AS Mathematics: Quadratic Functions – Key Points Explained | AS 数学：二次函数考点精讲

📚 AS Mathematics: Quadratic Functions – Key Points Explained | AS 数学：二次函数考点精讲

Quadratic functions are a cornerstone of AS Mathematics. A solid grasp of their algebra, graphs, and the connections between them is essential for exam success. This guide pulls together the key ideas you must master, from standard form to inequalities and applied optimisation.

二次函数是 AS 数学的基石。牢固掌握其代数运算、图像以及它们之间的联系是考试成功的关键。本文汇集了你必须掌握的核心概念，从标准形到不等式再到实际优化应用。

1. Definition and Standard Form | 定义与标准形式

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. If a = 0, the function becomes linear, not quadratic.

二次函数是最高次数为 2 的多项式函数。其标准式为 f(x) = ax² + bx + c，其中 a、b、c 为常数且 a ≠ 0。若 a = 0，函数就退化为一次函数，不再是二次。

The coefficient a is called the leading coefficient and determines the shape and direction of the graph. The constant term c gives the y-intercept (0, c).

系数 a 称为首项系数，它决定了图像的形状与开口方向。常数项 c 则给出了 y 轴截距 (0, c)。

2. The Parabola Graph | 抛物线图像

The graph of any quadratic function is a parabola. If a > 0, the parabola opens upwards (like a ∪); if a < 0, it opens downwards (like a ∩).

任何二次函数的图像都是一条抛物线。当 a > 0 时，抛物线开口向上（像 ∪ 形）；当 a < 0 时，开口向下（像 ∩ 形）。

The parabola is symmetric about a vertical line called the axis of symmetry. The y-intercept is simply c, but the x-intercepts (if any) are found by solving f(x) = 0.

抛物线关于一条垂直线对称，该直线称为对称轴。y 轴截距很简单就是 c，而 x 轴截距（如果存在）则需通过解方程 f(x) = 0 来求得。

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

By completing the square, f(x) = ax² + bx + c can be written in vertex form: f(x) = a(x – h)² + k. The point (h, k) is the vertex of the parabola, and the axis of symmetry is the line x = h.

通过配方法，可将 f(x) = ax² + bx + c 写成顶点式：f(x) = a(x – h)² + k。点 (h, k) 就是抛物线的顶点，对称轴为直线 x = h。

Directly from standard form, the x-coordinate of the vertex is h = -b / (2a). The y-coordinate is then k = f(h).

直接从标准式出发，顶点的 x 坐标为 h = -b/(2a)，相应的 y 坐标为 k = f(h)。

4. The Discriminant | 判别式

For the quadratic equation ax² + bx + c = 0, the discriminant is Δ = b² – 4ac. It reveals the nature of the roots without solving the equation.

对于二次方程 ax² + bx + c = 0，判别式 Δ = b² – 4ac。它能够在不求解方程的情况下揭示根的性质。

Condition	Nature of roots (中文)
Δ > 0: two distinct real roots	Δ > 0：两个不相等的实根
Δ = 0: one repeated real root (equal roots)	Δ = 0：一个重根（两个相等实根）
Δ < 0: no real roots	Δ < 0：无实根

The sign of Δ also tells you how many times the parabola crosses the x-axis: twice for Δ > 0, touches it once for Δ = 0, and never crosses for Δ < 0.

判别式的符号也告诉你抛物线与 x 轴相交的次数：Δ > 0 时相交两次，Δ = 0 时相切一次，Δ < 0 时没有交点。

5. Solving by Factoring | 因式分解法求根

When the quadratic expression can be factorised, it offers the quickest route to the roots. Write ax² + bx + c as a product of two binomials and set each factor to zero.

当二次表达式可以因式分解时，这是求根最快捷的方法。将 ax² + bx + c 写成两个一次因式的乘积，再分别令每个因式等于零。

For example, x² – 5x + 6 = 0 factorises to (x – 2)(x – 3) = 0, giving roots x = 2 and x = 3.

例如，x² – 5x + 6 = 0 可分解为 (x – 2)(x – 3) = 0，得出根 x = 2 和 x = 3。

Remember that not all quadratics factorise neatly over integers; then you must use other methods.

记住，并非所有二次式都能在整数范围内漂亮地分解；此时则需要使用其他方法。

6. Completing the Square | 配方法

Completing the square transforms a quadratic into the form a(x – h)² + k. For a monic quadratic x² + bx, you add (b/2)² to obtain (x + b/2)².

配方法将二次式变为 a(x – h)² + k 的形式。对于首一二次式 x² + bx，加上 (b/2)² 即可得到 (x + b/2)²。

To solve x² + 6x + 5 = 0: rewrite as x² + 6x = -5, add (6/2)² = 9 to both sides, giving (x + 3)² = 4. Then x + 3 = ±2 → x = -1 or x = -5.

解 x² + 6x + 5 = 0：移项为 x² + 6x = -5，两边加上 (6/2)² = 9，得 (x + 3)² = 4。于是 x + 3 = ±2，解得 x = -1 或 x = -5。

This technique is also used to find the vertex and to derive the quadratic formula.

这一技巧也常用于求顶点以及推导二次公式。

7. Quadratic Formula | 二次公式

The quadratic formula solves any equation ax² + bx + c = 0:

二次公式可以求解任何形式为 ax² + bx + c = 0 的方程：

x = (-b ± √(b² – 4ac)) / (2a)

Pay close attention to the sign of a and the order of operations. The expression under the square root is the discriminant.

要特别注意 a 的符号以及运算顺序。根号下的表达式就是判别式。

When Δ is negative, the formula yields complex numbers; at AS level, you usually stop at stating ‘no real roots’.

当 Δ 为负时，公式给出复数根；在 AS 阶段，通常只需说明 ‘无实根’。

8. Intersections with the x-axis | 与 x 轴的交点

The real solutions of f(x) = 0 correspond exactly to the x-intercepts of the graph y = f(x). Thus, a quadratic graph can cross the x-axis 0, 1 or 2 times.

方程 f(x) = 0 的实数解恰好对应图像 y = f(x) 与 x 轴的交点。因此，二次函数图像可以与 x 轴相交 0 次、1 次或 2 次。

If the vertex lies on the x-axis, the discriminant is zero and the parabola touches the axis at exactly one point.

如果顶点恰好落在 x 轴上，则判别式为零，抛物线正好在一点与轴相切。

9. Quadratic Inequalities | 二次不等式

To solve an inequality like x² – 5x + 6 > 0, first solve the equation x² – 5x + 6 = 0 to get the critical values x = 2 and x = 3.

解不等式如 x² – 5x + 6 > 0 时，先解方程 x² – 5x + 6 = 0 得到临界值 x = 2 和 x = 3。

Since a > 0, the parabola opens upward and is above the x-axis outside the interval [2, 3]. The solution is x < 2 or x > 3, written as (-∞, 2) ∪ (3, ∞).

由于 a > 0，抛物线开口向上，在区间 [2, 3] 之外图像位于 x 轴上方。解为 x < 2 或 x > 3，记作 (-∞, 2) ∪ (3, ∞)。

If the inequality were ≤ 0, the solution would include the roots: [2, 3].

若不等式为 ≤ 0，则解集会包含端点：[2, 3]。

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

The vertex holds the extreme value of a quadratic function. When a > 0, the function has a minimum value of k at x = h; when a < 0, it has a maximum value of k.

顶点给出了二次函数的极值。当 a > 0 时，函数在 x = h 处取得最小值 k；当 a < 0 时，函数在 x = h 处取得最大值 k。

In optimisation problems, you often model a quantity as a quadratic and find its vertex to maximise or minimise the outcome.

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AS Mathematics: Quadratic Functions – Key Points Explained | AS 数学：二次函数 考点精讲