Mastering IGCSE Practice Book G-5-100: Quadratic Graphs, Roots and Problem Solving | 攻克 IGCSE 练习册 G-5-100：二次函数图像、根与问题求解

📚 Mastering IGCSE Practice Book G-5-100: Quadratic Graphs, Roots and Problem Solving | 攻克 IGCSE 练习册 G-5-100：二次函数图像、根与问题求解

Practice Book item G-5-100 is a classic IGCSE Mathematics task that links quadratic algebra with graphical interpretation. It asks you to sketch a quadratic curve, find its roots, vertex and intercepts, then use the graph to solve related equations.

练习册第 G-5-100 题是一道经典 IGCSE 数学题，将二次代数与图像解读结合起来。题目要求画出二次曲线、求根、顶点和截距，并利用图像解相关方程。

1. Understanding the Task | 理解题意

² - 4x + 3. The question asks you to complete a table of values, plot the curve, and read solutions from the graph.

在 G-5-100 题中，通常会给你一个二次函数，例如 f(x) = x² - 4x + 3。题目要求你完成数值表、绘制曲线，并从图像中读取解。

This type of question tests three skills at once: algebraic manipulation, coordinate geometry, and graphical interpretation.

这类题同时考查三种技能：代数变形、坐标几何和图像解读。

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2. Key Vocabulary and Symbols | 关键术语与符号

tandard form of a quadratic is y = ax² + bx + c, where a, b and c are constants and a ≠ 0.

二次函数的标准形式为 y = ax² + bx + c，其中 a、b、c 为常数且 a ≠ 0。

The graph of a quadratic is called a parabola. If a > 0, the parabola opens upward and has a minimum point. If a < 0, it opens downward and has a maximum point.

二次函数的图像称为抛物线。若 a > 0，抛物线开口向上且有最小值点；若 a < 0，开口向下且有最大值点。

Term	中文术语
Root	根
Vertex	顶点
Axis of symmetry	对称轴
Discriminant	判别式

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3. Writing the Quadratic in Standard Form | 将二次式写成标准形式

writing the given expression in the form f(x) = x² - 4x + 3. Here a = 1, b = -4 and c = 3.

在 G-5-100 中，先将给定表达式写成 f(x) = x² - 4x + 3。这里 a = 1，b = -4，c = 3。

e square is often useful: x² - 4x + 3 = (x - 2)² - 1. This immediately reveals the vertex.

配方法通常很有用：x² - 4x + 3 = (x - 2)² - 1。这样可以立即看出顶点。

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4. Finding the y-intercept | 求 y 轴截距

0 into the function: f(0) = 0² - 4(0) + 3 = 3. So the curve crosses the y-axis at (0, 3).

求 y 轴截距时，将 x = 0 代入函数：f(0) = 0² - 4(0) + 3 = 3。因此曲线经过 y 轴上的点 (0, 3)。

This point is important because it gives one exact anchor for the sketch.

这一点很重要，因为它为画图提供了一个准确的定位点。

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5. Solving the Quadratic Equation for Roots | 解二次方程求根

To find the roots, set f(x) = 0: x² - 4x + 3 = 0. Factorise to get (x - 1)(x - 3) = 0.

求根时令 f(x) = 0：x² - 4x + 3 = 0。因式分解得到 (x - 1)(x - 3) = 0。

acket to zero gives x = 1 and x = 3. These are the points where the curve cuts the x-axis.

令每个括号等于零，得到 x = 1 和 x = 3。这些就是曲线与 x 轴相交的点。

Alternatively, you can use the quadratic formula.

也可以使用求根公式。

x = (-b ± √b² - 4ac) ÷ 2a

-4, c = 3, the discriminant is 16 - 12 = 4, so the roots are (4 ± 2) ÷ 2, giving 3 and 1.

代入 a = 1，b = -4，c = 3，判别式为 16 - 12 = 4，因此根为 (4 ± 2) ÷ 2，得到 3 和 1。

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6. Locating the Vertex | 确定顶点

+ bx + c, the x-coordinate of the vertex is given by x = -b ÷ 2a. Here x = -(-4) ÷ 2 = 2.

对于二次函数 y = ax² + bx + c，顶点的 x 坐标为 x = -b ÷ 2a。这里 x = -(-4) ÷ 2 = 2。

2 back into the function: f(2) = 2² - 4(2) + 3 = 4 - 8 + 3 = -1. So the vertex is (2, -1).

将 x = 2 代回函数：f(2) = 2² - 4(2) + 3 = 4 - 8 + 3 = -1。所以顶点为 (2, -1)。

is the axis of symmetry. Any point on the curve has a reflected partner across this line.

直线 x = 2 是对称轴。曲线上的任意一点都有关于这条直线的对称点。

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7. Sketching the Parabola | 绘制抛物线

Plot the key points: roots (1, 0) and (3, 0), y-intercept (0, 3), and vertex (2, -1). Then draw a smooth U-shaped curve through them.

标出关键点：根 (1, 0) 和 (3, 0)，y 轴截距 (0, 3)，以及顶点 (2, -1)。然后用平滑的 U 形曲线穿过这些点。

ens upward. Do not join the points with straight line segments; a parabola is smooth and continuous.

由于 a = 1 为正，曲线开口向上。不要用直线段连接各点；抛物线是平滑且连续的。

Label the curve with its equation y = x² - 4x + 3.

在曲线上标出方程 y = x² - 4x + 3。

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8. Using the Graph to Solve f(x) = k | 利用图像解 f(x) = k

3 = 0, read the x-values where the curve meets the line y = 0. These are x = 1 and x = 3.

画好图像后，你可以解相关方程。例如，解 x² - 4x + 3 = 0 时，读取曲线与直线 y = 0 交点的 x 值，即 x = 1

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