📚 Quadratic Functions: Graphs and Properties | 二次函数图像与性质
Quadratic functions are fundamental in IB Mathematics, appearing in topics from algebra to calculus and real-world modelling. Their parabolic graphs have distinctive features that can be analysed algebraically and geometrically.
二次函数是IB数学中的基础内容，从代数到微积分以及实际建模中都会出现。它们的抛物线图像具有独特的特征，可以通过代数和几何方法进行分析。
1. Standard Form and General Form | 标准形式与一般形式
A quadratic function can be written in several forms. The general form is f(x) = ax2 + bx + c, where a ≠ 0. The standard form, also called vertex form, is f(x) = a(x-h)2 + k, where (h,k) is the vertex of the parabola.
二次函数可以写成多种形式。一般形式为 f(x) = ax2 + bx + c，其中 a ≠ 0。标准形式也称为顶点式，为 f(x) = a(x-h)2 + k，其中 (h,k) 是抛物线的顶点。
The coefficient a determines the direction and width of the parabola. If a > 0, the parabola opens upward; if a < 0, it opens downward. The larger |a| is, the narrower the parabola becomes.
系数 a 决定抛物线的开口方向和宽度。若 a > 0，抛物线开口向上；若 a < 0，开口向下。|a| 越大，抛物线越窄。
2. The Vertex | 顶点
The vertex is the turning point of the parabola. In vertex form a(x-h)2 + k, the vertex is directly (h,k). In general form, the x-coordinate of the vertex is given by h = -b2a, and the y-coordinate is k = f(h) or k = 4ac - b2⁄4a.
顶点是抛物线的转折点。在顶点式 a(x-h)2 + k 中，顶点就是 (h,k)。在一般形式中，顶点的横坐标为 h = -b2a，纵坐标为 k = f(h) 或 k = 4ac - b2⁄4a。
For a parabola opening upward, the vertex represents the minimum value of the function. For a parabola opening downward, the vertex represents the maximum value. This makes the vertex essential for optimisation problems.
对于开口向上的抛物线，顶点对应函数的最小值；对于开口向下的抛物线，顶点对应最大值。这使得顶点在优化问题中至关重要。
3. Axis of Symmetry | 对称轴
Every parabola is symmetric about a vertical line passing through its vertex. This line is called the axis of symmetry, with equation x = h. In general form, the axis of symmetry is x = -b2a.
每条抛物线都关于通过其顶点的一条竖直线对称。这条直线称为对称轴，方程为 x = h。在一般形式中，对称轴为 x = -b2a。
The axis of symmetry implies that if a point (x1, y) lies on the parabola, then its mirror image (2h - x1, y) also lies on the parabola. This property is often used to find the second root when one root is known.
对称轴意味着如果点 (x1, y) 在抛物线上，那么它的镜像点 (2h - x1, y) 也在抛物线上。这一性质常用于在已知一个根时求另一个根。
4. Roots and the Discriminant | 根与判别式
The roots of a quadratic equation ax2 + bx + c = 0 are the x-intercepts of the graph. They can be found by factorisation, completing the square, or using the quadratic formula x = -b ± √b2 - 4ac⁄2a.
二次方程 ax2 + bx + c = 0 的根就是图像与x轴的交点。可以通过因式分解、配方法或求根公式 x = -b ± √b2 - 4ac⁄2a 求得。
The discriminant Δ = b2 - 4ac determines the number of real roots:
判别式 Δ = b2 - 4ac 决定实根的个数：
| Discriminant 判别式 | Number of real roots 实根个数 | Graph interpretation 图像含义 |
| Δ > 0 | Two distinct real roots 两个不等实根 | Parabola crosses x-axis twice 与x轴有两个交点 |
| Δ = 0 | One repeated real root 一个重根 | Parabola touches x-axis once 与x轴相切 |
| Δ < 0 | No real roots 无实根 | Parabola does not meet x-axis 与x轴不相交 |
5. y-Intercept | y轴截距
The y-intercept is the point where the graph crosses the y-axis. Since all points on the y-axis have x = 0, the y-intercept is simply f(0) = c. Therefore, in the general form y = ax2 + bx + c, the constant term c gives the y-intercept directly.
y轴截距是图像与y轴的交点。由于y轴上所有点的 x = 0，所以y轴截距就是 f(0) = c。因此，在一般形式 y = ax2 + bx + c 中，常数项 c 直接给出y轴截距。
For example, the function y = 2x2 - 3x + 5 has y-intercept (0,5). The y-intercept is always a single point, because a function can have at most one y-value for x=0.
例如，函数 y = 2x2 - 3x + 5 的y轴截距为 (0,5)。y轴截距总是唯一一个点，因为函数在 x=0 处最多只有一个y值。
6. Sketching the Graph | 绘制图像
To sketch a quadratic graph accurately, follow these steps:
要准确绘制二次函数图像，请按以下步骤进行：
- Determine the direction of opening from the sign of a. 根据 a 的符号确定开口方向。
- Find the vertex using h = -b/(2a) and k = f(h). 用 h = -b/(2a) 和 k = f(h) 求顶点。
- Find the y-intercept (0,c). 求y轴截距 (0,c)。
- Find the roots by solving ax2+bx+c=0, if they exist. 解方程 ax2+bx+c=0 求根（若存在）。
- Plot the points and draw a smooth curve through them. 描点并画出光滑曲线。
Remember that the parabola is symmetric, so once you know one side of the axis of symmetry, you can reflect points to the other side.
记住抛物线是对称的，因此只要知道对称轴一侧的点，就可以将其对称反射到另一侧。
7. Transformations of Quadratic Functions | 二次函数的变换
The vertex form y = a(x-h)2 + k clearly shows transformations from the basic graph y = x2:
顶点式 y = a(x-h)2 + k 清楚地显示了从基本图像 y = x2 的变换：
- a: vertical stretch (if |a|>1) or compression (if 0<|a|<1), and reflection in x-axis if a<0. a：纵向拉伸（|a|>1）或压缩（0<|a|<1），若 a<0 则关于x轴反射。
- h: horizontal translation. The graph shifts right by h units if h>0, left if h<0. h：水平平移。若 h>0，图像向右平移 h 个单位；若 h<0 则向左。
- k: vertical translation. The graph shifts up by k units if k>0, down if k<0. k：纵向平移。若 k>0，图像向上平移 k 个单位；若 k<0 则向下。
These transformations are essential for understanding how parameters affect the parabola without plotting many points.
这些变换对于理解参数如何影响抛物线而无需描很多点至关重要。
8. Completing the Square | 配方法
Completing the square converts the general form to vertex form. Given y = ax2 + bx + c:
配方法将一般形式转换为顶点式。给定 y = ax2 + bx + c：
y = a(x + b2a)2 + 4ac-b2⁄4a
This is derived by factoring out a, halving the coefficient of x, and adjusting the constant term. The result directly gives the vertex coordinates.
该式通过提取 a、将 x 系数减半并调整常数项得到。结果直接给出顶点坐标。
For example, y = 2x2 + 8x + 5 can be written as y = 2(x+2)2 - 3, so the vertex is (-2,-3).
例如，y = 2x2 + 8x + 5 可写成 y = 2(x+2)2 - 3，因此顶点为 (-2,-3)。
9. Quadratic Inequalities | 二次不等式
The graph of a quadratic function helps solve inequalities like ax2+bx+c > 0 or ax2+bx+c ≤ 0. First find the roots, then determine the sign of the quadratic in each interval.
二次函数的图像有助于求解形如 ax2+bx+c > 0 或 ax2+bx+c ≤ 0 的不等式。先求根，再判断每个区间内二次式的符号。
If a>0, the quadratic is positive outside the roots and negative between them. If a<0, the sign pattern reverses. For example, x2 - 3x + 2 > 0 has roots 1 and 2, so the solution is x < 1 or x > 2.
若 a>0，二次式在两根之外为正，两根之间为负。若 a<0，符号模式相反。例如，x2 - 3x + 2 > 0 的根为1和2，因此解集为 x < 1 或 x > 2。
10. Sum and Product of Roots | 根的和与积
For a quadratic equation ax2 + bx + c = 0 with roots α and β, the following relationships hold:
对于二次方程 ax2 + bx + c = 0，若其根为 α 和 β，则有以下关系：
α + β = -ba, \quad αβ = ca
These relations are useful for checking roots or forming a quadratic equation when the roots are known. For instance, if roots are 3 and -2, the equation is x2 - (3-2)x + (3)(-2) = x2 - x - 6 = 0.
这些关系可用于检验根，或在已知根时构造二次方程。例如，若根为3和-2，则方程为 x2 - (3-2)x + (3)(-2) = x2 - x - 6 = 0。
11. Applications in Real-World Contexts | 实际应用
Quadratic functions model many real-world phenomena, such as projectile motion, area optimisation, and revenue problems. The vertex often represents a maximum height or minimum cost.
二次函数可模拟许多现实世界现象，如抛体运动、面积优化和利润问题。顶点通常代表最大高度或最小成本。
For a projectile launched upward, the height h(t) as a function of time t is typically given by h(t) = -gt2 + v0 t + h0, where g is gravity, v0 is initial velocity, and h0 is initial height. The maximum height occurs at the vertex t = v0/(2g).
对于向上发射的抛体，高度 h(t) 随时间 t 变化的函数通常为 h(t) = -gt2 + v0 t + h0，其中 g 为重力加速度，v0 为初速度，h0 为初始高度。最大高度出现在顶点 t = v0/(2g) 处。
12. Common Exam Pitfalls | 常见考试易错点
Many students make avoidable errors when working with quadratic functions. Here are some common pitfalls:
许多学生在处理二次函数时会犯一些可以避免的错误。以下是常见易错点：
- Forgetting that a cannot be zero; otherwise the function is linear. 忘记 a 不能为零，否则函数就是一次函数。
- Misidentifying the vertex sign when reading y = a(x-h)2 + k: the vertex is (h,k), not (-h,k). 读取 y = a(x-h)2 + k 时弄错顶点符号：顶点是 (h,k)，不是 (-h,k)。
- Using the discriminant on the equation ax2+bx+c=0 before ensuring the coefficient a is constant. 在确保系数 a 为常数之前就使用判别式。
- Forgetting to consider the direction of the inequality when multiplying by a negative number. 乘以负数时忘记改变不等式方向。
- Confusing the axis of symmetry with the roots: the axis of symmetry is the average of the roots: x = x1+x2⁄2. 将对称轴与根混淆：对称轴是两根的平均值：x = x1+x2⁄2。
Mastering these details will help you avoid careless marks lost in exams.
掌握这些细节将帮助你在考试中避免因粗心而失分。
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