📚 Quadratic Functions | 二次函数
A quadratic function is one of the most fundamental algebraic objects in the IB Mathematics curriculum. It appears in equations, graphs, modelling, and optimisation, and its U‑shaped graph – the parabola – describes the motion of projectiles, the shape of satellite dishes, and the profit curve in business. Understanding quadratics deeply gives you the tools to factorise, complete the square, use the discriminant, solve inequalities, and connect algebraic expressions to geometric pictures.
二次函数是IB 数学课程中最基础且最重要的代数对象之一。它出现在方程、图像、建模和优化问题中，其 U 形图像——抛物线——能够描述抛体运动、卫星天线形状以及商业中的利润曲线。深入理解二次函数，将为你提供因式分解、配方法、使用判别式、解不等式以及将代数表达式与几何图像相连接的有力工具。
1. What is a Quadratic Function? | 什么是二次函数？
tes into a linear function. The highest power of the variable x is 2, which gives quadratics their characteristic curvature.
二次函数是指可以写成 f(x) = ax² + bx + c 形式的任何函数，其中 a、b 和 c 为实数，且 a ≠ 0。a ≠ 0 这一条件是本质要求；若 a = 0，函数将退化为一次函数。变量 x 的最高次幂为 2，这赋予了二次函数独特的弯曲特征。
The name “quadratic” comes from the Latin word quadratus, meaning square. The squared term ax² dominates the behaviour of the function for large values of |x|. In IB exams, you will meet quadratics in standard form, vertex form, and factored form, and you need to move flexibly between them.
“二次” 一词来源于拉丁语 quadratus，意为平方。平方项 ax² 主导了 |x| 很大时的函数行为。在 IB 考试中，你会碰到标准形式、顶点形式和因式形式的二次函数，并且需要灵活地在它们之间进行转换。
2. Standard Form and Coefficients | 标准形式与系数
eading coefficient, b is the linear coefficient, and c is the constant term. The sign of a controls the direction of opening: if a > 0 the parabola opens upwards (a “smile”), while if a < 0 it opens downwards (a “frown”).
二次函数的标准形式是 f(x) = ax² + bx + c。其中 a 是二次项系数，b 是一次项系数，c 是常数项。a 的符号控制抛物线的开口方向：若 a > 0，抛物线开口向上（呈 “微笑” 状）；若 a < 0，则开口向下（呈 “皱眉” 状）。
he coefficient a also affects the width of the parabola. Larger absolute values of a (e.g. |a| = 5) make the parabola steeper and narrower, while smaller absolute values (e.g. |a| = 0.2) make it wider and flatter. The constant term c gives the y‑ nts b and c together with a determine the position of the vertex and the axis of symmetry.
系数 a 还影响抛物线的宽窄。|a| 的绝对值越大（如 |a| = 5），抛物线越陡越窄；|a| 越小（如 |a| = 0.2），抛物线越平越宽。常数项 c 给出了 y 轴截距，因为 f(0) = c。b 和 c 一起与 a 共同决定了顶点和对称轴的位置。
Standard form: f(x) = ax² + bx + c, a ≠ 0
标准形式：f(x) = ax² + bx + c, a ≠ 0
| Coefficient | Meaning | 系数 | 含义 |
|---|---|---|---|
| a | Direction & width of parabola | a | 抛物线的开口方向与宽窄 |
| b | Position of axis of symmetry | b | 对称轴的位置 |
| c | y‑intercept | c | y 轴截距 |
3. Vertex Form and Completing the Square | 顶点形式与配方法
Any quadratic function in standard form can be rewritten in vertex form: f(x) = a(x – eved by completing the square. This technique is a core algebraic skill in IB Mathematics.
任何标准形式的二次函数都可以改写为顶点形式：f(x) = a(x – h)² + k，其中 (h, k) 是抛物线的顶点。从标准形式到顶点形式的转换通过配方法来实现，这项技巧是 IB 数学中的一项核心代数技能。
n add and subtract (b/(2a))² inside the brackets. The result is f(x) = a(x + b/(2a))² + (c – b²/(4a)). Thus the vertex is at (–b/(2a), c – b²/(4a)). This matches the formula h = –b/(2a), which is the x‑coordinate of the vertex.
对 f(x) = ax² + bx + c 进行配方时，首先从 x² 与 x 项中提取 a：f(x) = a[x² + (b/a)x] + c。然后在括号内加上并减去 (b/(2a))²。最终得到 f(x) = a(x + b/(2a))² + (c – b²/(4a))。因此顶点坐标为 (–b/(2a), c – b²/(4a))。这与 h = –b/(2a) 这一顶点 x 坐标公式一致。
The vertex form immediately reveals the maximum or minimum value of the quadratic. If a > 0, the vertex is a minimum and k is the minimum value; if a < 0, it is a maximum. This is invaluable for optimisation problems.
顶点形式能够立即显示二次函数的最大值或最小值。若 a > 0，顶点为最小值点，k 即为最小值；若 a < 0，顶点为最大值点。这对优化问题极有价值。
Vertex form: f(x) = a(x – h)² + k
顶点形式：f(x) = a(x – h)² + k
4. Factored Form and Roots | 因式形式与根
If a quadratic function crosses or touches the x‑axis, it can be expressed in factored form using its real roots r₁ and r₂: f(x) = a(x – r₁)(x – cs have real roots; the discriminant (see next section) tells us whether real roots exist.
如果二次函数与 x 轴相交或相切，则可以利用它的实根 r₁ 和 r₂ 表示为因式形式：f(x) = a(x – r₁)(x – r₂)。这些根是方程 f(x) = 0 的解。并非所有二次函数都有实根，判别式（见下一节）会告诉我们实根是否存在。
en a quadratic has a double root (also called a repeated root), the parabola touches the x‑ Mathematically, the two roots coincide: r₁ = r₂. The factored form then becomes f(x) = a(x – r)².
当二次函数有二重根（也称为重根）时，抛物线仅在一点与 x 轴相切。从数学上讲，两根重合：r₁ = r₂。此时因式形式变为 f(x) = a(x – r)²。
uadratic depends on recognising special patterns, such as the difference of two squares a² – b² = (a – nd add to b. IB questions often expect you to factorise mentally for simple integer roots.
能否对二次三项式进行因式分解，取决于能否识别特殊模式，例如平方差公式 a² – b² = (a – b)(a + b)，或者能否找到两个数，使其乘积为 a×c 且和为 b。IB 试题常要求你心算分解简单的整数根。
5. The Discriminant Δ | 判别式 Δ
letter Delta (Δ), is the expression under the square root in the quadratic formula: Δ = b² – ure of the roots of the quadratic equation ax² + bx + c = 0 without solving it explicitly.
判别式用希腊字母 Δ 表示，是求根公式中平方根号下的表达式：Δ = b² – 4ac。它无需显式求解，便能决定二次方程 ax² + bx + c = 0 的根的数量和性质。
If Δ > 0, the quadratic has two distinct real roots. The parabola cuts the x‑axis at two points.
若 Δ > 0，二次方程有两个不同的实根。抛物线与 x 轴交于两点。
If Δ = 0, there is exactly one real root (a repeated root). The parabola touches the x‑axis at the vertex.
若 Δ = 0，恰好有一个实根（重根）。抛物线的顶点与 x 轴相切。
If Δ < 0, the quadratic has no real roots; the two roots are complex conjugates. The parabola does not intersect the x‑axis.
若 Δ < 0，二次方程没有实根；两个根为共轭复数。抛物线与 x 轴不相交。
le, finding k such that f(x) = x² + (k+1)x + 4 has a repeated root requires setting Δ = 0.
判别式还经常用于 IB 考试中要求较高的题目，例如求未知参数的取值范围，使二次方程没有实根、有重根或有两个不同的实根。例如，要找到使得 f(x) = x² + (k+1)x + 4 有重根的 k，就需要令 Δ = 0。
6. Graph of a Quadratic (Parabola) | 二次函数的图像（抛物线）
The graph of any quadratic function is a parabola – a smooth, symmetric curve. The parabola is symmetric about a vertical line called the axis of symmetry, which passes through the vertex. All parabolas have the same basic shape, but they can be stretched, compressed, reflected, or translated.
任何二次函数的图像都是一条抛物线——一条光滑、对称的曲线。抛物线关于一条称为对称轴的竖直线对称，该直线穿过顶点。所有抛物线都具有相同的基本形状，但可以被拉伸、压缩、对称反射或平移。
, the parabola has a minimum point at the vertex; for a < 0, it has a maximum point. The y‑intercept is simply the constant term c, while the x‑intercepts (if they exist) are the roots of the equation ax² + bx + c = 0.
抛物线顶点代表转折点。当 a > 0 时，抛物线在顶点处有最小值点；当 a < 0 时，有最大值点。y 轴截距就是常数项 c，而 x 轴截距（如果存在）是方程 ax² + bx + c = 0 的根。
Sketching a parabola accurately requires knowledge of five key features: the direction of opening, the y‑intercept, the vertex, the axis of symmetry, and the x‑intercepts (or the fact that there are none). IB exam questions often award marks for correctly labelling these elements on a graph.
准确地画出抛物线草图，需要掌握五个关键特征：开口方向、y 轴截距、顶点、对称轴以及 x 轴截距（或确认没有 x 轴截距）。IB 考试题中，正确地在图上标出这些元素通常可以得分。
7. Axis of Symmetry and Vertex | 对称轴与顶点
The axis of symmetry is the vertical line x = –b/(2a). This line divides the parabola into two mirror‑ and its equation can be derived directly from standard form or from completing the square.
对称轴是垂直线 x = –b/(2a)。这条线将抛物线分成两个互为镜像的部分。每一个二次函数都恰有一条对称轴，其方程可以直接从标准形式或通过配方法推导得出。
The vertex lies on the axis of symmetry, so its x‑coordinate is h = –b/(2a). Substituting this value back into the function gives the y‑coordinate k = f(–b/(2a)). The vertex is therefore the point (h, k) = (–b/(2a), f(–b/(2a))).
顶点位于对称轴上，因此其 x 坐标为 h = –b/(2a)。将该值代回原函数可得 y 坐标 k = f(–b/(2a))。因此顶点为点 (h, k) = (–b/(2a), f(–b/(2a)))。
range of the quadratic function. For a > 0, the range is [k, ∞); for a < 0, the range is (–∞, k]. This is frequently tested in IB questions involving domain and range.
知道了顶点，就可以确定二次函数的值域。若 a > 0，值域为 [k, ∞)；若 a < 0，值域为 (–∞, k]。这在涉及定义域和值域的 IB 题目中经常考查。
8. Intercepts: x‑intercepts and y‑intercept | 截距：x 截距和 y 截距
Intercepts are the points where the graph crosses the coordinate axes. The y‑ , yielding the point (0, c). This is often the easiest point to plot when sketching a parabola.
截距是图像与坐标轴相交的点。令 x = 0 即可求得 y 轴截距，从而得到点 (0, c)。在绘制抛物线草图时，这往往是最容易描出的点。
The x‑ ts or zeros) are found by solving f(x) = 0. Depending on Δ, there can be two, one, or no x‑intercepts. When the quadratic is in factored form, a(x – r₁)(x – r₂) = 0, the x‑intercepts are simply (r₁, 0) and (r₂, 0).
x 轴截距（也称为根或零点）通过求解 f(x) = 0 得到。根据 Δ 的不同，可能存在两个、一个或没有 x 轴截距。当二次函数写成因式形式 a(x – r₁)(x – r₂) = 0 时，x 轴截距就是 (r₁, 0) 和 (r₂, 0)。
If the discriminant is negative, the quadratic does not cross the x‑axis; however, you can still find its minimum or maximum value and its y‑intercept to sketch it. IB questions may ask you to “find the coordinates of any points where the graph cuts the axes”, so always consider both axes.
若判别式为负，二次函数不与 x 轴相交；但你仍然可以找到它的最小值或最大值以及 y 轴截距来画图。IB 题目或许会要求你 “求出图像与坐标轴相交的所有点的坐标”，因此务必同时考虑两轴。
9. Solving Quadratic Equations | 解二次方程
ting the square, and the quadratic formula. Factorisation is fastest when a, b, and c are integers and the roots are rational. Completing the square is often used when the quadratic cannot be easily factorised, and it also helps to derive the vertex form.
解二次方程 ax² + bx + c = 0 主要有三种方法：因式分解法、配方法和求根公式法。当 a、b、c 为整数且根为有理数时，因式分解最为快速。配方法常在无法轻松因式分解时使用，同时也有助于导出顶点形式。
The quadratic formula is universal and always works: x = (–b ± √Δ) / (2a), where Δ = b² – ex roots. When Δ < 0, the formula gives complex solutions using i (or i in IB) to denote √–1.
求根公式是通用的且总能求解：x = (–b ± √Δ) / (2a)，其中 Δ = b² – 4ac。学生必须熟记该公式。它能给出精确解，包括无理根和复根。当 Δ < 0 时，公式使用 i （IB 中用 i）表示 √–1 从而给出复数解。
x = (–b ± √b² – 4ac) / (2a)
x = (–b ± √b² – 4ac) / (2a)
10. Quadratic Inequalities | 二次不等式
sidering the shape of the parabola and its roots. The first step is always to move all terms to one side so that you compare to zero. Then find the roots of the corresponding quadratic equation.
诸如 ax² + bx + c > 0、≥ 0、< 0 或 ≤ 0 的二次不等式，可通过考虑抛物线形状及其根来求解。第一步总是将所有项移到一边，使其与零比较。然后求出相应二次方程的根。
Once the roots are found (real and distinct, or a double root), the parabola divides the x‑axis into intervals. The sign of the quadratic in each interval is determined by testing a sample value or by using the “parabola above‑below x‑axis” rule. The solution set is expressed as inequalities or in interval notation.
一旦求出根（不同实根或二重根），抛物线便将 x 轴分成若干区间。二次函数在每个区间的符号，可通过检验某个样值或运用 “抛物线与 x 轴上下关系” 规则来确定。解集用不等式或区间记号表示。
For example, to solve x² – 5x + 6 > 0, factorise to (x – 2)(x – roots are x = 2 and x = 3. Since a = 1 > 0, the parabola opens upwards and is above the x‑ s x < 2 or x > 3. IB questions often require graphical methods to justify these solutions.
例如，解 x² – 5x + 6 > 0，因式分解得 (x – 2)(x – 3) > 0。根为 x = 2 和 x = 3。由于 a = 1 > 0，抛物线开口向上，并在两根之间的区间外位于 x 轴上方。因此解为 x < 2 或 x > 3。IB 题目常要求用图像方法为这些解提供依据。
11. Sum and Product of Roots (Vieta's Formulas) | 根的和与积（韦达定理）
For the quadratic equation ax² + bx + c = 0 with roots r₁ and r₂, Vieta's formulas state: r₁ + r₂ = – ing your factorisation, or solving problems where the roots are not required individually.
对于有根 r₁ 和 r₂ 的二次方程 ax² + bx + c = 0，韦达定理指出：r₁ + r₂ = –b/a 且 r₁ × r₂ = c/a。这些关系在根据给定根构造二次方程、检验因式分解或求解无需单独求出根的问题时都非常有用。
If a quadratic is monic (a = 1), then the sum of the roots is simply – akes mental factorisation easier: look for two numbers whose product is c and whose sum is – xtended to problems involving symmetric functions of roots, such as r₁² + r₂² = (r₁ + r₂)² – 2r₁r₂.
若二次方程是首一的（a = 1），那么根的和就是 –b，根的积就是 c。这使得心算因式分解更加容易：寻找乘积为 c 且和为 –b 的两个数。此外，韦达定理可以延伸到根的对称函数问题，例如 r₁² + r₂² = (r₁ + r₂)² – 2r₁r₂。
the value of an expression involving the roots without solving the equation. For example, “Given that the roots of 2x² – 3x + 5 = 0 are α and β, find α²β + αβ².” Factorising αβ(α + β) and applying Vieta yields the answer quickly.
IB 常考查这些关系，要求学生根据根求出二次方程，或者在不求解方程的情况下，求出涉及根的某个表达式的值。例如，“已知 2x² – 3x + 5 = 0 的根为 α 和 β，求 α²β + αβ²。” 提取 αβ(α + β) 并运用韦达定理，即可迅速得出答案。
12. Applications and Modelling | 实际应用与建模
Quadratic functions model many real‑world scenarios: the trajectory of a ball under gravity, the area of a rectangular enclosure, profit maximisation in business, and the shape of a suspension bridge’s cables. In IB, you are expected to translate a word problem into a quadratic function, find its maximum or minimum, and interpret the results in context.
二次函数可以对众多现实情形建模：重力作用下的球的飞行轨迹、矩形围场的面积、商业中的利润最大化以及悬索桥缆绳的形状。在 IB 中，你需要将应用题翻译为二次函数，求出其最大值或最小值，并在上下文中解释结果。
A typical projectile problem gives the height h(t) of an object as h(t) = – of the parabola can also be used: the time to reach the maximum height is exactly half of the total flight time when landing at the same vertical level.
一个典型的抛体问题会给出物体的高度 h(t) = –5t² + vt + h₀（IB 中使用米和秒为单位）。求最大高度需要定位顶点，求何时落地则需要解 h(t) = 0。抛物线的对称性也可以利用：当物体落回同一水平高度时，达到最大高度所用的时间恰好是总飞行时间的一半。
In optimisation problems, you often set up a quadratic expression for the quantity to be maximised (e.g., area or profit), then find the vertex. The x‑coordinate of the vertex gives the optimal input, and the y‑coordinate gives the optimal value. Always remember to check that your solution is within the feasible domain of the problem.
在优化问题中，你通常需要为待最大化的量（如面积或利润）建立一个二次表达式，然后找到顶点。顶点的 x 坐标给出最优输入，y 坐标给出最优值。请始终记得检查所得解是否落在问题的可行域内。
Overall, quadratics are a cornerstone of the IB syllabus. Mastering them requires fluidity between forms, a deep understanding of the discriminant, and the ability to link algebraic solutions to graphical behaviour. With practice, they become a reliable tool rather than a source of confusion.
总而言之，二次函数是 IB 课程大纲的基石。要掌握它们，需要能灵活转换不同形式、深入理解判别式，并能够将代数解法与图像行为联系起来。通过练习，它们将成为你可靠的工具，而不再是困惑的来源。
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