Quadratic Functions: Graphs and Properties Analysis | 二次函数图像与性质分析

📚 Quadratic Functions: Graphs and Properties Analysis | 二次函数图像与性质分析

Quadratic functions are one of the most fundamental topics in A-Level Mathematics. Understanding their graphs and algebraic properties is essential for solving equations, inequalities, and optimisation problems. This article provides a comprehensive analysis of quadratic functions, from basic definitions to advanced exam techniques.

二次函数是A-Level数学中最基础的内容之一。理解其图像与代数性质，对于求解方程、不等式和优化问题至关重要。本文将全面分析二次函数，从基本定义到高阶应试技巧。

1. Standard Form and Basic Definition | 标准形式与基本定义

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

二次函数是次数为2的多项式函数，其标准形式为：

f(x) = ax² + bx + c, where a, b, c are real constants and a ≠ 0

f(x) = ax² + bx + c，其中 a、b、c 为实数常数，且 a ≠ 0

The leading coefficient a must not be zero; if a = 0, the function degenerates into a linear function. The coefficient b contributes to the linear term, while c is a constant that determines the vertical position of the graph.

首项系数 a 必须不为零；若 a = 0，函数就会退化为一次函数。系数 b 贡献一次项，而 c 为常数项，决定了图像在竖直方向的位置。

All quadratic functions share a common graphical shape called a parabola. This section lays the groundwork for exploring how each coefficient affects the curve.

所有二次函数的图像都具有统一形状——抛物线。本小节为后续探究各系数对曲线的影响奠定基础。

2. Shape and Opening Direction | 抛物线的形状与开口方向

The graph of a quadratic function is a parabola. When a > 0, the parabola opens upward, forming a ‘U’ shape; when a < 0, it opens downward, forming an inverted 'U'. The sign of a is therefore the very first thing to check when analysing any quadratic.

二次函数的图像是一条抛物线。当 a > 0 时，抛物线开口向上，呈“U”形；当 a < 0 时，抛物线开口向下，呈倒“U”形。因此，分析任何二次函数时首先应确认 a 的符号。

The magnitude |a| controls the width of the parabola. A larger |a| produces a narrower curve, while a smaller |a| produces a wider, flatter curve. For instance, y = 5x² is much narrower than y = ½x².

|a| 的大小控制着抛物线的宽窄。|a| 越大，曲线越窄；|a| 越小，曲线越宽、越平坦。例如，y = 5x² 远比 y = ½x² 窄。

Every parabola is symmetric about a vertical line known as the axis of symmetry. This line passes through the vertex, which is the highest or lowest point of the curve depending on the sign of a.

每条抛物线都关于一条称为对称轴的竖直线对称。对称轴经过顶点，而顶点是曲线的最高点或最低点，具体取决于 a 的符号。

3. Axis of Symmetry and Vertex | 对称轴与顶点

For a quadratic function y = ax² + bx + c, the axis of symmetry is given by:

对于二次函数 y = ax² + bx + c，其对称轴为：

x = -b/(2a)

The vertex lies on this axis, so its x-coordinate is also -b/(2a). Substituting this value back into the function gives the y-coordinate of the vertex:

顶点位于对称轴上，因此其横坐标也是 -b/(2a)。将此值代回函数，可得到顶点的纵坐标：

y = (4ac – b²)/(4a)

The vertex is indispensable for sketching graphs, locating extrema and solving optimisation problems. For a > 0, it represents the minimum point; for a < 0, it represents the maximum point.

顶点在画图、求极值和解决优化问题时不可或缺。当 a > 0 时，它为最小值点；当 a < 0 时，它为最大值点。

4. Completing the Square | 配方法

Completing the square is a powerful algebraic technique that rewrites a quadratic in vertex form:

配方法是一种强大的代数技巧，可将二次函数改写为顶点式：

f(x) = a(x – h)² + k

where h = -b/(2a) and k = (4ac – b²)/(4a). This form makes the vertex (h, k) immediately visible and is especially useful for identifying transformations.

其中 h = -b/(2a)，k = (4ac – b²)/(4a)。此形式使顶点 (h, k) 一目了然，特别有助于识别图像变换。

Worked example: f(x) = 2x² – 8x + 5. First factor out 2 to obtain 2(x² – 4x) + 5, then complete the square inside the bracket: (x – 2)² – 4. Hence f(x) = 2(x – 2)² – 8 + 5 = 2(x – 2)² – 3, so the vertex is (2, -3).

例题：f(x) = 2x² – 8x + 5。先提出 2 得 2(x² – 4x) + 5，再在括号内配方：(x – 2)² – 4。因此 f(x) = 2(x – 2)² – 8 + 5 = 2(x – 2)² – 3，所以顶点为 (2, -3)。

5. Discriminant and Nature of Roots | 判别式与根的性质

The discriminant of a quadratic equation ax² + bx + c = 0 is defined as:

二次方程 ax² + bx + c = 0 的判别式定义为：

Δ = b² – 4ac

It determines the nature of the roots as follows:

它决定根的性质，具体如下：

Δ > 0: two distinct real roots | 两个不同实根
Δ = 0: one repeated real root | 一个重根
Δ < 0: no real roots (two complex conjugate roots) | 无实根（两个共轭复根）

Graphically, when Δ > 0 the parabola crosses the x-axis at two points; when Δ = 0 it touches the x-axis tangentially at exactly one point; when Δ < 0 it never intersects the x-axis.

从图像来看：Δ > 0 时，抛物线与 x 轴有两个交点；Δ = 0 时，抛物线与 x 轴相切于一点；Δ < 0 时，抛物线与 x 轴没有交点。

6. Intersections with Coordinate Axes | 与坐标轴的交点

The x-intercepts of y = ax² + bx + c are the solutions of the equation ax² + bx + c = 0. The quadratic formula yields:

抛物线 y = ax² + bx + c 与 x 轴的交点就是方程 ax² + bx + c = 0 的解。求根公式给出：

x = (-b ± √Δ)/(2a)

The y-intercept is simply the constant term c, because substituting x = 0 gives y = c. Combining the y-intercept, the vertex and the x-intercepts provides a reliable framework for sketching the graph by hand.

y 轴截距就是常数项 c，因为代入 x = 0 得 y = c。将 y 轴截距、顶点与 x 轴交点结合，可为手动画图提供可靠框架。

Vieta’s formulas relate the roots to the coefficients: for roots α and β, α + β = -b/a and αβ = c/a. These identities are frequently tested in A-Level examination questions.

韦达定理将根与系数联系起来：对根 α 与 β，有 α + β = -b/a，αβ = c/a。这些恒等式在A-Level考试题目中频繁出现。

7. Transformation of Quadratic Graphs | 二次函数图像的变换

Starting from the basic graph y = x², several standard transformations can be applied:

以基本图像 y = x² 为起点，可进行以下几种标准变换：

y = x² + k: vertical translation by k units | 竖直平移 k 个单位
y = (x – h)²: horizontal translation by h units | 水平平移 h 个单位
y = ax²: vertical stretch/compression by factor a | 竖直伸缩，倍数为 a
y = -x²: reflection in the x-axis | 关于 x 轴翻折

Combining these transformations produces the general vertex form. For example, y = 2(x – 3)² + 4 is obtained from y = x² by shifting right 3 units, stretching vertically by a factor of 2, and then shifting up 4 units.

这些变换组合起来就得到一般顶点式。例如，y = 2(x – 3)² + 4 可由 y = x² 先向右平移 3 个单位、竖直拉伸 2 倍、再向上平移 4 个单位得到。

When a question describes a transformation, always track the vertex as an anchor point. The vertex moves exactly according to the translation vectors, while the shape changes according to the stretch factors.

当题目描述变换时，始终以顶点作为参照点。顶点按照平移向量精确移动，而曲线形状根据伸缩因子改变。

8. Maximum and Minimum Values | 最大值与最小值

The vertex gives the extremum of a quadratic function. If a > 0, the minimum value is:

顶点给出二次函数的极值。若 a > 0，最小值为：

f(-b/(2a)) = (4ac – b²)/(4a)

If a < 0, this same value represents the maximum. Since the parabola extends to ±∞, there is no global maximum when a > 0 and no global minimum when a < 0.

若 a < 0，该值即为最大值。由于抛物线向 ±∞ 延伸，当 a > 0 时没有全局最大值，当 a < 0 时没有全局最小值。

In optimisation problems, the x-coordinate of the vertex represents the quantity that optimises the outcome, while the y-coordinate gives the optimal value. This principle is widely applied in economics, physics and engineering contexts.

在优化问题中，顶点横坐标代表使结果最优的量，纵坐标就是最优值。这一原理广泛应用于经济、物理与工程等场景。

9. Quadratic Inequalities | 二次不等式

To solve ax² + bx + c > 0 or ax² + bx + c < 0, sketch the parabola and read off the solution intervals. Assuming a > 0 and Δ > 0 with roots α < β:

求解 ax² + bx + c > 0 或 ax² + bx + c < 0 时，先画抛物线草图，再读出解区间。若 a > 0 且 Δ > 0，两根为 α < β，则：

ax² + bx + c > 0 holds for x < α or x > β | 解为 x < α 或 x > β
ax² + bx + c < 0 holds for α < x < β | 解为 α < x < β

If the parabola does not cross the x-axis, the inequality reduces to a sign check of the entire curve. For a > 0 and Δ < 0, the quadratic expression is always positive; for a < 0 and Δ < 0, it is always negative.

若抛物线与 x 轴无交点，则不等式问题简化为判断整条曲线的符号。当 a > 0 且 Δ < 0 时，二次表达式恒为正；当 a < 0 且 Δ < 0 时，恒为负。

Students are advised to draw a quick sketch before writing down intervals, as visualisation greatly reduces sign errors.

建议学生在写下解区间之前先画一个快速草图，因可视化能大幅减少符号错误。

10. Applications and Exam Tips | 实际应用与考试要点

Quadratic models appear in projectile motion, area optimisation, revenue maximisation and many other real-world problems. A-Level exams often combine quadratics with coordinate geometry, differentiation and curve sketching.

二次模型常见于抛体运动、面积优化、收益最大化等实际问题。A-Level 考试常常将二次函数与解析几何、微分和曲线作图结合起来考查。

Key exam tips for tackling quadratic questions:

解答二次函数题目的关键要点如下：

Always check the sign of a before doing anything else | 动笔前先确认 a 的符号
Complete the square when the question asks for the vertex or extremum | 题目要求顶点或最值时优先使用配方法
Use the discriminant to classify roots without fully solving | 用判别式判断根的类型，无需完整求解
Sketch a rough parabola before solving quadratic inequalities | 解二次不等式前先画抛物线草图
Verify the y-intercept by substituting x = 0 | 代入 x = 0 验证 y 轴截距

Mastering the link between the algebraic form and the graphical interpretation is the single most important skill for this topic. With consistent practice, quadratic graph analysis becomes one of the most scoring sections in the A-Level Mathematics paper.

掌握代数形式与图像解释之间的联系，是本主题最重要的技能。通过持续练习，二次函数图像分析将成为A-Level数学试卷中得分率最高的板块之一。

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Quadratic Functions: Graphs and Properties Analysis | 二次函数图像与性质分析