Quadratic Functions and Their Graphs | 二次函数及其图像

📚 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, graph sketching, problem solving, and even in later calculus courses. In this article, you will learn everything you need to know about quadratic functions, from their standard form to their graphs, roots, and real-world applications, with clear bilingual explanations to support your revision.

二次函数是IGCSE数学中最重要的话题之一。它出现在代数、画图、解题以及后续的微积分课程中。在这篇文章中，你将学到关于二次函数的所有必备知识，从标准形式到图像、求根和实际应用，并通过清晰的中英双语解释来支持你的复习。

1. What Is a Quadratic Function? | 什么是二次函数？

A quadratic function is a polynomial of degree 2. Its general form is written as:

二次函数是次数为2的多项式。它的一般形式可以写成：

y = ax² + bx + c

Here, a, b and c are constants, and a ≠ 0. The highest power of x is 2, which gives the function its curved shape. If a = 0, the function becomes linear, not quadratic.

这里，a、b和c是常数，且a ≠ 0。x的最高次数是2，这使得函数呈现弯曲的形状。如果a = 0，函数就变成线性的，而不是二次的了。

Examples of quadratic functions include y = x², y = 2x² + 3x − 1, and y = −x² + 4.

二次函数的例子包括 y = x²，y = 2x² + 3x − 1 和 y = −x² + 4。

2. Standard Form and Key Features | 标准形式及关键特征

The standard form y = ax² + bx + c is the most common way to write a quadratic function. Each coefficient affects the shape and position of the graph.

标准形式 y = ax² + bx + c 是书写二次函数最常见的方式。每个系数都会影响图像的形状和位置。

a determines the direction of the parabola. If a > 0, the parabola opens upwards. If a < 0, it opens downwards.
a决定抛物线的开口方向。如果 a > 0，抛物线开口向上；如果 a < 0，开口向下。
b affects the position of the axis of symmetry and the vertex.
b影响对称轴和顶点的位置。
c is the y-intercept, meaning the point where the graph crosses the y-axis, at (0, c).
c是y截距，即图像与y轴的交点，坐标为 (0, c)。

Coefficient	Effect on Graph	系数的作用
a > 0	Opens upward, minimum point	开口向上，有最小值
a < 0	Opens downward, maximum point	开口向下，有最大值
c	y-intercept at (0, c)	y轴截距为 (0, c)

3. The Graph of a Quadratic Function (Parabola) | 二次函数图像（抛物线）

The graph of a quadratic function is called a parabola. It has a symmetrical curved shape with one turning point, which is either a maximum or a minimum.

二次函数的图像称为抛物线。它具有对称的弯曲形状，并且有一个转向点，这个点要么是最大值要么是最小值。

To sketch a parabola, you need three key pieces of information:

要画出抛物线草图，你需要三个关键信息：

The direction of the opening (determined by the sign of a).
开口方向（由a的符号决定）。
The vertex (the turning point).
顶点（转向点）。
The x-intercepts (roots) and y-intercept.
x轴截距（根）和y轴截距。

For example, to sketch y = x² − 2x − 3, find the y-intercept at (0, −3), factorise to get (x − 3)(x + 1), so the roots are x = 3 and x = −1. The vertex is at x = 1, with y = −4.

例如，画出 y = x² − 2x − 3 的草图，先找到y轴截距 (0, −3)，再因式分解得 (x − 3)(x + 1)，所以根是 x = 3 和 x = −1。顶点在 x = 1 处，此时 y = −4。

4. Finding Roots (Solutions) | 求根（解）

The roots of a quadratic equation are the values of x that make y = 0. Geometrically, they are the x-coordinates where the parabola crosses the x-axis.

二次方程的根是使 y = 0 的 x 值。在几何上，它们是抛物线与x轴交点的x坐标。

There are three common methods to find roots:

求根有三种常用方法：

Factorisation (factoring the expression into two binomials).
因式分解法（将表达式分解为两个二项式）。
Completing the square (rewriting in the form a(x − h)² + k).
配方法（改写为 a(x − h)² + k 的形式）。
Using the quadratic formula (used when factorisation is difficult or impossible).
使用求根公式（当因式分解困难或无法分解时使用）。

Not all quadratic equations have real roots. Some have two distinct real roots, some have one repeated root, and some have no real roots.

并非所有二次方程都有实数根。有些有两个不同的实数根，有些有一个重根，有些则没有实数根。

5. The Quadratic Formula | 求根公式

The quadratic formula works for every quadratic equation in the form ax² + bx + c = 0. It is given by:

求根公式适用于所有形式为 ax² + bx + c = 0 的二次方程，公式如下：

x = (−b ± √(b² − 4ac)) ÷ (2a)

Here, the symbol ± means you take two separate values: one with a plus sign and one with a minus sign.

这里的 ± 符号表示你要取两个不同的值：一个用加号，一个用减号。

For example, solve 2x² − 4x − 6 = 0 using the formula. Here a = 2, b = −4, c = −6. Substituting gives:

例如，用公式解 2x² − 4x − 6 = 0。这里 a = 2，b = −4，c = −6。代入得到：

x = (4 ± √(16 + 48)) ÷ 4 = (4 ± √64) ÷ 4 = (4 ± 8) ÷ 4

So x = (4 + 8) ÷ 4 = 3, and x = (4 − 8) ÷ 4 = −1. The roots are 3 and −1.

所以 x = (4 + 8) ÷ 4 = 3，x = (4 − 8) ÷ 4 = −1。根是3和−1。

6. Factorisation Method | 因式分解法

Factorisation is often the fastest method, but it only works when the equation can be written as a product of two linear expressions.

因式分解通常是最快的方法，但它只适用于方程可以写成两个线性表达式乘积的情况。

To factorise x² + bx + c, look for two numbers whose product is c and whose sum is b.

要对 x² + bx + c 进行因式分解，需要找到两个数，它们的乘积是 c，和是 b。

Example: Solve x² − 5x + 6 = 0. We need two numbers that multiply to 6 and add to −5. The numbers are −2 and −3. So:

例子：解 x² − 5x + 6 = 0。我们需要两个数，乘积为6，和为−5。这两个数是−2和−3。因此：

(x − 2)(x − 3) = 0

Then either x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.

于是 x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。

Remember: if the product of two factors is zero, at least one factor must be zero. This is called the zero product property.

记住：如果两个因式的乘积为零，那么至少有一个因式必须为零。这称为零积性质。

7. Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form a(x − h)² + k. This form directly shows the vertex of the parabola: (h, k).

配方法将二次表达式改写为 a(x − h)² + k 的形式。这种形式直接显示抛物线的顶点：(h, k)。

Here is the process for x² + 6x + 5:

下面是对 x² + 6x + 5 进行配方的过程：

Take half of the coefficient of x, which is 6 ÷ 2 = 3.
取x系数的一半，即 6 ÷ 2 = 3。
Square it: 3² = 9.
将其平方：3² = 9。
Rewrite: x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4.
重写：x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4。

So the vertex of y = x² + 6x + 5 is at (−3, −4).

因此 y = x² + 6x + 5 的顶点在 (−3, −4)。

This method is especially useful for solving equations and finding the maximum or minimum value of a function.

这种方法在解方程以及求函数最大值或最小值时特别有用。

8. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root sign: Δ = b² − 4ac. It tells us how many real roots a quadratic equation has.

判别式是求根公式中根号下的部分：Δ = b² − 4ac。它告诉我们二次方程有多少个实数根。

Value of Δ	Number of Real Roots	实数根的数量
Δ > 0	Two distinct real roots	两个不同的实数根
Δ = 0	One repeated root (two equal roots)	一个重根（两个相等的根）
Δ < 0	No real roots	没有实数根

For example, for x² + 4x + 5 = 0, Δ = 16 − 20 = −4 < 0, so the equation has no real roots.

例如，对于 x² + 4x + 5 = 0，Δ = 16 − 20 = −4 < 0，因此方程没有实数根。

The discriminant also tells you whether the parabola crosses the x-axis, touches it at one point, or never touches it.

判别式还告诉你抛物线是与x轴相交、在一点相切，还是永不相交。

9. Vertex and Axis of Symmetry | 顶点和对称轴

Every parabola has a vertical line of symmetry. This line passes through the vertex, and its equation is given by:

每条抛物线都有一条垂直对称轴。这条线通过顶点，其方程为：

x = −b ÷ (2a)

The x-coordinate of the vertex is exactly this value. To find the y-coordinate, substitute this x-value back into the original equation.

顶点的x坐标正是这个值。要找到y坐标，将这个x值代回原方程即可。

For example, for y = 2x² − 8x + 1, the axis of symmetry is x = 8 ÷ (2 × 2) = 2. Substituting x = 2 gives y = 2(4) − 16 + 1 = −7. So the vertex is (2, −7).

例如，对于 y = 2x² − 8x + 1，对称轴是 x = 8 ÷ (2 × 2) = 2。代入 x = 2 得到 y = 2(4) − 16 + 1 = −7。所以顶点是 (2, −7)。

Remember: if a > 0, the vertex is a minimum point; if a < 0, it is a maximum point.

记住：如果 a > 0，顶点是最小值点；如果 a < 0，顶点是最大值点。

10. Real-World Applications | 实际应用

Quadratic functions model many real-world situations, including projectile motion, profit maximisation, and area optimisation.

二次函数可以模拟许多现实情境，包括抛体运动、利润最大化和面积优化等。

For example, when an object is thrown upwards, its height h after time t can be modelled by h = −5t² + 20t + 2. The negative coefficient of t² means the parabola opens downward, so the vertex gives the maximum height.

例如，当一个物体被向上抛出时，其高度 h 与时间 t 的关系可以建模为 h = −5t² + 20t + 2。t²的系数为负意味着抛物线开口向下，因此顶点就是最大高度。

Using the vertex formula x = −b ÷ (2a), we get t = −20 ÷ (2 × −5) = 2 seconds. Then h = −5(4) + 40 + 2 = 22 metres. The maximum height is 22 metres.

使用顶点公式 x = −b ÷ (2a)，得到 t = −20 ÷ (2 × −5) = 2秒。然后 h = −5(4) + 40 + 2 = 22米。最大高度为22米。

Quadratic models are also used to find break-even points in business, where profit equals zero.

二次模型还用于计算商业中的盈亏平衡点，即利润为零时的点。

11. Exam Tips for IGCSE | IGCSE考试建议

To succeed in IGCSE quadratic function questions, follow these practical tips.

要在IGCSE二次函数问题中取得好成绩，请遵循以下实用建议。

Always check whether a is positive or negative before sketching the graph.
画图前先检查a是正还是负。
When factorising, always expand your answer to check that it is correct.
因式分解后，一定要展开你的答案来检查是否正确。
If a quadratic does not factorise easily, use the quadratic formula instead of guessing.
如果二次式不容易因式分解，就使用求根公式，不要猜测。
Remember to write down both solutions when using the ± sign.
使用 ± 符号时，务必写出两个解。
When solving word problems, translate the words into a quadratic equation first.
解决应用题时，先把文字转化为二次方程。
Use your calculator to check your roots if allowed.
如果允许，用计算器检查你的根。

Practice sketching graphs by identifying the y-intercept, roots, and vertex in that order.

练习画图时，按照y轴截距、根、顶点的顺序依次确定关键点。

Also, be careful with signs when substituting negative values into the quadratic formula.

另外，在将负值代入求根公式时要特别注意符号。

12. Summary and Final Checklist | 总结与最终清单

A quadratic function has the form y = ax² + bx + c with a ≠ 0. Its graph is a parabola, which opens upward or downward depending on the sign of a.

二次函数的表达式为 y = ax² + bx + c，且 a ≠ 0。它的图像是抛物线，根据 a 的正负决定开口向上或向下。

The roots of a quadratic equation can be found by factorisation, completing the square, or the quadratic formula. The discriminant Δ = b² − 4ac determines the nature of the roots.

二次方程的根可以通过因式分解、配方法或求根公式来求解。判别式 Δ = b² − 4ac 决定根的性质。

The vertex of the parabola is found at x = −b ÷ (2a), and it represents the maximum or minimum value of the function.

抛物线的顶点位于 x = −b ÷ (2a)，它表示函数的最大值或最小值。

Use this checklist before your exam:

考试前请使用这份清单：

Can you write a quadratic in standard form?
你能写出二次函数的标准形式吗？
Can you calculate the discriminant and interpret it?
你能计算判别式并解释其含义吗？
Can you factorise and solve a quadratic equation?
你能因式分解并解二次方程吗？
Can you complete the square to find the vertex?
你能通过配方法找到顶点吗？
Can you sketch a parabola using its key features?
你能利用关键特征画出抛物线草图吗？

Mastering quadratic functions opens the door to more advanced topics such as inequalities, graphs of polynomials, and optimisation. Keep practising, and remember: every parabola has a turning point, just like every learner finds their turning point in understanding.

掌握二次函数为更高级的话题打开了大门，如不等式、多项式图像和优化问题。继续练习，并记住：每条抛物线都有一个转向点，就像每位学习者都会在理解中找到自己的转折点一样。

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Quadratic Functions and Their Graphs | 二次函数及其图像