📚 IB Mathematics: Quadratic Functions and Their Graphs — Key Exam Points | IB数学：二次函数图像与性质考点梳理
Quadratic functions form the backbone of algebraic modelling in the IB Mathematics curriculum. Whether you are taking Analysis and Approaches (AA) or Applications and Interpretation (AI), the ability to sketch, interpret, and manipulate \( y = ax^2 + bx + c \) is essential. This article consolidates the most frequently tested concepts—vertex form, discriminant analysis, transformations, inequalities, and applications—so that you can approach any quadratic problem with confidence.
二次函数是 IB 数学课程中代数建模的核心基础。无论你选择 Analysis and Approaches (AA) 还是 Applications and Interpretation (AI)，熟练绘制、解读和变换 y = ax² + bx + c 的图像都是必备技能。本文系统梳理考试中最常出现的考点——顶点式、判别式分析、图像变换、不等式及应用问题——帮助你从容应对各类二次函数题目。
1. Standard, Vertex, and Factored Forms | 一般式、顶点式与因式式
A quadratic function can be expressed in three useful forms, each revealing specific features of its graph. The standard form \( y = ax^2 + bx + c \) directly shows the y-intercept and the direction of opening. The vertex form \( y = a(x – h)^2 + k \) makes the turning point (h, k) immediately visible. The factored form \( y = a(x – r_1)(x – r_2) \) explicitly displays the x-intercepts or roots \( r_1 \) and \( r_2 \).
二次函数有三种常用表达形式，每种形式都直接揭示图像的不同特征。一般式 y = ax² + bx + c 直接给出 y 截距和开口方向；顶点式 y = a(x − h)² + k 使顶点 (h, k) 一目了然；因式式 y = a(x − r₁)(x − r₂) 则直接呈现两根 r₁ 和 r₂。
In IB exam questions, you will often be asked to convert between these forms using the method of completing the square. The conversion is not merely an algebraic exercise—it is the key to finding maximum or minimum values, solving optimisation problems, and sketching graphs accurately.
在 IB 考试中，经常要求你通过配方法在不同形式之间转换。这种转换不仅是代数操作，更是求解最大值或最小值、解决优化问题以及准确作图的关键。
Vertex: \( h = -b/2a \), \( k = c – b^2/(4a) \) | 顶点坐标: \( h = -b/2a \)，k = c − b²/(4a)
| Form / 形式 | Expression / 表达式 | Feature revealed / 揭示特征 |
| Standard / 一般式 | y = ax² + bx + c | y-intercept, concavity / y 截距、开口方向 |
| Vertex / 顶点式 | y = a(x − h)² + k | Vertex (h, k), line of symmetry / 顶点与对称轴 |
| Factored / 因式式 | y = a(x − r₁)(x − r₂) | Roots & x-intercepts / 根与 x 截距 |
2. Axis of Symmetry and the Vertex | 对称轴与顶点
The axis of symmetry is the vertical line that divides the parabola into two mirror images. Its equation is always \( x = h \), where \( h \) is the x-coordinate of the vertex. For a quadratic in standard form, this is computed directly from \( x = -b/(2a) \). This line is fundamental because every key feature—roots, intercepts, and turning point—can be analysed relative to it.
对称轴是把抛物线分为左右镜像的竖直直线，其方程始终为 x = h，其中 h 是顶点的 x 坐标。对于一般式二次函数，可直接通过 x = −b/(2a) 计算。对称轴至关重要，因为根、截距和顶点等所有关键特征都可以围绕它来分析。
To find the vertex, substitute \( x = -b/(2a) \) into the original equation to obtain the y-coordinate. Alternatively, complete the square: \( y = a(x – h)^2 + k \). The vertex is (h, k). For example, given \( y = 2x^2 – 8x + 5 \), we complete the square as \( y = 2(x – 2)^2 – 3 \), so the vertex is (2, −3).
求顶点坐标时，将 x = −b/(2a) 代入原方程即可得到 y 坐标。或者使用配方法：y = a(x − h)² + k，顶点即 (h, k)。例如，对于 y = 2x² − 8x + 5，配方得到 y = 2(x − 2)² − 3，因此顶点为 (2, −3)。
In the IB exams, identify whether the vertex is a maximum or a minimum by the sign of a. If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum.
在 IB 考试中，判断顶点是最大值还是最小值只需看 a 的符号。若 a > 0，抛物线开口向上，顶点为最小值；若 a < 0，开口向下，顶点为最大值。
3. The Discriminant and the Nature of Roots | 判别式与根的性质
The discriminant, \( \Delta = b^2 – 4ac \), determines how many real roots a quadratic equation \( ax^2 + bx + c = 0 \) has. This is one of the most frequently tested concepts in the IB examinations, because it bridges algebra, graph sketching, and problem-solving.
判别式 Δ = b² − 4ac 决定了一元二次方程 ax² + bx + c = 0 有多少个实数根。这是 IB 考试中最常考查的概念之一，因为它将代数、作图和问题求解连接在一起。
There are three cases to remember. If \( \Delta > 0 \), the equation has two distinct real roots, and the graph crosses the x-axis at two points. If \( \Delta = 0 \), there is exactly one real root (a repeated root), and the graph touches the x-axis at the vertex. If \( \Delta < 0 \), there are no real roots, and the graph does not intersect the x-axis at all.
需要记住三种情况。若 Δ > 0，方程有两个不同的实数根，图像与 x 轴相交于两点；若 Δ = 0，方程只有一个实数根（重根），图像在顶点处与 x 轴相切；若 Δ < 0，方程无实数根，图像与 x 轴无交点。
| Discriminant / 判别式 | Roots / 根的情况 | Graph intersection / 图像交点 |
| Δ > 0 | Two distinct real roots / 两个不等实根 | Crosses x-axis twice / 与 x 轴交于两点 |
| Δ = 0 | One repeated real root / 一个重根 | Touches x-axis once / 与 x 轴相切于一点 |
| Δ < 0 | No real roots / 无实数根 | Does not meet x-axis / 与 x 轴无交点 |
4. Roots, Sum and Product of Roots | 根、根之和与根之积
For the quadratic \( ax^2 + bx + c = 0 \) with roots \( r_1 \) and \( r_2 \), Vieta’s formulas give \( r_1 + r_2 = -b/a \) and \( r_1 r_2 = c/a \). These relationships are powerful because they allow you to determine properties of the roots without actually solving the equation. IB questions often ask for the sum and product of roots to construct a quadratic with given properties.
对于方程 ax² + bx + c = 0，设两根为 r₁ 和 r₂，韦达定理给出 r₁ + r₂ = −b/a，r₁ × r₂ = c/a。这些关系非常有用，因为可以不解方程直接判断根的性质。IB 题目常要求利用根之和与根之积构造满足特定条件的二次方程。
For example, if the roots of \( 3x^2 – 6x + 2 = 0 \) are \( \alpha \) and \( \beta \), then \( \alpha + \beta = 2 \) and \( \alpha\beta = 2/3 \). From these, you can find \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 – 2\alpha\beta = 4 – 4/3 = 8/3 \). Such derivations appear regularly in Paper 1 and Paper 2.
例如，若 3x² − 6x + 2 = 0 的两根为 α 和 β，则 α + β = 2，αβ = 2/3。由此可得 α² + β² = (α + β)² − 2αβ = 4 − 4/3 = 8/3。这类推导在 Paper 1 和 Paper 2 中经常出现。
Another common application is constructing a new equation whose roots are transformed in a known way, such as \( 2\alpha \) and \( 2\beta \), or \( \alpha + 1 \) and \( \beta + 1 \). In each case, compute the new sum and product, then rebuild the quadratic as \( x^2 – Sx + P = 0 \).
另一个常见应用是构造一个根经过某种变换的新方程，例如新根为 2α 和 2β，或 α + 1 和 β + 1。每种情况都需要计算新的和与积，再通过 x² − Sx + P = 0 重建方程。
5. Y-Intercept and X-Intercepts | y 截距与 x 截距
The y-intercept is the point where the graph crosses the y-axis. Since \( x = 0 \) at this point, it is simply \( y = c \) in the standard form. This is the easiest feature to read off from any quadratic equation, and it serves as an anchor point when sketching the graph.
y 截距是图像与 y 轴的交点。因为此处 x = 0，所以一般式中 y 截距就是 c。这是从二次方程中最容易读取的特征，也是作图时的基准点。
The x-intercepts are found by setting \( y = 0 \). In factored form, they are \( x = r_1 \) and \( x = r_2 \). If the quadratic is not easily factorable, use the quadratic formula:
x 截距通过令 y = 0 求解。在因式式中，它们就是 x = r₁ 和 x = r₂。若二次式不易因式分解，可使用二次方程求根公式：
x = (−b ± √(b² − 4ac)) / (2a)
In IB, you may be asked to sketch a graph from given information. A complete sketch should clearly label the vertex, the axis of symmetry, the y-intercept, and any real x-intercepts. The quadratic formula must be used when the coefficients are not integers or when the roots are irrational.
在 IB 考试中，你可能需要根据已知信息绘制图像。完整的草图应清晰标出顶点、对称轴、y 截距以及全部实数 x 截距。当系数不是整数或根为无理数时，必须使用求根公式。
6. Graph Transformations | 图像变换
Transformations of the graph \( y = x^2 \) are a fundamental component of the IB syllabus. The general vertex form \( y = a(x – h)^2 + k \) encodes all four types of transformation. The parameter \( a \) controls vertical stretch or compression and reflection in the x-axis. The parameter \( h \) controls horizontal translation, and \( k \) controls vertical translation.
y = x² 的图像变换是 IB 教学大纲的核心内容。顶点式 y = a(x − h)² + k 包含了全部四类变换：参数 a 控制纵向伸缩以及对 x 轴的反射；参数 h 控制水平平移；参数 k 控制竖直平移。
For horizontal shifts, it is essential to understand the direction: \( y = (x – 3)^2 \) shifts the graph 3 units to the right, while \( y = (x + 3)^2 \) shifts it 3 units to the left. A negative \( a \), such as \( y = -(x – 1)^2 + 2 \), reflects the parabola across the x-axis and then translates it up by 2 units.
关于水平平移，必须注意方向：y = (x − 3)² 将图像向右平移 3 个单位，而 y = (x + 3)² 向左平移 3 个单位。若 a 为负，例如 y = −(x − 1)² + 2，则先沿 x 轴翻折，再向上平移 2 个单位。
When describing transformations in the exam, use the correct order: horizontal translation, reflection/stretch, then vertical translation. Marks are often awarded for the precise wording of these transformations.
在考试中描述变换时，要注意正确顺序：先水平平移，再翻折或伸缩，最后竖直平移。对变换的精确描述通常占有多道题目的得分点。
7. Domain and Range | 定义域与值域
In the IB curriculum, the domain of a quadratic function is normally the set of all real numbers, \(\mathbb{R}\). However, in many applications and in multiple-choice questions, the domain may be restricted—for example, to model the height of a projectile for the time it is in the air.
在 IB 课程中，二次函数的定义域通常是全体实数集 ℝ。然而在许多应用题和选择题中，定义域可能受到限制——例如用于模拟抛射物在空中的高度时，定义域仅限于其在空中的时间段。
The range depends on the direction of the parabola. If \( a > 0 \), the range is \( [k, \infty) \), where \( k \) is the y-coordinate of the vertex. If \( a < 0 \), the range is \( (-\infty, k] \). For restricted domains, evaluate the function at the endpoints as well as at the vertex to find the exact range.
值域取决于抛物线的开口方向。若 a > 0，值域为 [k, +∞)，其中 k 是顶点的 y 坐标；若 a < 0，值域为 (−∞, k]。对于受限定义域，需同时计算端点值和顶点值来确定精确的值域。
Example: For \( f(x) = (x – 2)^2 – 1 \) with domain \( 0 \leq x \leq 5 \), the vertex is at (2, −1) and the endpoints give f(0) = 3 and f(5) = 8. Therefore the range is [−1, 8], since the maximum is at x = 5.
例题：设 f(x) = (x − 2)² − 1，定义域为 0 ≤ x ≤ 5。顶点在 (2, −1)，端点为 f(0) = 3、f(5) = 8，因此值域为 [−1, 8]，最大值在 x = 5 处取得。
8. Solving Quadratic Inequalities | 求解二次不等式
Quadratic inequalities such as \( ax^2 + bx + c > 0 \) or \( ax^2 + bx + c \leq 0 \) require you to combine algebraic solving with a graphical interpretation. The most reliable strategy is to first solve the corresponding equation to find the boundary points, then sketch a quick sign graph or number line to determine the solution intervals.
二次不等式如 ax² + bx + c > 0 或 ax² + bx + c ≤ 0 需要将代数求解与图像理解结合起来。最可靠的策略是：先解对应方程找到分界点，再通过快速的符号图或数轴来确定解区间。
For example, solve \( x^2 – 3x – 4 \leq 0 \). Factor to obtain \( (x – 4)(x + 1) \leq 0 \). The roots are \( x = -1 \) and \( x = 4 \). Because the parabola opens upward, the expression is negative between the roots. Therefore the solution is \( -1 \leq x \leq 4 \).
例如，解 x² − 3x − 4 ≤ 0。因式分解得 (x − 4)(x + 1) ≤ 0，根为 x = −1 和 x = 4。由于抛物线开口向上，表达式在两根之间为负，因此解集为 −1 ≤ x ≤ 4。
For a quadratic inequality involving a double root, such as \( (x – 2)^2 > 0 \), the solution is all real numbers except \( x = 2 \). Pay careful attention to whether the inequality is strict or non-strict, as this affects whether endpoints are included.
对于含有重根的二次不等式，例如 (x − 2)² > 0，解集为除 x = 2 以外的全体实数。还要特别注意不等式是严格还是非严格，因为这直接影响端点是否包含在解集中。
9. Quadratic Optimisation Problems | 二次函数优化问题
Optimisation using quadratic functions appears in both AA and AI examinations. In these problems, you construct a quadratic model from a real-life situation, then find the vertex to locate the maximum or minimum value. This is particularly common in business contexts (profit maximisation) and geometric contexts (maximising area).
利用二次函数解决优化问题在 AA 和 AI 考试中均经常出现。这类题目要求你根据实际情况建立二次模型，然后通过顶点求出最大值或最小值。这在商业情境（利润最大化）和几何情境（面积最大化）中尤为常见。
The general formula for the x-coordinate of the vertex, \( x = -b/(2a) \), provides a quick shortcut. Once the optimal x is obtained, substitute back to find the optimal value of the objective function. In exam questions, be careful to interpret the result in the context of the problem.
顶点 x 坐标公式 x = −b/(2a) 提供了一条快捷途径。得到最优 x 后，将其代回原函数即可求出目标函数的最优值。在考试题目中，要注意结合题意解释结果的实际含义。
Worked example: A farmer has 200 metres of fencing to enclose a rectangular field along a river. If the side along the river requires no fencing, let \( x \) be the width perpendicular to the river. Then the length parallel to the river is \( 200 – 2x \), and the area is \( A = x(200 – 2x) = -2x^2 + 200x \). The vertex occurs at \( x = -200/(2 \times -2) = 50 \). The maximum area is \( A = 50 \times 100 = 5000 \) m².
例题：一位农民有 200 米围栏，要沿河圈出一块矩形田地。河边一侧不需要围栏。设垂直于河岸的宽度为 x，则平行于河岸的长度为 200 − 2x，面积为 A = x(200 − 2x) = −2x² + 200x。顶点为 x = −200/(2 × (−2)) = 50。最大面积为 A = 50 × 100 = 5000 平方米。
10. Projectile Motion and Quadratic Modelling | 抛体运动与二次建模
Projectile motion is one of the most common applications of quadratic functions in the IB Applications and Interpretation course. The height of an object launched upward is modelled by \( h(t) = -4.9t^2 + v_0 t + h_0 \), where \( v_0 \) is the initial velocity and \( h_0 \) is the initial height. The negative coefficient of \( t^2 \) reflects the constant downward acceleration due to gravity.
抛体运动是 IB Applications and Interpretation 课程中二次函数最常见的应用之一。物体向上抛出的高度通常用 h(t) = −4.9t² + v₀t + h₀ 建模，其中 v₀ 是初速度，h₀ 是初始高度。t² 的负系数反映了重力引起的恒定向下加速度。
Exam questions typically ask for: the maximum height (found at the vertex), the time at which the object hits the ground (found by solving \( h(t) = 0 \)), and the range of times for which the height is above a given level (solving a quadratic inequality). These questions reward both algebraic fluency and contextual interpretation.
这类考题通常要求：求最大高度（在顶点处取得）、物体落地时间（解 h(t) = 0）、物体高度超过某个给定值的时间范围（解二次不等式）。此类问题既考查代数运算能力，也考查对实际情境的解释能力。
A common trap is forgetting that time \( t \) cannot be negative in a projectile problem. Always check whether the negative root of the quadratic equation is valid in the given context, and clearly state any restrictions on the domain.
一个常见陷阱是忘记在抛体问题中时间 t 不能为负。务必检查二次方程的负根在给定情境下是否有效，并明确写出定义域的限制条件。
11. Quadratic Functions in Inverse Relationships | 二次函数与反函数
One of the critical distinctions in IB Mathematics AA is understanding that a quadratic function does not have an inverse on its entire domain unless the domain is restricted. Because a parabola fails the horizontal line test, its inverse is not a function over \(\mathbb{R}\). To define an inverse, you must restrict the domain to either \( x \geq h \) or \( x \leq h \).
IB Mathematics AA 中一个重要的区别是：二次函数在其整个定义域上没有反函数，除非限制定义域。由于抛物线无法通过水平线测试，其反函数在 ℝ 上不是函数。要定义反函数，必须将定义域限制为 x ≥ h 或 x ≤ h。
For example, \( f(x) = (x – 1)^2 \) with domain \( x \geq 1 \) has an inverse \( f^{-1}(x) = \sqrt{x} + 1 \). If the domain is \( x \leq 1 \), the inverse is \( f^{-1}(x) = 1 – \sqrt{x} \). Questions on this topic frequently ask you to identify the correct restriction and then find the inverse.
例如，f(x) = (x − 1)² 在定义域 x ≥ 1 上存在反函数 f⁻¹(x) = √x + 1；若定义域为 x ≤ 1，则反函数为 f⁻¹(x) = 1 − √x。这类题目经常要求你判断正确的定义域限制，然后求反函数。从一般式转换为顶点式时，不要忘记提取系数 a 时每一项都要正确处理。
12. Common Mistakes and Exam Tips | 常见错误与应试技巧
The most common errors that IB students make in quadratic problems include: forgetting to set the discriminant to zero for tangent conditions, reversing the direction of horizontal translations, misidentifying the vertex coordinates from vertex form, and ignoring the possibility that a quadratic equation may have no real roots when solving application problems.
IB 学生在二次函数问题中最常见的错误包括：在相切条件下忘记令判别式等于零、搞反水平平移的方向、从顶点式中读取顶点坐标时出错，以及在解决应用问题时忽略方程可能没有实数根的情况。
Another frequent issue is arithmetic errors when completing the square. A reliable check is to expand your vertex form back to standard form to confirm that the coefficients match the original expression. When using the quadratic formula, write out the values of a, b, and c first, then substitute carefully.
另一个常见问题是配方时的计算错误。一个有效的检查方法是把顶点式展开回一般式，确认系数与原式一致。使用求根公式时，先列出 a、b、c 的值，再仔细代入。
Avoid losing easy marks by always checking the sign of \( a \) to determine whether the stationary point is a maximum or minimum. Draw a rough sketch whenever possible—a correct diagram can help you catch sign errors and interval mistakes in the final answer.
为避免不必要的失分，务必检查 a 的符号以确定极值是最大值还是最小值。只要条件允许就画一个草图——准确的图像能帮助你发现符号错误和最后答案中的区间失误。
Finally, practise with past paper questions and pay attention to the command terms. In IB, “find”, “determine”, and “hence” each carry different expectations. “Hence” requires you to use the result from the previous part, so read the question structure carefully before attempting the solution.
最后，多做历年真题并注意指令词的含义。在 IB 中，find、determine 和 hence 各有不同的要求。Hence 意味着你必须使用上一问的结论，所以在动笔前要仔细阅读题目结构。
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