G-2 Workbook 230: Mastering Quadratic Functions | G-2练习册230：掌握二次函数

📚 G-2 Workbook 230: Mastering Quadratic Functions | G-2练习册230：掌握二次函数

Welcome to this focused revision guide for IGCSE Mathematics. In this article, we explore the key ideas behind quadratic functions, which frequently appear in exercises such as those found in Workbook G-2, page 230. We will break down the essential skills step by step, using clear examples and exam-style tips.

欢迎阅读这本IGCSE数学专题复习指南。在本文中，我们深入探讨二次函数这一核心概念，它在G-2练习册第230页等习题中频繁出现。我们将逐步分解关键技能，结合清晰示例和考试建议进行讲解。

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

A quadratic function is a polynomial of degree 2. Its standard form is f(x) = ax² + bx + c, where a, b and c are constants, and a ≠ 0.

二次函数是最高次数为2的多项式函数。它的标准形式为 f(x) = ax² + bx + c，其中 a、b、c 为常数，且 a ≠ 0。

The graph of a quadratic function is called a parabola. The sign of a determines whether the parabola opens upward (a > 0) or downward (a < 0).

二次函数的图像称为抛物线。a 的符号决定抛物线的开口方向：当 a > 0 时开口向上，当 a < 0 时开口向下。

In the exercises on page 230, you will often be asked to identify a, b and c from a given equation before proceeding to graphing or solving.

在第230页的练习中，你通常需要先从给定方程中识别出 a、b、c，然后再进行画图或求解。

Degree 2: the highest power of x is x².
最高次数为2：x 的最高次幂为 x²。
Leading coefficient a: controls the direction and width of the parabola.
首项系数 a：控制抛物线的开口方向和宽度。
Constant term c: gives the y-intercept.
常数项 c：给出与 y 轴的交点。

2. Graphs of Quadratic Functions | 二次函数的图像

To sketch a parabola, you need three key features: the y-intercept, the x-intercepts (if they exist), and the vertex.

画抛物线草图需要三个关键要素：与 y 轴的交点、与 x 轴的交点（若存在）以及顶点。

The y-intercept is found by setting x = 0. This gives y = c. So the point (0, c) always lies on the graph.

令 x = 0 即可找到 y 轴截距，此时 y = c。因此点 (0, c) 始终在图像上。

To find x-intercepts, set f(x) = 0 and solve the equation ax² + bx + c = 0. The number of real solutions tells you how many times the curve crosses the x-axis.

求 x 轴截距时，令 f(x) = 0，并解方程 ax² + bx + c = 0。实数解的个数告诉你曲线与 x 轴相交的次数。

For example, the graph of f(x) = x² – 4x + 3 has a y-intercept at (0, 3). Solving x² – 4x + 3 = 0 gives x = 1 and x = 3, so the parabola crosses the x-axis at these two points.

例如，函数 f(x) = x² – 4x + 3 的图像与 y 轴交于点 (0, 3)。解方程 x² – 4x + 3 = 0 得 x = 1 和 x = 3，因此抛物线在这两点穿过 x 轴。

3. The Vertex and Axis of Symmetry | 顶点与对称轴

Every parabola has a vertical line of symmetry that passes through its vertex. The x-coordinate of the vertex is given by:

每条抛物线都有一条穿过顶点的竖直对称轴。顶点的 x 坐标由下式给出：

x = -b / (2a)

Once you know this x-value, substitute it back into the original equation to find the y-coordinate of the vertex.

求得 x 值后，将其代回原方程即可得到顶点的 y 坐标。

For f(x) = x² – 4x + 3, we have a = 1 and b = -4. Thus the axis of symmetry is x = 2. Substituting x = 2 gives f(2) = 4 – 8 + 3 = -1, so the vertex is (2, -1).

对于 f(x) = x² – 4x + 3，a = 1，b = -4。因此对称轴为 x = 2。将 x = 2 代入得 f(2) = 4 – 8 + 3 = -1，所以顶点为 (2, -1)。

If a > 0, this vertex is the minimum point. If a < 0, the vertex is the maximum point.

若 a > 0，该顶点为最低点；若 a < 0，则顶点为最高点。

4. Solving by Factoring | 因式分解法求解

One of the quickest ways to solve ax² + bx + c = 0 is by factoring the quadratic expression into two linear factors.

解 ax² + bx + c = 0 最快的方法之一是将二次表达式因式分解为两个一次因式的乘积。

The principle is: if p × q = 0, then p = 0 or q = 0. This is the zero product property.

其原理是：若 p × q = 0，则 p = 0 或 q = 0。这就是零乘积性质。

Example: solve x² – 5x + 6 = 0. We look for two numbers that multiply to 6 and add to -5. These are -2 and -3. So the equation becomes (x – 2)(x – 3) = 0. Hence x = 2 or x = 3.

例：解 x² – 5x + 6 = 0。我们寻找两个数，它们相乘得 6，相加得 -5。这两个数是 -2 和 -3。因此方程变为 (x – 2)(x – 3) = 0。于是 x = 2 或 x = 3。

In Workbook G-2, you are often asked to factorise first, then solve. Always check if the equation is in standard form before factorising.

在G-2练习册中，你常被要求先因式分解再求解。因式分解前务必检查方程是否已化为标准形式。

5. Using the Quadratic Formula | 二次公式法

When factorising is difficult or impossible, use the quadratic formula. For ax² + bx + c = 0, the solutions are:

当因式分解困难或无法进行时，使用二次公式。对于 ax² + bx + c = 0，其解为：

x = ( -b ± √(b² – 4ac) ) / (2a)

Simplify the expression under the square root first, then compute the two possible values of x.

先化简根号内的表达式，再计算 x 的两个可能值。

Example: solve 2x² + 3x – 2 = 0. Here a = 2, b = 3, c = -2. The discriminant is b² – 4ac = 9 + 16 = 25. Therefore x = (-3 ± √25) / (4) = (-3 ± 5) / 4. So x = 0.5 or x = -2.

例：解 2x² + 3x – 2 = 0。这里 a = 2，b = 3，c = -2。判别式为 b² – 4ac = 9 + 16 = 25。因此 x = (-3 ± √25) / 4 = (-3 ± 5) / 4。所以 x = 0.5 或 x = -2。

Always write the formula clearly and show your substitution steps. This helps you avoid sign errors.

务必清楚写出公式并展示代入步骤，这有助于避免符号错误。

6. Completing the Square and the Discriminant | 配方法与判别式

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This is useful for finding the vertex and solving equations.

配方法将 ax² + bx + c 改写为 a(x + p)² + q 的形式。这在求顶点和解方程时非常有用。

For x² + bx, the missing constant is (b/2)². For example, x² – 6x + 5 = (x – 3)² – 4.

对于 x² + bx，所需补上的常数为 (b/2)²。例如，x² – 6x + 5 = (x – 3)² – 4。

The discriminant, Δ = b² – 4ac, tells us the nature of the roots:

判别式 Δ = b² – 4ac 告诉我们根的性质：

Δ > 0: two distinct real roots.
Δ > 0：有两个不相等的实数根。
Δ = 0: one repeated real root.
Δ = 0：有一个二重实数根。
Δ < 0: no real roots.
Δ < 0：没有实数根。

In Workbook 230 problems, the discriminant often determines whether you need to sketch, solve, or state the number of roots.

在第230页的练习中，判别式常常决定你需要画图、求解，还是直接说明根的个数。

7. Applications of Quadratic Functions | 二次函数的应用

Quadratic functions model many real-world situations, such as projectile motion, area optimisation and profit calculations.

二次函数可用于模拟许多现实情境，例如抛体运动、面积最优以及利润计算。

For a rectangle with fixed perimeter, the area can be written as a quadratic function of side length. The maximum area occurs at the vertex of the parabola.

对于周长固定的矩形，其面积可以表示为边长的二次函数。最大面积出现在抛物线的顶点处。

Example: a rectangular field is to be fenced with 200 m of fencing. If one side is x metres, the other is 100 – x metres. The area is A(x) = x(100 – x) = 100x – x². The maximum occurs at x = 50, giving an area of 2500 m².

例：用200米围栏围一个矩形场地。若一边长为 x 米，则另一边为 100 – x 米。面积为 A(x) = x(100 – x) = 100x – x²。最大值出现在 x = 50 处，此时面积为 2500 平方米。

Always define your variables clearly when tackling word problems from the workbook.

在解答练习册中的文字题时，一定要先清晰定义变量。

8. How to Tackle Workbook 230 Problems | 如何应对第230页的练习题

The exercises on page 230 typically combine graphing with algebra. Start by reading the question carefully and deciding whether you need the vertex, roots, intercepts, or all three.

第230页的练习通常将画图与代数结合。先仔细审题，判断题目要求的是顶点、根、截距，还是三者都要。

A reliable method is:

一个可靠的方法是：

Write the equation in standard form ax² + bx + c.
将方程化为标准形式 ax² + bx + c。
Find the axis of symmetry using x = -b/(2a).
用 x = -b/(2a) 求对称轴。
Calculate the vertex by substituting this x-value.
代入该 x 值计算顶点。
Find the y-intercept and, if possible, the x-intercepts.
求 y 截距；如果可能，再求 x 截距。
Sketch the parabola using these points.
用这些点画出抛物线草图。

Check your solutions by substituting them back into the original equation. This takes only a few seconds and catches many errors.

将解代回原方程进行检验。这只需几秒钟，却能避免许多错误。

If the problem gives the graph and asks for the equation, read the vertex and one other point from the graph, then use the vertex form or standard form to determine the coefficients.

如果题目给出图像并要求写出方程，从图像中读出顶点和另一个点，然后利用顶点式或标准式确定系数。

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G-2 Workbook 230: Mastering Quadratic Functions | G-2练习册230：掌握二次函数