Solving Quadratic Equations & Graphing Quadratic Functions | 解二次方程与绘制二次函数图像

📚 Solving Quadratic Equations & Graphing Quadratic Functions | 解二次方程与绘制二次函数图像

Quadratic equations and their graphs appear frequently in IGCSE Mathematics, testing both algebraic manipulation and visual interpretation. This article covers the standard methods for solving quadratic equations and the key steps for sketching accurate quadratic graphs.

二次方程及其图像在 IGCSE 数学中频繁出现，考查代数运算与图像解读能力。本文将涵盖解二次方程的标准方法，以及准确绘制二次函数图像的关键步骤。

1. General Form & Standard Form | 一般式与标准式

A quadratic equation is any equation of the form ax² + bx + c = 0, where a, b, and c are constants, with a ≠ 0. The corresponding quadratic function is given by f(x) = ax² + bx + c.

二次方程是任何形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。对应的二次函数为 f(x) = ax² + bx + c。

For graphing, the vertex form f(x) = a(x − h)² + k is also useful. Here, (h, k) is the vertex of the parabola.

在绘图时，顶点式 f(x) = a(x − h)² + k 也很有用，其中 (h, k) 是抛物线的顶点。

2. Solving by Factorisation | 因式分解法

When the quadratic expression factorises neatly, this is the quickest method. For example, solve x² − 5x + 6 = 0. We find two numbers that multiply to 6 and add to −5: −2 and −3. Thus (x − 2)(x − 3) = 0, so x = 2 or x = 3.

当二次表达式能够整齐分解时，因式分解法是最快捷的方法。例如，解 x² − 5x + 6 = 0。我们找到两个数相乘得 6、相加得 −5：−2 和 −3。于是 (x − 2)(x − 3) = 0，所以 x = 2 或 x = 3。

Always expand your factors to check, and remember: if the coefficient of x² is not 1, use the method of splitting the middle term or factor by grouping.

务必展开因式进行检验；若 x² 的系数不为 1，可使用拆分中间项或分组分解的方法。

3. Solving by Completing the Square | 配方法

Completing the square rewrites x² + bx + c as (x + p)² + q. For example, solve x² + 6x + 2 = 0:

配方法将 x² + bx + c 改写为 (x + p)² + q。例如，解 x² + 6x + 2 = 0：

x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7

Set (x + 3)² = 7, so x + 3 = ±√7, and x = −3 ± √7.

令 (x + 3)² = 7，则 x + 3 = ±√7，所以 x = −3 ± √7。

4. Solving by the Quadratic Formula | 公式法

The quadratic formula works for every quadratic equation, including those that do not factorise. For ax² + bx + c = 0:

二次公式适用于所有二次方程，包括无法因式分解的情形。对于 ax² + bx + c = 0：

x = [−b ± √(b² − 4ac)] / (2a)

The discriminant, Δ = b² − 4ac, determines the number of real roots:

判别式 Δ = b² − 4ac 决定实数根的数量：

Δ > 0: two distinct real roots
Δ > 0：两个不同的实数根
Δ = 0: one repeated real root
Δ = 0：一个重根
Δ < 0: no real roots
Δ < 0：没有实数根

5. The Graph of a Quadratic Function | 二次函数的图像

The graph of f(x) = ax² + bx + c is a parabola. If a > 0 it opens upwards (U-shaped); if a < 0 it opens downwards (∩-shaped). The line of symmetry is vertical, through the vertex.

二次函数 f(x) = ax² + bx + c 的图像是抛物线。当 a > 0 时开口向上（U 形）；当 a < 0 时开口向下（∩ 形）。对称轴是过顶点的竖直线。

Three key features are usually required: the y-intercept, the x-intercepts (roots), and the vertex.

通常需要确定三个关键特征：y 截距、x 截距（根）和顶点。

6. Finding the y-intercept | 求 y 截距

The y-intercept is the point where the curve crosses the y-axis. This occurs when x = 0. Substituting x = 0 gives f(0) = c. Therefore the y-intercept is (0, c).

y 截距是曲线与 y 轴的交点，此时 x = 0。代入 x = 0 得 f(0) = c。因此 y 截距为 (0, c)。

7. Finding the x-intercepts (Roots) | 求 x 截距（根）

The x-intercepts are the solutions of f(x) = 0, i.e. the roots of the quadratic equation. Use factorisation, completing the square, or the quadratic formula to find them. If the discriminant is negative, the curve does not cross the x-axis.

x 截距是方程 f(x) = 0 的解，即二次方程的根。可以使用因式分解、配方或公式法求解。若判别式为负，则曲线不与 x 轴相交。

8. Finding the Vertex | 求顶点

The x-coordinate of the vertex is the midpoint of the two roots, or can be found directly from the coefficients:

顶点的 x 坐标是两根的中点，也可以直接由系数求得：

x_vertex = −b / (2a)

Substitute this x-value into f(x) to find the y-coordinate. In vertex form f(x) = a(x − h)² + k, the vertex is simply (h, k).

将此 x 值代入 f(x) 即可得到 y 坐标。在顶点式 f(x) = a(x − h)² + k 中，顶点就是 (h, k)。

9. Sketching the Graph | 绘制草图

Follow these steps to sketch a quadratic graph accurately:

按以下步骤准确绘制二次函数草图：

Determine the direction of opening from the sign of a.
根据 a 的正负确定开口方向。
Plot the y-intercept (0, c).
标出 y 截距 (0, c)。
Solve f(x) = 0 to plot the x-intercepts (if any).
解 f(x) = 0 以标出 x 截距（若存在）。
Calculate and plot the vertex, then draw the axis of symmetry.
计算并标出顶点，然后画出对称轴。
Draw a smooth curve through these points.
用平滑曲线连接这些点。

10. Example: Sketch y = x² − 2x − 3 | 示例：绘制 y = x² − 2x − 3

Here a = 1 > 0, so the parabola opens upwards. The y-intercept is (0, −3). Solving x² − 2x − 3 = 0 gives (x − 3)(x + 1) = 0, so the x-intercepts are (3, 0) and (−1, 0).

此处 a = 1 > 0，抛物线开口向上。y 截距为 (0, −3)。解 x² − 2x − 3 = 0 得 (x − 3)(x + 1) = 0，所以 x 截距为 (3, 0) 和 (−1, 0)。

The vertex’s x-coordinate is x = −(−2)/(2×1) = 1. Substituting x = 1 gives y = 1 − 2 − 3 = −4. So the vertex is (1, −4). The axis of symmetry is x = 1.

顶点的 x 坐标为 x = −(−2)/(2×1) = 1。代入 x = 1 得 y = 1 − 2 − 3 = −4。所以顶点为 (1, −4)。对称轴为 x = 1。

Feature	Value	特征	数值
Direction	Upward	开口方向	向上
y-intercept	(0, −3)	y 截距	(0, −3)
x-intercepts	(3, 0), (−1, 0)	x 截距	(3, 0), (−1, 0)
Vertex	(1, −4)	顶点	(1, −4)

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

Quadratic equations model projectile motion, area optimisation, and profit functions. For example, the height h(t) of an object thrown upward is often h(t) = −5t² + 20t + 2. The maximum height occurs at the vertex.

二次方程可用于建模抛体运动、面积最优问题及利润函数。例如，物体上抛的高度 h(t) 常用 h(t) = −5t² + 20t + 2 表示，最大高度出现在顶点处。

When solving word problems, define your variable clearly, set up the quadratic equation, solve it, and check whether the answer makes sense in context.

解应用题时，请明确设定未知数，建立二次方程，求解后再检查答案是否符合实际情境。

12. Common Mistakes & Exam Tips | 常见错误与考试技巧

Do not forget that a quadratic equation may have two, one, or zero real solutions.
不要忘记二次方程可能有两个、一个或零个实数解。
When using the quadratic formula, write down a, b, c first and substitute carefully, especially with negative values.
使用公式法时，先写出 a、b、c，再仔细代入，尤其注意负号。
When sketching, always label the vertex, intercepts, and axis of symmetry.
画图时务必标出顶点、截距和对称轴。
If the coefficient of x² is negative, the graph opens downward — a common oversight.
若 x² 的系数为负，图像开口向下——这是常见疏忽。
Check your graph by testing an extra point, such as x = −2 or x = 3.
通过测试额外点（如 x = −2 或 x = 3）来检验图像。

By mastering these techniques, you can confidently solve any quadratic equation and sketch its graph accurately. Practice with past papers, and remember to show all working steps clearly.

掌握这些技巧后，你就能自信地求解任何二次方程并准确绘制其图像。请结合历年真题练习，并记住清晰展示完整的解题步骤。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Solving Quadratic Equations & Graphing Quadratic Functions | 解二次方程与绘制二次函数图像