminant, transformations, and application problems. Each section pairs an English explanation with a Chinese one to help you learn effectively.
在这篇复习文章中，我们将核心概念分为几个清晰的小节，包括标准形式、抛物线形状、顶点、根、判别式、图像变换和应用题。每节内容采用英文讲解与中文翻译配对的方式，帮助你高效学习。
1. Standard Form and General Form | 标准形式与一般形式
A quadratic function can be written in two common forms. The general form is \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a \neq 0 \). The term \( ax^2 \) is called the quadratic term.
二次函数有两种常见写法。一般形式为 y = ax² + bx + c，其中 a、b、c 为常数，且 a ≠ 0。 ax² 项称为二次项。
The vertex form is another useful representation: \( y = a(x – h)^2 + k \), where the vertex of the parabola is at the point \( (h, k) \). This form makes it easy to read the turning point directly.
顶点形式是另一种有用的表达：y = a(x – h)² + k，其中抛物线的顶点坐标为 (h, k)。这种形式能让你直接读出转折点。
The sign of \( a \) determines the direction of the parabola. If \( a > 0 \), the parabola opens upward and has a minimum point. If \( a < 0 \), it opens downward and has a maximum point.
a 的正负决定抛物线的开口方向。若 a > 0，抛物线开口向上，有最小值；若 a < 0，开口向下，有最大值。
2. The Shape of a Parabola | 抛物线的形状
The graph of a quadratic function is always a smooth, symmetric curve called a parabola. It has one turning point, called the vertex, and a vertical line of symmetry passing through the vertex.
二次函数的图像始终是一条平滑且对称的曲线，称为抛物线。它有一个转折点，称为顶点，并且有一条穿过顶点的竖直对称轴。
The simplest quadratic graph is \( y = x^2 \). Its vertex is at the origin, and its line of symmetry is the y-axis. Every other parabola is a transformation of this basic graph.
最简单的二次函数图像是 y = x²。它的顶点在原点，对称轴为 y 轴。所有其他抛物线都是这个基本图像的变换。
When drawing a parabola, you should plot at least five points: the vertex, two points to the left, and two points to the right. This ensures the curve is accurate and symmetrical.
在画抛物线时，至少需要描五个点：顶点、左侧两个点和右侧两个点。这样可以确保曲线准确且对称。
3. The Vertex and Axis of Symmetry | 顶点与对称轴
The vertex is the highest or lowest point of the parabola. You can find its x-coordinate using the formula \( x = -b/(2a) \). Once you have this value, substitute it back into the equation to find the y-coordinate.
顶点是抛物线的最高点或最低点。你可以使用公式 x = -b/(2a) 求顶点的 x 坐标。得到该值后，将其代回原方程即可求出 y 坐标。
For example, consider \( y = 2x^2 – 8x + 3 \). Here \( a = 2 \) and \( b = -8 \), so the x-coordinate of the vertex is \( x = -(-8)/(2 \times 2) = 2 \).
例如，对于 y = 2x² – 8x + 3，其中 a = 2，b = -8，所以顶点的 x 坐标为 x = -(-8)/(2×2) = 2。
Substituting \( x = 2 \) into the equation gives \( y = 2(2)^2 – 8(2) + 3 = 8 – 16 + 3 = -5 \). Therefore, the vertex is at \( (2, -5) \), and the axis of symmetry is the line \( x = 2 \).
将 x = 2 代入方程得 y = 2(2)² – 8(2) + 3 = 8 – 16 + 3 = -5。因此，顶点坐标为 (2, -5)，对称轴为直线 x = 2。
4. Finding Roots by Factorisation | 用因式分解求根
The roots of a quadratic equation are the x-values where the graph crosses the x-axis, i.e. where \( y = 0 \). One method to solve \( ax^2 + bx + c = 0 \) is factorisation.
二次方程的根是图像与 x 轴交点处的 x 值，即 y = 0 时的 x 值。求解 ax² + bx + c = 0 的方法之一是因式分解。
For example, solve \( x^2 – 5x + 6 = 0 \). We look for two numbers that multiply to 6 and add to -5. These numbers are -2 and -3, so we write \( (x – 2)(x – 3) = 0 \).
例如，解 x² – 5x + 6 = 0。我们需要寻找两个数，它们相乘得 6，相加得 -5。这两个数是 -2 和 -3，因此可写成 (x – 2)(x – 3) = 0。
Setting each factor equal to zero gives \( x = 2 \) or \( x = 3 \). These are the roots of the equation, and the graph crosses the x-axis at these two points.
令每个因式等于零，可得 x = 2 或 x = 3。这两个就是方程的根，图像在这两点与 x 轴相交。
5. The Quadratic Formula | 求根公式
Not every quadratic equation can be factorised easily. In such cases, you should use the quadratic formula, which works for any quadratic equation in the form \( ax^2 + bx + c = 0 \).
并非所有二次方程都能轻松因式分解。在这种情况下，你应该使用求根公式，它适用于任何形式为 ax² + bx + c = 0 的二次方程。
x = (-b ± √(b² – 4ac)) / (2a)
To use this formula, identify \( a \), \( b \), and \( c \) from the equation, then substitute them carefully. The symbol ± means you will get two possible values: one with addition and one with subtraction.
使用该公式时，先从方程中确定 a、b、c 的值，然后仔细代入。± 符号表示你会得到两个可能的值：一个用加法，一个用减法。
For example, solve \( 2x^2 + 3x – 2 = 0 \). Here \( a = 2 \), \( b = 3 \), and \( c = -2 \). Substituting into the formula gives \( x = 0.5 \) or \( x = -2 \).
例如，解 2x² + 3x – 2 = 0。其中 a = 2，b = 3，c = -2。代入公式可得 x = 0.5 或 x = -2。
6. The Discriminant | 判别式
The discriminant is the part of the quadratic formula under the square root, written as \( b^2 – 4ac \). It tells us how many real roots a quadratic equation has without solving it fully.
判别式是求根公式中根号内的部分，写作 b² – 4ac。它无需完整求解，就能告诉我们二次方程有多少个实根。
If \( b^2 – 4ac > 0 \), the equation has two distinct real roots. If \( b^2 – 4ac = 0 \), the equation has exactly one real root, also called a repeated root. If \( b^2 – 4ac < 0 \), the equation has no real roots.
若 b² – 4ac > 0，方程有两个不同的实根；若 b² – 4ac = 0，方程恰好有一个实根，也称为重根；若 b² – 4ac < 0，方程没有实根。
Graphically, this tells you how many times the parabola intersects the x-axis: twice, once, or not at all. You should memorise these three cases for the exam.
从图像上看，这告诉你抛物线与 x 轴相交的次数：两次、一次或零次。你应该在考试前牢记这三种情况。
7. Completing the Square | 配方法
Completing the square is another algebraic technique used to rewrite a quadratic expression in vertex form. It is especially useful for finding the maximum or minimum value of a function.
配方法是另一种代数技巧，用于将二次表达式改写为顶点形式。它在求函数最大值或最小值时特别有用。
To complete the square for \( x^2 + 6x + 5 \), take half of 6, which is 3. Write \( (x + 3)^2 – 9 + 5 = (x + 3)^2 – 4 \). Thus the vertex is at \( (-3, -4) \).
要对 x² + 6x + 5 配方，取 6 的一半即 3。写出 (x + 3)² – 9 + 5 = (x + 3)² – 4。因此顶点坐标为 (-3, -4)。
For expressions where the coefficient of \( x^2 \) is not 1, factor out that coefficient first, then complete the square inside the bracket. Practice this method until you are confident.
当 x² 的系数不为 1 时，先提出该系数，再在括号内配方。反复练习此方法，直到你变得自信为止。
8. Graph Transformations | 图像变换
Quadratic graphs can be transformed by changing their equation. Adding a constant shifts the graph vertically, while adding a value inside the bracket shifts it horizontally.
二次函数图像可以通过改变方程进行变换。在方程外加上常数会使图像上下移动，而在括号内加上数值则使其左右移动。
For example, \( y = x^2 + 2 \) shifts the basic parabola upward by 2 units, while \( y = (x – 3)^2 \) shifts it to the right by 3 units. A negative sign in front of \( a \) reflects the graph across the x-axis.
例如，y = x² + 2 将基本抛物线向上移动 2 个单位，而 y = (x – 3)² 将其向右移动 3 个单位。a 前面的负号会使图像关于 x 轴上下翻转。
These transformations are easy to test with specific points. Pick a known point on \( y = x^2 \), such as \( (1, 1) \), and see where it moves under the new rule.
这些变换可以通过选取特定点轻松检验。在 y = x² 上选取已知点，如 (1, 1)，观察它在新规则下移动到哪里。
9. Solving Quadratic Inequalities | 解二次不等式
Quadratic inequalities are solved by first finding the roots of the corresponding equation, then using a number line or a sketch of the graph to determine the solution interval.
解二次不等式时，首先求出对应方程的根，然后利用数轴或函数图像草图来确定解区间。
For example, solve \( x^2 – x – 6 < 0 \). First factorise to get \( (x - 3)(x + 2) < 0 \). The roots are \( x = 3 \) and \( x = -2 \).
例如，解 x² – x – 6 < 0。先因式分解得 (x - 3)(x + 2) < 0。两根为 x = 3 和 x = -2。
Since the parabola opens upward, the graph is below the x-axis between the roots. Therefore the solution is \( -2 < x < 3 \). Remember to check the direction of the inequality sign carefully.
因为抛物线开口向上，图像在两个根之间位于 x 轴下方。因此解为 -2 < x < 3。记得仔细检查不等号的方向。
10. Applications in Real-World Problems | 实际应用问题
Quadratic functions appear in many real-world situations, such as projectile motion, area optimisation, and profit calculations. IGCSE questions often frame these in word problem format.
二次函数出现在许多现实情境中，例如抛体运动、面积优化和利润计算。IGCSE 考题通常以应用题形式呈现。
For example, the height \( h \) of a ball after \( t \) seconds can be modelled by \( h = -5t^2 + 20t + 1 \). To find the maximum height, complete the square or use the vertex formula.
例如，球在 t 秒后的高度 h 可以用 h = -5t² + 20t + 1 建模。要求最大高度，可以使用配方法或顶点公式。
When solving such problems, clearly define the variables, write the equation, and interpret your result in the context of the question. A correct numerical answer without context may lose marks.
解答此类问题时，要明确变量含义，写出方程，并结合问题情境解释结果。只给出正确数值但不结合情境，可能会丢分。
11. Common Exam Mistakes | 常见考试错误
Many students lose marks on quadratic questions due to small but avoidable errors. One common mistake is forgetting that \( a \) cannot be zero in a quadratic equation.
许多学生在二次函数题目上丢分是因为一些细小但可避免的错误。一个常见错误是忘记二次方程中 a 不能为零。
Another mistake is incorrectly substituting negative values into the quadratic formula. Always write down your substitution step with brackets to avoid sign errors.
另一个错误是把负值代入求根公式时出错。务必写出带括号的代入步骤，以避免符号错误。
Finally, when solving word problems, do not forget to check whether your answer makes sense. For instance, a negative time or a negative length is impossible in most contexts.
最后，在解应用题时，不要忘记检查答案是否合理。例如，负时间或负长度在大多数情境中都是不可能的。
12. Exam Tips and Summary | 考试技巧与总结
Begin by identifying which form your quadratic equation is in. If you need the vertex, complete the square or use the formula \( x = -b/(2a) \). If you need roots, factorise or use the quadratic formula.
首先判断你的二次方程属于哪种形式。如果需要顶点，就配方或使用公式 x = -b/(2a)。如果需要根，就因式分解或使用求根公式。
Always draw a quick sketch of the parabola if the question allows. A simple graph can help you visualise the vertex, roots, and the direction of opening, reducing errors in algebraic manipulation.
只要题目允许，务必画一个抛物线草图。简单的图形能帮助你直观地看到顶点、根和开口方向，从而减少代数运算中的错误。
Practise past papers and familiarise yourself with the wording of quadratic questions. The key to mastering this topic is consistent practice and checking each step carefully.
多做历年真题，熟悉二次函数题目的表述方式。掌握这一主题的关键在于持续练习，并仔细检查每一步。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply