The Quadratic Function | 二次函数

📚 The Quadratic Function | 二次函数

Quadratic functions are a central topic in IGCSE mathematics, appearing in everything from simple algebraic manipulation to advanced graph sketching and real-world optimisation problems. Mastering this topic requires a clear understanding of the standard form, the vertex form, and the powerful technique of completing the square.

二次函数是IGCSE数学中的核心话题，从简单的代数运算到复杂的图像描绘以及现实世界中的最优化问题，它无处不在。掌握这一主题需要清楚理解标准形式、顶点形式以及配方法这一强大技巧。

1. The Standard Form of a Quadratic Function | 二次函数的标准形式

A quadratic function is typically written in the standard form: y = ax² + bx + c, where a, b and c are constants and a ≠ 0. The coefficient ‘a’ determines the direction of the parabola; if a > 0, the parabola opens upwards, and if a < 0, it opens downwards.

二次函数通常写成标准形式：y = ax² + bx + c，其中a、b和c为常数，且a ≠ 0。系数a决定抛物线的开口方向；当a > 0时，抛物线开口向上；当a < 0时，抛物线开口向下。

The constant term ‘c’ represents the y-intercept, which is the point where the graph crosses the y-axis. The coefficient ‘b’ influences the position of the axis of symmetry and the vertex.

常数项c表示y轴截距，即图像与y轴相交的点。系数b影响对称轴和顶点的位置。

If a > 0, the graph has a minimum point.
如果a > 0，图像有一个最低点。
If a < 0, the graph has a maximum point.
如果a < 0，图像有一个最高点。

2. The Graph of y = ax² + bx + c | y = ax² + bx + c 的图像

The graph of a quadratic function is a smooth curve called a parabola. It is symmetric about a vertical line known as the axis of symmetry. The point where the parabola turns is called the vertex.

二次函数的图像是一条平滑曲线，称为抛物线。它关于一条垂直直线对称，这条直线称为对称轴。抛物线转向的点称为顶点。

To sketch the graph accurately, you need the y-intercept, the roots (if they exist), the axis of symmetry, and the vertex. For example, the axis of symmetry can be found using the formula: x = -b / (2a).

要准确画出图像，你需要y轴截距、根（若存在）、对称轴以及顶点。例如，对称轴可以通过公式x = -b / (2a)求得。

x = -b / (2a)

3. Completing the Square | 配方法

Completing the square is a method used to rewrite a quadratic expression from standard form into vertex form. This is extremely useful because it reveals the vertex and the axis of symmetry directly.

配方法是一种将二次表达式从标准形式改写为顶点形式的方法。这非常有用，因为它能直接揭示顶点和对称轴。

For a quadratic expression x² + bx, we add and subtract (b/2)². For example, given y = x² + 6x + 5, we complete the square as follows:

对于二次表达式x² + bx，我们加上并减去(b/2)²。例如，已知y = x² + 6x + 5，我们按如下方式配方：

x² + 6x becomes (x + 3)² – 9.
x² + 6x变成(x + 3)² – 9。
Then y = (x + 3)² – 4.
因此y = (x + 3)² – 4。

4. The Vertex Form | 顶点形式

The vertex form of a quadratic function is written as: y = a(x – h)² + k, where the vertex is at the point (h, k). The value of ‘a’ still tells you the direction of the parabola and whether the vertex is a maximum or a minimum.

二次函数的顶点形式写成：y = a(x – h)² + k，其中顶点位于点(h, k)。a的值仍然表明抛物线的开口方向，以及顶点是最低点还是最高点。

For instance, the function y = 2(x – 3)² + 1 has a vertex at (3, 1) and opens upwards because a = 2 > 0. This form also makes it easy to describe graph transformations.

例如，函数y = 2(x – 3)² + 1的顶点在(3, 1)，开口向上，因为a = 2 > 0。这种形式也便于描述图像变换。

5. Vertex Form and Graph Transformations | 顶点形式与图像变换

Starting from the basic graph y = x², you can apply a series of translations to obtain any quadratic graph. In vertex form, y = a(x – h)² + k, the value ‘h’ shifts the graph horizontally, and ‘k’ shifts it vertically.

我们从基本图像y = x²出发，可以应用一系列平移从而得到任意二次图像。在顶点形式y = a(x – h)² + k中，值h使图像水平移动，值k使图像垂直移动。

If h > 0, the graph shifts to the right of the origin.
如果h > 0，图像向右移动。
If h < 0, the graph shifts to the left.
如果h < 0，图像向左移动。
If k > 0, the graph shifts upwards.
如果k > 0，图像向上移动。

The coefficient ‘a’ can also cause a vertical stretch or compression. If |a| > 1, the graph becomes narrower; if 0 < |a| < 1, the graph becomes wider.

系数a还会引起垂直拉伸或压缩。如果|a| > 1，图像变窄；如果0 < |a| < 1，图像变宽。

6. Finding Roots by Factorising | 通过因式分解求根

The roots of a quadratic equation are the x-values where the graph crosses the x-axis, i.e. where y = 0. One method of solving a quadratic equation is by factorising, which works when the expression can be written as a product of two linear factors.

二次方程的根是图像与x轴相交时的x值，即y = 0处的x值。解二次方程的一种方法是因式分解，当表达式可以写成两个线性因子的乘积时，该方法有效。

Consider the equation x² – 5x + 6 = 0. This can be factorised to (x – 2)(x – 3) = 0, giving the solutions x = 2 or x = 3. These are the roots of the quadratic.

考虑方程x² – 5x + 6 = 0。它可以因式分解为(x – 2)(x – 3) = 0，得到解x = 2或x = 3。这就是二次方程的根。

7. The Quadratic Formula | 公式法

Not all quadratics can be factorised easily. In such cases, the quadratic formula can always be used to find the roots. The formula is given by:

并非所有二次方程都能轻松因式分解。在这种情况下，公式法总能用来求根。公式如下：

x = (-b ± √(b² – 4ac)) / (2a)

To use the formula, simply identify the values of a, b and c from the quadratic equation ax² + bx + c = 0, then substitute them into the formula. Carefully evaluate the discriminant b² – 4ac before taking the square root.

使用公式时，只需从二次方程ax² + bx + c = 0中确认a、b和c的值，然后将其代入公式即可。在开平方之前，请仔细计算判别式b² – 4ac。

8. The Discriminant and the Nature of Roots | 判别式与根的性质

The discriminant, denoted by Δ, is the part of the quadratic formula under the square root: Δ = b² – 4ac. It tells us how many real roots a quadratic equation has without solving it fully.

判别式用Δ表示，是二次公式中根号下的部分：Δ = b² – 4ac。它告诉我们二次方程拥有多少个实数根，而无需完全求解。

If Δ > 0, there are two distinct real roots.
如果Δ > 0，有两个不同的实数根。
If Δ = 0, there is exactly one repeated real root.
如果Δ = 0，有一个重根。
If Δ < 0, there are no real roots.
如果Δ < 0，没有实数根。

This concept also appears in problems involving the intersection of a line and a parabola, where the discriminant helps determine whether they intersect at two points, one point, or do not intersect at all.

这一概念也出现在涉及直线与抛物线交点的问题中，其中判别式有助于判断它们是相交于两点、一点，还是完全不相交。

9. Applications in Problem Solving | 二次函数在解题中的应用

Quadratic functions are used to model many real-life situations, such as the path of a projectile, profit functions, and the shape of archways. In IGCSE exams, you may be asked to set up a quadratic equation from a word problem and then solve it.

二次函数常用于模拟许多真实情境，例如投射物的路径、利润函数以及拱门的形状。在IGCSE考试中，你可能会被要求根据文字题建立二次方程并求解。

For example, a rectangle has a length of (x + 4) cm and a width of x cm. If its area is 45 cm², we can write the equation x(x + 4) = 45, which simplifies to x² + 4x – 45 = 0. Solving this gives the positive value x = 5.

例如，一个矩形的长为(x + 4) cm，宽为x cm。若其面积为45 cm²，我们可以写出方程x(x + 4) = 45，化简得x² + 4x – 45 = 0。解这个方程可得正数解x = 5。

10. Summary and Revision Tips | 总结与复习建议

The quadratic function is a versatile and heavily tested topic in IGCSE mathematics. The key is to be comfortable switching between the standard form, the vertex form, and the factorised form depending on what the problem requires.

二次函数是IGCSE数学中一个多变且频繁考查的话题。关键在于熟能生巧地在标准形式、顶点形式和因式分解形式之间切换，具体取决于题目的要求。

Always check the sign of the coefficient ‘a’ first to know whether the graph opens upwards or downwards. Then find the y-intercept, the roots, and the vertex to sketch the graph correctly. Practice completing the square until you can do it quickly and accurately.

做题时请先检查系数a的符号，以判断图像开口向上还是向下。然后求y轴截距、根和顶点，从而正确画出图像。多加练习配方法，直到你能够快速且准确地完成。

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The Quadratic Function | 二次函数