📚 Quadratic Functions | 二次函数
A quadratic function is a polynomial function of degree 2. It is one of the most fundamental and widely studied functions in mathematics, forming the backbone of many algebraic and geometric concepts. From the simple parabola to advanced optimisation problems, mastering quadratics is essential for success in IB Mathematics. This article will guide you through all key aspects: forms, graphing, solving, discriminant analysis, and real-life applications.
二次函数是一种次数为 2 的多项式函数。它是数学中最为基础且被广泛研究的函数之一，构成了许多代数与几何概念的支柱。从简单的抛物线到复杂的优化问题，掌握二次函数对于 IB 数学取得优异成绩至关重要。本文将带你逐一攻克所有关键环节：不同表达式、作图、求解、判别式分析以及实际应用。
1. Definition and Standard Form | 定义与标准形式
stants and a ≠ 0. This is called the general or standard form. The coefficient a determines the direction and width of the parabola, b affects the axis of symmetry, and c is the y-intercept.
二次函数定义为 f(x) = ax² + bx + c，其中 a、b 和 c 为常数且 a ≠ 0。这称为一般形式或标准形式。系数 a 决定了抛物线的开口方向和宽窄，b 影响对称轴的位置，c 是图像与 y 轴交点的纵坐标。
- The domain of any quadratic function is all real numbers, written as x ∈ ℝ.
- 任何二次函数的定义域都是全体实数，记作 x ∈ ℝ。
- The graph of a quadratic function is called a parabola.
- 二次函数的图像称为抛物线。
- If a > 0, the parabola opens upwards and has a minimum point.
- 若 a > 0，抛物线开口向上，有一个最低点。
- If a < 0, the parabola opens downwards and has a maximum point.
- 若 a < 0，抛物线开口向下，有一个最高点。
2. Graph of a Quadratic Function: The Parabola | 二次函数的图像：抛物线
c). The x-intercepts, if they exist, are the real roots of the equation ax² + bx + c = 0.
理解抛物线的形状与特征至关重要。图像关于一条称为对称轴的竖直线对称。y 轴截距在 x = 0 时取得，即点 (0, c)。如果存在 x 轴截距，它们就是方程 ax² + bx + c = 0 的实根。
- The vertex is the turning point where the function changes direction.
- 顶点是函数改变方向的转折点。
- The axis of symmetry passes through the vertex, given by x = -b/(2a).
- 对称轴穿过顶点，其方程为 x = -b/(2a)。
3. Vertex and Axis of Symmetry | 顶点与对称轴
ined by substituting h into the function. The axis of symmetry is the vertical line x = h.
利用标准形式可以找到顶点坐标 (h, k)。对于 f(x) = ax² + bx + c，顶点的横坐标为 h = -b/(2a)。然后将 h 代入函数即可求得 k = f(h)。对称轴为竖直线 x = h。
h = -b / (2a)
k = f(h) = a(-b/(2a))² + b(-b/(2a)) + c
elpful, as it immediately gives the axis of symmetry and the x-coordinate of the extremum point.
记住 h = -b/(2a) 非常有用，因为它能立即给出对称轴和最值点的横坐标。
4. Vertex Form and Completing the Square | 顶点式与配方法
the vertex. This form makes it easy to identify the vertex, the direction of opening, and to perform transformations. Converting from standard form to vertex form is done by completing the square.
二次函数的顶点式为 f(x) = a(x - h)² + k，其中 (h, k) 为顶点。此形式能方便地识别顶点、开口方向以及进行图像变换。从标准形式变为顶点式需要通过配方法实现。
Example: Express y = 2x² + 8x + 5 in vertex form.
示例：将 y = 2x² + 8x + 5 化为顶点式。
- Factor out a from the first two terms: y = 2(x² + 4x) + 5
- 从前两项中提出 a：y = 2(x² + 4x) + 5
- t: (x² + 4x + 4) = (x + 2)², so add and subtract 4 inside. Adjust: y = 2[(x + 2)² - 4] + 5
- 在括号内完成配方：(x² + 4x + 4) = (x + 2)²，因此内部加 4 并减 4。调整：y = 2[(x + 2)² - 4] + 5
- Simplify: y = 2(x + 2)² - 8 + 5 = 2(x + 2)² - 3. Vertex is (-2, -3).
- 化简：y = 2(x + 2)² - 8 + 5 = 2(x + 2)² - 3。顶点为 (-2, -3)。
5. Factored Form and Roots | 因式分解式与根
re p and q are the roots or zeros of the function. The factored form is extremely useful for solving quadratic equations and sketching graphs quickly.
如果二次函数与 x 轴交于 p 和 q 两点，则可将其写成因式分解式 f(x) = a(x - p)(x - q)。这里的 p 和 q 即为函数的根或零点。因式分解式对于解二次方程以及快速绘制图像极为有用。
Not all quadratics factor neatly with integers; sometimes roots are irrational or complex. Factoring by inspection, grouping, or using special products (difference of squares, perfect square trinomials) are common methods.
并非所有二次式都能用整数整洁分解；有些根是无理数或复数。常见的方法包括观察法、分组法，或利用特殊乘积（平方差、完全平方三项式）。
- x² - 5x + 6 = (x - 2)(x - 3) → roots 2 and 3
- x² - 5x + 6 = (x - 2)(x - 3) → 根为 2 和 3
- 4x² - 9 = (2x - 3)(2x + 3) → roots 3/2 and -3/2
- 4x² - 9 = (2x - 3)(2x + 3) → 根为 3/2 和 -3/2
6. The Discriminant | 判别式
. It provides critical information about the number and type of roots without actually solving the equation.
二次方程 ax² + bx + c = 0 的判别式为 Δ = b² - 4ac。它提供了关于根的个数和类型的关键信息，而无需求解方程。
Δ = b² - 4ac
This concept is heavily tested in IB Mathematics, especially in problems involving intersections of graphs and determining the nature of solutions.
这一概念在 IB 数学考试中出现频率极高，尤其是在涉及图像交点个数以及判断解的性质的问题中。
7. Nature of Roots | 根的性质
The discriminant determines the nature of the roots as follows:
判别式对根的性质判定如下：
| Discriminant (Δ) | Nature of Roots | 根的性质 |
|---|---|---|
| Δ > 0 and a perfect square | Two distinct real rational roots | 两个不等实有理根 |
| Δ > 0 but not a perfect square | Two distinct real irrational roots | 两个不等实无理根 |
| Δ = 0 | One real repeated root (two equal real roots) | 一个实数重根（两个相等实根） |
| Δ < 0 | No real roots (two complex conjugate roots) | 没有实根（一对共轭复数根） |
Understanding this table helps in questions that ask 'for what values of k does the equation have two distinct real roots?' Such problems involve setting Δ > 0 and solving the inequality.
理解此表有助于解答诸如“k 取何值时方程有两个不等实根？”的问题。这类问题需要设 Δ > 0 并求解不等式。
8. Quadratic Formula | 二次公式
equation ax² + bx + c = 0. It is derived from completing the square on the standard form.
当因式分解不可行或不方便时，二次公式可以求解任何二次方程 ax² + bx + c = 0。它是通过对标准形式配方推导而来的。
x = [ -b ± √b² - 4ac ] / (2a)
Always make sure the equation is set equal to zero before identifying a, b, and c. The '±' generally two solutions, corresponding to the two x-intercepts of the parabola (if real).
在使用公式前，务必确保方程已设为等于零，然后确定 a、b 和 c。’±‘ 符号表明通常有两解，对应于抛物线的两个 x 轴截距（若为实数）。
9. Solving Quadratic Equations | 求解二次方程
l when linking to vertex form), and taking square roots (for equations like (x - p)² = q).
IB 考试中的题目常常要求选择最高效的方法求解二次方程。主要策略有：因式分解（当可能时）、使用二次公式、配方法（在需要联系顶点式时很有用）以及直接开平方（用于形如 (x - p)² = q 的方程）。
Example using square root method:
平方根法示例：
- Solve 2(x - 3)² - 8 = 0
- 求解 2(x - 3)² - 8 = 0
- Add 8: 2(x - 3)² = 8
- 加 8：2(x - 3)² = 8
- Divide by 2: (x - 3)² = 4
- 除以 2：(x - 3)² = 4
- Take square root: x - 3 = ±2
- 开平方：x - 3 = ±2
- x = 3 ± 2, giving x = 5 or x = 1.
- x = 3 ± 2，得出 x = 5 或 x = 1。
10. Applications and Word Problems | 应用与文字题
Quadratic functions are used to model projectile motion, area optimisation, revenue and cost analysis, and many other real-world scenarios. A typical problem: find the maximum area of a rectangular enclosure with a fixed perimeter, or determine the maximum height of a ball thrown upwards.
二次函数可用于建立抛体运动、面积优化、收益成本分析以及许多其他实际场景的模型。典型问题：求固定周长下矩形围栏的最大面积，或者求向上抛出小球的最大高度。
In such problems, identify the quantity to be maximised or minimised, express it as a quadratic function of one variable, and then use the vertex (h, k) to find the optimum value.
在此类问题中，先确定需要最大或最小化的量，将其表示成关于一个变量的二次函数，然后利用顶点 (h, k) 求得最优值。
- Area A = x(50 - x) = -x² + 50x. Vertex gives maximum area at x = 25.
- 面积 A = x(50 - x) = -x² + 50x。顶点给出当 x = 25 时面积最大。
- The y-coordinate of the vertex represents the maximum (or minimum) value in context.
- 顶点的纵坐标在对应情境中代表最大值（或最小值）。
11. Inequalities and Quadratic Functions | 不等式与二次函数
Quadratic inequalities such as ax² + bx + c > 0 are solved by first finding the roots of the corresponding equation and then testing intervals on a number line, or by analysing the graph. The parabola is above the x-axis where the inequality holds.
对于像 ax² + bx + c > 0 这样的二次不等式，求解方法是先求出对应方程的根，接着在数轴上检验各个区间，或者通过分析图像来判断。抛物线位于 x 轴上方的那部分即满足不等式。
- Solve x² - x - 6 < 0: roots are -2 and 3. The parabola opens upwards, so the inequality holds between the roots: -2 < x < 3.
- 解 x² - x - 6 < 0：根为 -2 和 3。抛物线开口向上，因此在两根之间不等式成立：-2 < x < 3。
nequality is strict (<, >) or inclusive (≤, ≥) and adjust interval boundaries accordingly.
务必留意不等式是严格不等号（<, >）还是包含等号（≤, ≥），并据此调整区间端点。
12. Transformations of Quadratic Graphs | 二次函数图像的变换
ives horizontal translation (shift right if h > 0, left if h < 0), k gives vertical translation (up if k > 0, down if k < 0), and a gives a vertical stretch/compression and possibly a reflection over the x-axis if a is negative.
利用顶点式 f(x) = a(x - h)² + k，可以轻松描述由母函数 y = x² 所经过的变换：h 给出水平平移（h > 0 时右移，h < 0 时左移），k 给出竖直平移（k > 0 时上移，k < 0 时下移），a 给出垂直拉伸或压缩，若 a 为负则还包含关于 x 轴的反射。
Combined transformations follow the order: horizontal shift, scaling/reflection, vertical shift. Being comfortable with these helps in sketching graphs rapidly and solving graph-matching questions.
组合变换遵循顺序：水平平移、伸缩/反射、竖直平移。熟练掌握这些有助于快速绘制图像并解答图像匹配题。
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