📚 IB CIE Mathematics: Quadratic Functions Key Points | IB CIE 数学：二次函数 考点精讲
Quadratic functions form the bedrock of algebra in both IB and CIE mathematics. Mastery of this topic is essential for success in topics ranging from equation solving and graph sketching to optimisation and calculus applications. This article synthesises the most tested concepts, common pitfalls, and efficient revision strategies to help you excel in exams.
二次函数是 IB 和 CIE 数学中代数的基石。掌握该专题不仅对解方程和图像绘制至关重要，更与优化问题、微积分应用紧密相连。本文整合了最常考的要点、常见错误和高效复习策略，助你考试夺优。
1. Quadratic Function Definition | 二次函数的定义
A quadratic function is a polynomial function of degree two, which can be written in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The condition a ≠ 0 ensures that the function is genuinely quadratic and not linear.
二次函数是二次多项式函数，可写作 f(x) = ax² + bx + c，其中 a、b、c 为常数且 a ≠ 0。a ≠ 0 这一条件确保函数确为二次而非线性。
2. Standard Form and Coefficients | 标准形式与系数
The standard form is f(x) = ax² + bx + c. The leading coefficient a determines the opening direction of the parabola: if a > 0, the parabola opens upwards (a ‘smile’); if a < 0, it opens downwards (a 'frown'). The constant term c is the y-intercept, where the graph crosses the y-axis at (0, c).
标准形式为 f(x) = ax² + bx + c。首项系数 a 决定抛物线开口方向：若 a > 0，抛物线开口向上（“微笑”状）；若 a < 0，则开口向下（“皱眉”状）。常数项 c 是 y 轴截距，图像与 y 轴交于 (0, c)。
The coefficient b, together with a, influences the position of the axis of symmetry, but exam questions often focus on the sign of a and value of c for quick graph analysis.
系数 b 与 a 共同影响对称轴的位置，但考试中常通过对 a 的符号和 c 的值进行快速图像分析。
3. Vertex Form and Completing the Square | 顶点式与配方法
The vertex form of a quadratic is f(x) = a(x – h)² + k, where (h, k) is the vertex of the parabola. This form is particularly useful for identifying the maximum or minimum value of the function: if a > 0, the minimum value is k; if a < 0, the maximum is k.
二次函数的顶点式为 f(x) = a(x – h)² + k，其中 (h, k) 为抛物线的顶点。该形式极利于确定函数的最值：若 a > 0，最小值为 k；若 a < 0，最大值为 k。
To convert from standard form to vertex form, we use the method of completing the square. For f(x) = ax² + bx + c, we write f(x) = a(x + b/(2a))² + (c – b²/(4a)). The vertex is then (-b/(2a), c – b²/(4a)), which is equivalent to (-b/(2a), (4ac – b²)/(4a)).
将标准形式化为顶点式需使用配方法。对于 f(x) = ax² + bx + c，可写为 f(x) = a(x + b/(2a))² + (c – b²/(4a))。于是顶点为 (-b/(2a), c – b²/(4a))，亦即 (-b/(2a), (4ac – b²)/(4a))。
Example: f(x) = 2x² – 8x + 5 → f(x) = 2(x – 2)² – 3. Vertex is (2, -3).
示例：f(x) = 2x² – 8x + 5 → f(x) = 2(x – 2)² – 3。顶点为 (2, -3)。
4. Axis of Symmetry and Vertex | 对称轴与顶点
The axis of symmetry is the vertical line that passes through the vertex, partitioning the parabola into two mirror images. Its equation is x = -b/(2a). The vertex lies at the intersection of this axis with the parabola.
对称轴是经过抛物线的顶点并将其分成镜像对称两部分的垂直线，其方程为 x = -b/(2a)。顶点即此轴与抛物线的交点。
In IB and CIE exams, you may be asked to find the axis of symmetry directly from standard form or to use it to deduce one unknown coordinate of the vertex. Memorising the formula x = -b/(2a) is essential for speed.
在 IB 和 CIE 考试中，你可能需要直接从标准形式求对称轴，或利用它推导顶点的一个未知坐标。熟记公式 x = -b/(2a) 对提高速度至关重要。
Axis of symmetry: x = -b/(2a)
对称轴：x = -b/(2a)
5. Discriminant and Nature of Roots | 判别式与根的性质
The discriminant Δ (Delta) is defined as Δ = b² – 4ac. It determines the number and type of real roots (x-intercepts) of the quadratic equation ax² + bx + c = 0:
判别式 Δ 定义为 Δ = b² – 4ac。它决定二次方程 ax² + bx + c = 0 的实根（x 轴交点）个数与类型：
| Δ Value | Nature of Roots | Graphical Implication |
|---|---|---|
| Δ > 0 | Two distinct real roots | Parabola intersects x-axis at two points |
| Δ = 0 | One repeated real root (double root) | Parabola touches x-axis (tangent) |
| Δ < 0 | No real roots | Parabola entirely above or below x-axis |
Δ > 0: two distinct real roots (图像与 x 轴交于两点)；Δ = 0: 一个重复实根 (图像与 x 轴相切)；Δ < 0: 无实根 (图像完全在 x 轴上方或下方)。
CIE Additional Maths and IB Analysis & Approaches frequently use the discriminant to find unknown coefficients or to prove that a line is tangent to a curve.
CIE 附加数学和 IB 分析与方法卷中常利用判别式求未知系数或证明某直线与曲线相切。
6. Sum and Product of Roots (Vieta’s Formulas) | 根的和与积（韦达定理）
For a quadratic equation ax² + bx + c = 0 with roots α and β (real or complex), Vieta’s formulas state:
对于有根 α 和 β（实根或复根）的二次方程 ax² + bx + c = 0，韦达定理指出：
Sum of roots: α + β = -b/a
Product of roots: αβ = c/a
根的和：α + β = -b/a
根的积：αβ = c/a
These relations allow you to form a quadratic equation if the roots are known, or to find symmetric expressions such as α² + β² and 1/α + 1/β without solving the equation explicitly.
这些关系使你能在已知根的情况下构造二次方程，或不必解出具体根就能求出对称表达式，如 α² + β² 和 1/α + 1/β。
A common exam trick is to use the identity α² + β² = (α + β)² – 2αβ. IB may ask for the equation whose roots are 2α+1 and 2β+1, requiring substitution.
常见的考试技巧是利用恒等式 α² + β² = (α + β)² – 2αβ。IB 可能会要求写出以 2α+1 和 2β+1 为根的方程，这需要代换。
7. Factored Form and x-Intercepts | 交点式与 x 轴交点
If a quadratic has real roots p and q, it can be written in factored form: f(x) = a(x – p)(x – q). The x-intercepts are then (p, 0) and (q, 0). This form is most convenient for graph sketching because the roots are immediately visible.
若二次函数有实根 p 和 q，则可写成交点式：f(x) = a(x – p)(x – q)。此时 x 轴交点为 (p, 0) 和 (q, 0)。该形式因根一目了然，最便于绘制图像。
When a quadratic cannot be factorised easily, completing the square or the quadratic formula x = [-b ± √(b² – 4ac)]/(2a) should be used to find the roots.
当二次式不易因式分解时，应使用配方法或求根公式 x = [-b ± √(b² – 4ac)]/(2a) 来求根。
Quadratic formula: x = [-b ± √Δ]/(2a)
求根公式：x = [-b ± √Δ]/(2a)
8. Graph Transformations | 图像变换
Quadratic graphs can be transformed using translations, reflections, and stretches. Understanding these transformations helps in quickly sketching transformed functions like f(x) = a(x – h)² + k.
二次函数图像可通过平移、对称和伸缩进行变换。理解这些变换有助于快速绘制如 f(x) = a(x – h)² + k 的变换图像。
- Vertical shift: f(x) → f(x) + d moves the graph up by d units. / 上下平移：f(x) → f(x) + d 图像上移 d 个单位。
- Horizontal shift: f(x) → f(x – h) shifts the graph right by h units (h positive). / 左右平移：f(x) → f(x – h) 图像右移 h 个单位（h 为正）。
- Reflection in x-axis: f(x) → -f(x) flips the parabola upside down. / x 轴对称：f(x) → -f(x) 将抛物线上下翻转。
- Vertical stretch: f(x) → a f(x) stretches the graph vertically by factor |a|. / 垂直伸缩：f(x) → a f(x) 图像纵向拉伸因子 |a|。
CIE exams frequently ask for the image equation after a combination of transformations. IB may present these in the context of composite functions.
CIE 考试频考复合变换后的图像方程。IB 可能结合复合函数的情境出题。
9. Quadratic Inequalities | 二次不等式
Solving a quadratic inequality such as ax² + bx + c > 0 involves finding the range of x-values for which the parabola lies above the x-axis. The critical method is:
求解二次不等式如 ax² + bx + c > 0，需找出使抛物线位于 x 轴上方的 x 值范围。关键方法为：
- Solve the corresponding equation ax² + bx + c = 0 to find the roots (if real). / 解对应方程 ax² + bx + c = 0 求根（若为实根）。
- Sketch the parabola using the sign of a and the roots. / 根据 a 的符号和根绘制抛物线草图。
- Read the solution intervals from the graph. For ‘> 0’, select x-values where the graph is above the x-axis; for ‘< 0', below. / 从图中读取解区间。对于‘> 0’，取图像在 x 轴上方的 x 值；对于‘< 0’，取下方。
Remember to use open intervals for strict inequalities (>, <) and closed intervals for non-strict (≥, ≤). In IB, inequalities often appear in calculus optimisation and domain determination.
切记严格不等式 (>, <) 用开区间，非严格 (≥, ≤) 用闭区间。在 IB 中，不等式常出现于微积分优化和定义域求解中。
10. Applications: Optimization and Modeling | 应用：最优化与建模
Quadratic functions are widely used to model real-world scenarios such as projectile motion, area maximisation, and profit/cost analysis. The vertex provides the optimal value: maximum profit, minimum cost, or maximum height.
二次函数广泛用于抛体运动、面积最大化、利润/成本分析等实际情境建模。顶点提供最优值：最大利润、最小成本或最大高度。
A typical problem: A farmer has 100 m of fencing and wants to enclose a rectangular field against a wall. Express the area as a quadratic in width x, then find x that maximises area. Here, writing the quadratic in vertex form quickly yields the maximum.
典型问题：某农夫有 100 米围栏，欲靠墙围一矩形场地。用宽 x 表示面积，并求使面积最大的 x。此时将二次式写为顶点式可迅速得出最大值。
CIE often embeds such questions in Paper 4 (Extended), requiring you to derive the quadratic and justify optimality. IB may ask for the domain of the quadratic model given the physical constraints.
CIE 常在卷四（扩展卷）中设置此类题目，要求推导二次式并论证最优化。IB 可能要求给定物理约束下二次模型的定义域。
11. Key IB/CIE Exam Tips | IB/CIE 考试技巧
Always check the sign of a before sketching a parabola. Many mistakes arise from drawing a frown when a > 0 or vice versa.
绘制抛物线前务必检查 a 的符号。许多错误源于 a > 0 时画成开口向下，或相反。
When completing the square, factor out a first if a ≠ 1: f(x) = a(x² + (b/a)x) + c, then complete the square inside the bracket. Never forget to multiply the added constant by a when subtracting outside.
配方法时，若 a ≠ 1 应先提取 a：f(x) = a(x² + (b/a)x) + c，然后对括号内进行配方。切勿忘记在减去常数时乘以 a。
For discriminant questions, set up an equation or inequality based on the required condition (e.g., tangent → Δ = 0). Always state the condition clearly in your working.
判别式问题中，根据所需条件（如相切 → Δ = 0）建立方程或不等式。解题步骤中务必明确陈述该条件。
In extended response questions, show substitution steps explicitly and verify that the found roots or vertex satisfy the original equation. IB values reasoning; CIE rewards method marks.
在解答题中，明确展示代入步骤并验证所求根或顶点满足原方程。IB 注重推理，CIE 奖励方法分。
12. Quick Reference: Key Formulas | 公式速查
| Quantity | Formula |
|---|---|
| Vertex x-coordinate | x = -b/(2a) |
| Vertex y-coordinate | y = (4ac – b²)/(4a) or f(-b/(2a)) |
| Discriminant | Δ = b² – 4ac |
| Roots (quadratic formula) | x = [-b ± √(b² – 4ac)]/(2a) |
| Sum of roots | α + β = -b/a |
| Product of roots | αβ = c/a |
| Vertex form | f(x) = a(x – h)² + k |
Keep this reference card handy during revision; these formulas cover 90% of quadratic function problems in IB and CIE exams.
复习时随身备好此速查卡；这些公式覆盖 IB 和 CIE 考试中 90% 的二次函数题目。
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