📚 Mastering Quadratic Functions: IB Math Exercise 4B | 掌握二次函数：IB数学练习4B
Quadratic functions lie at the very heart of the IB Mathematics curriculum. From sketching parabolas to solving real-world optimisation problems, the skills developed in Exercise 4B form the foundation for success in both Standard Level and Higher Level courses. This article unpacks every essential concept, technique, and common pitfall you will encounter, pairing clear English explanations with their Chinese translations to support bilingual mastery.
二次函数是IB数学课程的核心内容。从绘制抛物线到解决现实世界中的优化问题，练习4B中培养的技能为在标准水平和高级水平课程中取得成功奠定了基础。本文梳理了每个基本概念、技巧和常见陷阱，将清晰的英文解释与中文翻译配对，以支持双语掌握。
1. The General Form of a Quadratic | 二次函数的一般形式
x² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is called a parabola. The coefficient a determines whether the parabola opens upwards (a > 0) or downwards (a < 0).
二次函数是可以写成 f(x) = ax² + bx + c 形式的任何函数，其中 a、b 和 c 是常数且 a ≠ 0。二次函数的图像称为抛物线。系数 a 决定了抛物线是向上开口（a > 0）还是向下开口（a < 0）。
The constant term c gives the y-intercept of the graph, which is the point (0, c). The parameters a and b influence the position of the axis of symmetry and the vertex. Understanding how these three numbers interact is the first step in mastering Exercise 4B questions.
常数项 c 给出了图像的 y 轴截距，即点 (0, c)。参数 a 和 b 影响对称轴和顶点的位置。理解这三个数字如何相互作用是掌握练习4B题目的第一步。
- Example: f(x) = 2x² - 4x + 1 has a = 2, b = -4, c = 1.
- 示例：f(x) = 2x² - 4x + 1 中 a = 2，b = -4，c = 1。
2. Vertex Form and Completing the Square | 顶点式与配方法
k) is the vertex of the parabola. The process of converting from general form to vertex form is called completing the square. This technique not only reveals the vertex but also helps solve equations and sketch graphs quickly.
任何二次函数都可以改写成顶点式：f(x) = a(x - h)² + k，其中 (h, k) 是抛物线的顶点。从一般形式转换为顶点形式的过程称为配方法。这种方法不仅能揭示顶点，还有助于快速解方程和绘制图形。
expression clearly shows the horizontal shift h = -b/(2a) and the vertical shift k = f(h).
要对 f(x) = ax² + bx + c 进行配方，先从首两项提出 a，然后在括号内加上并减去 (b/(2a))²。得到的表达式清楚地显示了水平位移 h = -b/(2a) 和垂直位移 k = f(h)。
. The vertex is (2, -3). This tells us the minimum value of the function is -3 when x = 2.
例如，f(x) = 2x² - 8x + 5 变为 f(x) = 2(x - 2)² - 3。顶点是 (2, -3)。这告诉我们当 x = 2 时函数的最小值是 -3。
3. The Axis of Symmetry and Vertex Coordinates | 对称轴与顶点坐标
. This formula is derived directly from the quadratic formula or by completing the square.
每条抛物线都有一条通过其顶点的垂直对称线。对称轴的方程是 x = -b/(2a)。这个公式直接来源于二次公式或配方法。
Once you know the x-coordinate of the vertex, substituting it into the function yields the y-coordinate. So the vertex is (-b/(2a), f(-b/(2a))). In many IB problems, you are asked to identify the vertex without fully completing the square—using this quick formula saves time.
一旦知道了顶点的 x 坐标，将其代入函数即可得到 y 坐标。因此顶点是 (-b/(2a), f(-b/(2a)))。在许多IB问题中，你被要求在不进行完整配方的情况下确定顶点——使用这个快速公式可以节省时间。
) = (1, 5). Because a < 0, the parabola opens downwards and the vertex is a maximum point.
对于 f(x) = -3x² + 6x + 2，对称轴是 x = -6/(2×-3) = 1。顶点是 (1, f(1)) = (1, 5)。因为 a < 0，抛物线向下开口，顶点是一个最大值点。
4. Solving Quadratic Equations: Factoring | 解二次方程：因式分解法
becomes straightforward. Set each factor equal to zero and solve for x. This method works best when a, b, and c are integers and the discriminant is a perfect square.
当二次式可以写成两个一次因式的乘积时，解方程 ax² + bx + c = 0 就变得简单了。令每个因式等于零并解出 x。当 a、b 和 c 是整数且判别式为完全平方数时，这种方法效果最佳。
factor by grouping. Always check that your factors expand back to the original expression.
例如，x² - 5x + 6 = 0 因式分解为 (x - 2)(x - 3) = 0，得到解 x = 2 和 x = 3。如果 a ≠ 1，你可能需要用分组分解法。务必检查你的因式展开后是否能还原为原来的表达式。
Remember that not all quadratics factor nicely with integers. In those cases, you must use the quadratic formula or completing the square.
请记住，并非所有二次式都能用整数漂亮地因式分解。在那些情况下，你必须使用二次公式或配方法。
5. The Quadratic Formula and Discriminant | 二次公式与判别式
The quadratic formula, x = (-b ± √b² - 4ac = b² - 4ac, is called the discriminant. It determines the nature and number of the roots.
二次公式 x = (-b ± √b² - 4ac) / (2a) 提供了任何二次方程的解。平方根下的表达式 Δ = b² - 4ac 被称为判别式。它决定了根的性质和数量。
root (a repeated root), meaning the parabola touches the x-axis at its vertex. If Δ < 0, there are no real roots; the parabola does not intersect the x-axis, and the roots are complex conjugates.
如果 Δ > 0，有两个不同的实根。如果 Δ = 0，恰好有一个实根（重根），意味着抛物线在其顶点处与 x 轴相切。如果 Δ < 0，没有实根；抛物线不与 x 轴相交，根为共轭复数。
In IB exams, you will often be asked to “find the set of values of k for which the equation has two distinct real roots.” This requires setting Δ > 0 and solving the resulting inequality. Practice using the discriminant is a key part of Exercise 4B.
在IB考试中，你经常会被要求“找出使方程有两个不同实根的 k 值的集合”。这需要设定 Δ > 0 并解出由此产生的不等式。练习使用判别式是练习4B的关键部分。
6. Graphical Transformations of Parabolas | 抛物线的图形变换
ontal translations by h, and vertical translations by k combine to give y = a(x - h)² + k.
从基本的抛物线 y = x² 开始，我们可以应用变换将其映射到任何二次函数。以因子 a 进行垂直拉伸/压缩，水平平移 h，以及垂直平移 k，组合得到 y = a(x - h)² + k。
A negative sign in front of a reflects the parabola across the x-axis. Replacing x with (x - h) shifts the graph right by h units; adding k outside shifts it up by k units. Understanding these transformations allows you to sketch any quadratic quickly and accurately.
a 前面的负号会使抛物线关于 x 轴反射。用 (x - h) 替换 x 会使图像向右平移 h 个单位；在外面加上 k 会使图像向上平移 k 个单位。理解这些变换可以让你快速而准确地绘制任何二次函数的图像。
xis, a vertical compression by factor ½, a shift left by 3, and a shift up by 4. The vertex is at (-3, 4).
例如，y = -½(x + 3)² + 4 是从 y = x² 经过关于 x 轴反射、以因子 ½ 垂直压缩、向左平移3、向上平移4得到的。顶点在 (-3, 4)。
7. Intercepts: Finding Where the Parabola Meets the Axes | 截距：确定抛物线与坐标轴的交点
al whether the parabola cuts the x-axis (Δ > 0), touches it (Δ = 0), or avoids it (Δ < 0).
任何二次函数 f(x) = ax² + bx + c 的 y 轴截距就是 (0, c)。要找到 x 轴截距，令 f(x) = 0 并求解。x 轴截距是方程的实根，它们揭示了抛物线是穿过 x 轴 (Δ > 0)、接触 x 轴 (Δ = 0) 还是避开 x 轴 (Δ < 0)。
When sketching, always plot the vertex first, then the axis of symmetry, and mark the intercepts. The y-intercept is symmetric to another point on the parabola; use the axis of symmetry to find it. This gives you a quick five-point sketch.
在绘图时，始终先标出顶点，然后画出对称轴，并标出截距。y 轴截距与抛物线上的另一个点对称；利用对称轴找到它。这能让你快速完成五点草图。
ts come from 2x² - 4x - 6 = 0 → x² - 2x - 3 = 0 → (x - 3)(x + 1) = 0, so x = 3 and x = -1.
对于 f(x) = 2x² - 4x - 6，y 轴截距是 (0, -6)。x 轴截距来自 2x² - 4x - 6 = 0 → x² - 2x - 3 = 0 → (x - 3)(x + 1) = 0，因此 x = 3 和 x = -1。
8. Domain, Range, and Maximum/Minimum Values | 定义域、值域与最大/最小值
The domain of any quadratic function is all real numbers, ℝ e is [k, ∞). If a < 0, the function has a maximum at the vertex, and the range is (-∞, k].
任何二次函数的定义域都是全体实数 ℝ。然而，值域取决于顶点和开口方向。如果 a > 0，函数在顶点处有最小值，值域为 [k, ∞)。如果 a < 0，函数在顶点处有最大值，值域为 (-∞, k]。
Questions in Exercise 4B often ask for the minimum or maximum value of a quadratic expression defined on a restricted domain. You must check whether the vertex lies within the given interval; if not, the extreme value occurs at an endpoint.
练习4B中的题目经常要求在有限定义域上求二次表达式的最小值或最大值。你必须检查顶点是否位于给定区间内；如果不在，极值就会出现在端点处。
e the interval. Minimum f(2)=1; maximum at an endpoint: f(1)=2, f(5)=10, so maximum is 10.
例如，f(x) = (x - 2)² + 1 在定义域 1 ≤ x ≤ 5 上，顶点在 x = 2，位于区间内。最小值 f(2)=1；最大值在端点：f(1)=2，f(5)=10，所以最大值为10。
9. The Sum and Product of Roots | 根的和与积
c/a. These relationships come from comparing coefficients and are extremely useful for forming equations from given roots or solving symmetric problems.
对于有根 α 和 β 的二次方程 ax² + bx + c = 0，根的和为 α + β = -b/a，根的积为 αβ = c/a。这些关系来自系数比较，对于根据给定根构造方程或解决对称问题非常有用。
In IB questions, you might be told that the roots are in a certain ratio, or that one root is twice the other. Use sum and product relations to set up a system of equations and solve for the unknown coefficient. This avoids solving the full quadratic.
在IB题目中，你可能会被告知根成一定比例，或者一个根是另一个根的两倍。利用和与积的关系建立方程组，并求解未知系数。这避免了求解整个二次方程。
er, let roots be r and 2r. Then r + 2r = -k/2 and r·2r = 12/2 → 2r² = 6 → r² = 3, so r = ±√3. Then k = -2(3r) = -6r = ∓6√3.
示例：如果 2x² + kx + 12 = 0 的一个根是另一个根的两倍，设根为 r 和 2r。则 r + 2r = -k/2，且 r·2r = 12/2 → 2r² = 6 → r² = 3，所以 r = ±√3。然后 k = -2(3r) = -6r = ∓6√3。
10. Quadratic Inequalities | 二次不等式
Solving a quadratic inequality such as ax² + bx + c > 0 involves finding the intervals where the parabola is above the x-axis. First solve the equality to find critical values, then test intervals or use a sign diagram. The direction of the parabola (a > 0 or a < 0) determines the sign of the expression outside the roots.
求解二次不等式，如 ax² + bx + c > 0，需要找出抛物线在 x 轴上方的区间。首先解等式找到临界值，然后测试区间或使用符号表。抛物线的开口方向（a > 0 或 a < 0）决定了表达式在根外侧的符号。
For a > 0, the quadratic is positive outside the interval between the roots and negative inside. For a < 0, the opposite holds. Always draw a quick sketch to confirm your solution set. The answer is typically expressed as a union of intervals.
对于 a > 0，二次式在两根之间的区间外为正，在区间内为负。对于 a < 0，情况相反。务必快速画个草图来确认你的解集。答案通常表示为区间的并集。
re 2 and 3. Since a = 1 > 0, the solution is x < 2 or x > 3. In interval notation: (-∞, 2) ∪ (3, ∞).
示例：解 x² - 5x + 6 > 0。根为2和3。因为 a = 1 > 0，解为 x < 2 或 x > 3。区间表示法：(-∞, 2) ∪ (3, ∞)。
11. Applications in Modelling and Optimisation | 建模与优化中的应用
Many IB Exercise 4B problems model real-world situations: projectile motion, area maximisation, revenue and cost functions. The vertex of a quadratic model gives the maximum or minimum value of the quantity of interest. Identifying the correct domain and interpreting results in context is crucial.
许多IB练习4B的题目模拟了现实世界的情况：抛体运动、面积最大化、收益和成本函数。二次模型的顶点给出了所关注数量的最大值或最小值。确定正确的定义域并在上下文中解释结果至关重要。
rtex t = -b/(2a). The time when the ball hits the ground is the positive root of h(t) = 0.
例如，一个球被向上抛出遵循 h(t) = -4.9t² + vt + h₀，这是一个向下开口的抛物线。最大高度在顶点 t = -b/(2a) 处出现。球落地的时间是 h(t) = 0 的正根。
/2 - x) is quadratic. By completing the square, you can show that a square maximises area.
另一个经典问题是矩形围栏问题：用固定长度的围栏，面积 A = x(P/2 - x) 是二次的。通过配方，你可以证明正方形使面积最大化。
12. Common Mistakes and Exam Tips | 常见错误与应试技巧
Even strong students slip up on quadratics. A few frequent errors: forgetting to set the equation to zero before factoring; misapplying the discriminant (e.g., using it with inequalities incorrectly); losing the negative sign when moving terms; and confusing the axis of symmetry formula with the quadratic formula.
即使是优秀的学生在二次函数上也会出错。一些常见错误：在因式分解前忘记将方程设为零；误用判别式（例如，在不等式中错误使用）；移项时丢失负号；以及混淆对称轴公式和二次公式。
In the exam, always double-check your factoring by expanding mentally. Use the discriminant to verify the nature of roots before solving. Write down the vertex form alongside the general form to cross-check transformations. If a word problem asks for a maximum, confirm that indeed the coefficient of x² is negative.
在考试中，始终通过心算展开来复验你的因式分解。在求解之前使用判别式验证根的性质。在一般形式旁边写出顶点形式以交叉检查变换。如果一道文字题要求最大值，确认 x² 的系数确实是负的。
Finally, manage your time: quadratics often appear early in IB papers, so being fluent with these techniques builds confidence for the rest of the exam. Practice Exercise 4B thoroughly until each method becomes second nature.
最后，管理好你的时间：二次函数经常出现在IB试卷的前半部分，因此熟练掌握这些技巧能为其余考试内容建立信心。充分练习练习4B，直到每种方法都成为你的第二天性。
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