IB Mathematics: Quadratic Equations and Functions Core Concepts | IB数学：二次方程与二次函数核心考点精讲

📚 IB Mathematics: Quadratic Equations and Functions Core Concepts | IB数学：二次方程与二次函数核心考点精讲

Quadratic equations and functions form a cornerstone of IB Mathematics. They appear in algebra, graphing, optimization, and even in calculus contexts. Mastering this topic is essential for both Analysis and Approaches (AA) and Applications and Interpretation (AI).

二次方程与二次函数是IB数学的基石。它们出现在代数、图像、优化，甚至微积分相关背景中。掌握这一主题对于分析与方法（AA）和应用与解释（AI）两门课程都至关重要。

1. Standard Form and Key Features | 标准形式与关键特征

A quadratic function is any function of the form f(x) = ax² + bx + c, where a, b and c are constants and a ≠ 0. The graph of a quadratic function is a parabola.

二次函数是形如 f(x) = ax² + bx + c 的函数，其中 a、b、c 为常数，且 a ≠ 0。二次函数的图像是一条抛物线。

When a > 0, the parabola opens upwards and the vertex is a minimum point. When a < 0, the parabola opens downwards and the vertex is a maximum point. The sign of a also determines the range of the function.

当 a > 0 时，抛物线开口向上，顶点为最小值点；当 a < 0 时，抛物线开口向下，顶点为最大值点。a 的符号还决定了函数的值域。

The key features of a quadratic graph are: the vertex, the axis of symmetry, the y-intercept, the x-intercepts (roots), and the concavity. The axis of symmetry always passes through the vertex, and its equation is:

二次图像的关键特征包括：顶点、对称轴、y 轴截距、x 轴截距（根）以及开口方向。对称轴始终经过顶点，其方程为：

x = −b / (2a)

The x-coordinate of the vertex is x = −b / (2a). To find the y-coordinate, substitute this value back into the function: y = f(−b / (2a)).

顶点的 x 坐标为 x = −b / (2a)。要求顶点的 y 坐标，只需将这个值代回原函数：y = f(−b / (2a))。

Another useful form is the vertex form: f(x) = a(x − h)² + k, where (h, k) is the vertex. Converting from standard form to vertex form is done by completing the square.

另一种常用的形式是顶点式：f(x) = a(x − h)² + k，其中 (h, k) 为顶点。通过配方法可以将标准式转换为顶点式。

2. Solving Quadratic Equations | 解二次方程

Solving a quadratic equation means finding the values of x that satisfy ax² + bx + c = 0. There are four main methods in the IB syllabus.

解二次方程就是找到满足 ax² + bx + c = 0 的 x 值。IB 课程中主要涉及四种解法。

Method 1: Factorization. Write the quadratic as a product of two linear factors, then set each factor equal to zero.

方法一：因式分解。将二次表达式写成两个一次因式的乘积，然后令每个因式等于零。

x² − 5x + 6 = (x − 2)(x − 3) = 0, so x = 2 or x = 3

Factorization is often the fastest method, but it only works when the equation has rational roots and can be easily decomposed.

因式分解通常是最快的方法，但只有当方程具有有理数根且易于分解时才适用。

Method 2: Quadratic Formula. For ax² + bx + c = 0, the solutions are given by:

方法二：求根公式。对于 ax² + bx + c = 0，解为：

x = [−b ± √(b² − 4ac)] / (2a)

This formula works for all quadratic equations, including those with irrational or complex solutions. It is particularly useful when factorization is not obvious.

求根公式适用于所有二次方程，包括具有无理根或复数根的情况。当因式分解不明显时，它尤其有用。

Method 3: Completing the Square. Rewrite the equation in the form (x + p)² = q, then take square roots.

方法三：配方法。将方程改写为 (x + p)² = q 的形式，然后两边开平方。

x² + 6x + 5 = (x + 3)² − 4 = 0, so (x + 3)² = 4, hence x = −1 or x = −5

Method 4: GDC / Calculator. On the IB exam, a graphing calculator can be used to find roots quickly, especially when checking answers or dealing with messy decimals.

方法四：图形计算器（GDC）。在IB考试中，可以使用图形计算器快速求根，特别是在检验答案或处理复杂小数时。

3. The Discriminant | 判别式

The discriminant of a quadratic equation ax² + bx + c = 0 is defined as Δ = b² − 4ac. It appears inside the square root of the quadratic formula and tells us about the nature of the roots without solving the equation.

二次方程 ax² + bx + c = 0 的判别式定义为 Δ = b² − 4ac。它出现在求根公式的根号内，无需解方程即可判断根的性质。

If Δ > 0, the equation has two distinct real roots. The graph crosses the x-axis at two points.

若 Δ > 0，方程有两个不相等的实数根。图像与 x 轴相交于两点。
If Δ = 0, the equation has one repeated real root. The graph touches the x-axis at exactly one point, which is the vertex.

若 Δ = 0，方程有一个二重实数根。图像与 x 轴仅相切于一点，该点就是顶点。
If Δ < 0, the equation has no real roots. The graph does not cross or touch the x-axis. In HL, we say it has two complex conjugate roots.

若 Δ < 0，方程没有实数根。图像不与 x 轴相交或相切。在高级水平（HL）中，我们说它有两个共轭复根。

For IB AA HL, complex roots are often written in the form z = a ± bi. The discriminant still determines whether the roots are real or complex.

在IB AA HL中，复数根通常写成 z = a ± bi 的形式。判别式仍然决定根是实数还是复数。

4. Sum and Product of Roots | 根的和与积

If α and β are the two roots of a quadratic equation ax² + bx + c = 0, then we can express the sum and product of the roots directly in terms of the coefficients:

若 α 和 β 是二次方程 ax² + bx + c = 0 的两个根，则根的和与积可以直接用系数表示：

Sum: α + β = −b / a

Product: αβ = c / a

For example, for the equation 2x² + 3x − 5 = 0, the sum of the roots is −3/2 and the product is −5/2. These relationships are especially useful when you need to construct a quadratic equation from given roots.

例如，对于方程 2x² + 3x − 5 = 0，根的和为 −3/2，根的积为 −5/2。这些关系在已知根来构造二次方程时特别有用。

If the equation is monic, meaning the coefficient of x² is 1, then x² − (sum of roots)x + (product of roots) = 0. This can be written as:

如果方程为首一系数，即 x² 的系数为 1，那么 x² − (根的和)x + (根的积) = 0。可写为：

x² − Sx + P = 0, where S = α + β and P = αβ

5. Quadratic Functions and Graphs | 二次函数的图像

Understanding the graph of a quadratic function is central to many IB questions. The graph can be written in several forms, each revealing different information.

理解二次函数图像是许多IB题目的核心。图像可以写成几种不同形式，每种形式都揭示不同的信息。

Standard form: f(x) = ax² + bx + c. The y-intercept is c, and the axis of symmetry is x = −b/(2a).

标准式：f(x) = ax² + bx + c。y 轴截距为 c，对称轴为 x = −b/(2a)。

Vertex form: f(x) = a(x − h)² + k. The vertex is (h, k), and the range is k ≤ f(x) if a > 0, or f(x) ≤ k if a < 0.

顶点式：f(x) = a(x − h)² + k。顶点为 (h, k)。若 a > 0，值域为 f(x) ≥ k；若 a < 0，值域为 f(x) ≤ k。

Intercept form: f(x) = a(x − p)(x − q), where p and q are the roots. The axis of symmetry is x = (p + q)/2, and the x-coordinate of the vertex is the midpoint of the roots.

交点式：f(x) = a(x − p)(x − q)，其中 p 和 q 为根。对称轴为 x = (p + q)/2，顶点的 x 坐标是两根的中点。

When sketching a parabola, always identify these five features: concavity direction, vertex, axis of symmetry, y-intercept, and x-intercepts (if they exist).

画抛物线时，务必找出以下五个特征：开口方向、顶点、对称轴、y 轴截距以及 x 轴截距（若存在）。

6. Transformations of Parabolas | 抛物线的变换

Quadratic functions can be transformed by changing the values of a, h and k in the vertex form f(x) = a(x − h)² + k.

通过改变顶点式 f(x) = a(x − h)² + k 中的 a、h 和 k，可以对二次函数进行变换。

Vertical translation: f(x) + d shifts the graph up if d > 0 and down if d < 0.

纵坐标平移：f(x) + d 在 d > 0 时上移，在 d < 0 时下移。
Horizontal translation: f(x − c) shifts the graph right by c units if c > 0, and left by |c| units if c < 0.

横坐标平移：f(x − c) 在 c > 0 时向右平移 c 个单位，在 c < 0 时向左平移 |c| 个单位。
Vertical stretch/compression: a f(x) stretches the graph vertically when |a| > 1 and compresses it when 0 < |a| < 1.

纵向拉伸/压缩：a f(x) 在 |a| > 1 时纵向拉伸，在 0 < |a| < 1 时纵向压缩。
Reflection: −f(x) reflects the graph across the x-axis; f(−x) reflects it across the y-axis.

翻转：−f(x) 将图像关于 x 轴翻转；f(−x) 将图像关于 y 轴翻转。

In the vertex form f(x) = a(x − h)² + k, the parameter h controls horizontal shift, k controls vertical shift, and a controls the vertical stretch and reflection. Note that f(x − h) moves the graph to the right by h units when h > 0, which is a common source of sign confusion.

在顶点式 f(x) = a(x − h)² + k 中，h 控制水平平移，k 控制垂直平移，a 控制纵向拉伸和翻转。注意 f(x − h) 在 h > 0 时向右平移 h 个单位，这是常见的符号混淆点。

7. Quadratic Inequalities | 二次不等式

Quadratic inequalities are solved by finding the critical roots and then testing intervals on a number line or by sketching the parabola.

解二次不等式需要先求出临界根，然后在数轴上测试区间，或者画出抛物线草图。

Example: Solve x² − x − 6 ≥ 0.

示例：解不等式 x² − x − 6 ≥ 0。

First factorize: (x − 3)(x + 2) ≥ 0. The critical values are x = 3 and x = −2. Because the parabola opens upwards, the graph is above or on the x-axis when x ≤ −2 or x ≥ 3.

首先因式分解：(x − 3)(x + 2) ≥ 0。临界值为 x = 3 和 x = −2。由于抛物线开口向上，图像在 x ≤ −2 或 x ≥ 3 时位于 x 轴上方或与 x 轴相交。

x ∈ (−∞, −2] ∪ [3, ∞)

For a strict inequality such as x² − x − 6 > 0, the critical values are excluded, so the solution would be x < −2 or x > 3.

对于严格不等式，例如 x² − x − 6 > 0，临界值不能包含在内，因此解为 x < −2 或 x > 3。

Always check the direction of the inequality by testing one point in each interval. This avoids sign errors and ensures the correct union of intervals.

在解不等式时，务必在每个区间内取一个测试点验证不等式方向，这样可以避免符号错误并确保区间并集正确。

8. Applications in Word Problems | 应用题中的二次函数

Quadratic functions are widely used to model real-world situations such as projectile motion, area optimization, and profit maximization.

二次函数广泛用于建模现实问题，例如抛体运动、面积优化和利润最大化。

Example: Projectile motion. A ball is thrown upward with height given by h(t) = −5t² + 20t + 1, where h is in meters and t is in seconds. Find the maximum height.

示例：抛体运动。一个球被向上抛出，高度由 h(t) = −5t² + 20t + 1 给出，h 以米为单位，t 以秒为单位。求最大高度。

The vertex occurs at t = −b / (2a) = −20 / (2 × −5) = 2 seconds. The maximum height is h(2) = −5(2²) + 20(2) + 1 = 21 meters.

顶点出现在 t = −b / (2a) = −20 / (2 × −5) = 2 秒。最大高度为 h(2) = −5(2²) + 20(2) + 1 = 21 米。

Example: Area optimization. A rectangular enclosure is built with 100 m of fencing on three sides, with the fourth side along a wall. Let x be the width. The area is A(x) = x(100 − 2x) = 100x − 2x². The maximum area occurs at x = 25 m, giving A = 1250 m².

示例：面积优化。一个矩形围栏用 100 m 的篱笆围成三边，第四边靠墙。设宽为 x，面积为 A(x) = x(100 − 2x) = 100x − 2x²。最大面积出现在 x = 25 m，此时 A = 1250 m²。

When solving word problems, define the variables clearly, write the quadratic model, and state whether you are looking for a maximum or minimum. Always consider the domain restrictions based on the physical context.

在解答应用题时，请清晰定义变量，写出二次模型，并明确是求最大值还是最小值。同时务必考虑实际背景下的定义域限制。

9. Common IB Exam Pitfalls | 常见IB考试陷阱

Many students lose marks on quadratic questions due to avoidable errors. Here are the most frequent pitfalls and how to avoid them.

许多学生在二次函数题目上因可避免的错误而失分。以下是最常见的陷阱以及规避方法。

Forgetting a ≠ 0: If a = 0, the equation is linear, not quadratic. Always check the leading coefficient before applying quadratic methods.

忘记 a ≠ 0：若 a = 0，方程为一次方程，而不是二次方程。在使用二次方法前，务必检查最高次项系数。
Dropping negative roots: When taking square roots, remember both the positive and negative solutions. For example, x² = 9 gives x = 3 or x = −3.

遗漏负根：开平方时，务必同时考虑正负解。例如 x² = 9 得 x = 3 或 x = −3。
Not setting the equation to zero: When solving x² + 5x = 6, you must rewrite as x² + 5x − 6 = 0 before factorizing.

未将方程化为零：解 x² + 5x = 6 时，必须先改写为 x² + 5x − 6 = 0 再因式分解。
Confusing roots with x-intercepts: Roots are the solutions to the equation f(x) = 0; they are exactly the x-intercepts of the graph. Do not mix them with the y-intercept.

混淆根与 x 轴截距：根是方程 f(x) = 0 的解，也就是图像与 x 轴的交点；不要与 y 轴截距混淆。
Using the discriminant incorrectly for complex roots: When Δ < 0, do not say "no solutions"; say "no real solutions" in IB, and in HL describe the complex conjugate roots.

对复数根使用判别式错误：当 Δ < 0 时，不要说“无解”，而应说“无实数解”；在HL中应描述共轭复根。
Calculator rounding errors: In IB exams, if an answer is expected as an exact value, do not give only a rounded decimal. Use surds or fractions when necessary.

计算器舍入误差：在IB考试中，如果答案要求精确值，不要只给四舍五入的小数。必要时使用根号或分数形式。

10. Practice Questions | 巩固练习

Here are three practice questions that reflect typical IB exam style. Try to solve them before checking the short solutions below.

以下是三道典型IB考试风格的训练题。请先尝试解答，再看下方的简要解析。

Question 1: The quadratic function f(x) = 2x² + kx + 18 has exactly one real root. Find the positive value of k.

题目 1：已知二次函数 f(x) = 2x² + kx + 18 只有一个实数根。求 k 的正值。

Solution: Exactly one real root means Δ = 0. Thus k² − 4(2)(18) = 0, so k² = 144, and the positive value is k = 12.

解析：只有一个实数根意味着 Δ = 0。因此 k² − 4(2)(18) = 0，所以 k² = 144，正值为 k = 12。

Question 2: If α and β are the roots of x² − 6x + 2 = 0, find α + β and αβ.

题目 2：若 α 和 β 是 x² − 6x + 2 = 0 的两个根，求 α + β 和 αβ。

Solution: By Vieta’s formulas, α + β = 6 and αβ = 2. This is valid even when the roots are irrational, because the formulas use coefficients only.

解析：根据韦达定理，α + β = 6，αβ = 2。即使根是无理数，该公式依然成立，因为只使用系数。

Question 3: Solve 3x² − 2x − 1 = 0, giving your answers exactly.

题目 3：解 3x² − 2x − 1 = 0，给出精确答案。

Solution: Using the quadratic formula with a = 3, b = −2, c = −1:

解析：在求根公式中代入 a = 3，b = −2，c = −1：

x = [2 ± √(4 + 12)] / 6 = [2 ± √16] / 6 = (2 ± 4) / 6

Therefore x = 1 or x = −1/3.

因此 x = 1 或 x = −1/3。

Practice these types of questions until the methods become automatic. Quadratic concepts are guaranteed to appear in some form on the IB exam, whether in Paper 1, Paper 2, or in the internal assessment context.Published by TutorHao | IB Mathematics Revision Series | aleveler.com

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IB Mathematics: Quadratic Equations and Functions Core Concepts | IB数学：二次方程与二次函数核心考点精讲