IB Mathematics: A Comprehensive Guide to Quadratic Functions | IB数学：二次函数图像与性质归纳

📚 IB Mathematics: A Comprehensive Guide to Quadratic Functions | IB数学：二次函数图像与性质归纳

Quadratic functions, typically expressed in the form f(x) = ax² + bx + c (where a ≠ 0), form a cornerstone of the IB Mathematics curriculum. Mastering their graphs and properties is essential for success in both Analysis and Approaches (AA) and Applications and Interpretation (AI) courses. This guide provides a comprehensive yet concise summary of everything you need to know.

二次函数，通常表示为 f(x) = ax² + bx + c（其中 a ≠ 0），是 IB 数学课程中的基石。熟练掌握其图像与性质，对于在分析与方法（AA）和应用与解释（AI）两门课程中取得优异成绩都至关重要。本指南将为你提供一个全面而精炼的知识点归纳。

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1. Three Standard Forms of Quadratic Functions | 二次函数的三种标准形式

A quadratic function can be written in three main algebraic forms, each highlighting different features of its graph. Understanding how to convert between these forms is a critical skill for solving various types of IB questions.

二次函数可以写成三种主要的代数形式，每种形式都分别突出了其图像的不同特征。理解如何在三种形式之间进行转换，是解答各类 IB 题目的关键技能。

Form 形式	Equation 方程	Key Feature 关键特征
General 一般式	f(x) = ax² + bx + c	Y-intercept is c; easy for algebra 代数运算方便；y轴截距为c
Vertex 顶点式	f(x) = a(x – h)² + k	Vertex is (h, k); easy for transformations 顶点为(h,k)；便于图像变换
Factored 交点式	f(x) = a(x – p)(x – q)	X-intercepts are p and q; easy for roots x轴交点为p和q；便于求根

The general form is the most commonly used in algebraic manipulation. The vertex form is incredibly useful for identifying the maximum or minimum value of a function, which is a frequent optimization problem in IB. The factored form is ideal for quickly determining the roots and thus sketching the graph. Being able to move fluently between these forms, primarily through expanding brackets or completing the square, will give you a significant advantage.

一般式是代数运算中最常用的形式。顶点式在寻找函数最大值或最小值时非常有用，这也是 IB 中常见的优化问题。交点式则适合快速确定根并画出草图。能够熟练地在三种形式之间切换，主要是通过展开括号或因式分解，会让你在考试中占据显著优势。

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2. Direction of Opening and Y-Intercept | 开口方向与y轴截距

The leading coefficient ‘a’ (the coefficient of x²) determines the direction of the parabola. This is always the first thing you should check when analyzing a quadratic function.

首项系数 ‘a’（x²的系数）决定了抛物线的开口方向。这是分析二次函数时应该首先检查的关键点。

If a > 0, the parabola opens upwards (smile shape). The vertex is a minimum point.
If a < 0, the parabola opens downwards (frown shape). The vertex is a maximum point.

当 a > 0 时，抛物线开口向上（笑脸形状），顶点为最小值点。
当 a < 0 时，抛物线开口向下（哭脸形状），顶点为最大值点。

The constant term ‘c’ uniquely identifies the y-intercept. This is the point where the graph crosses the y-axis, which always has an x-coordinate of zero. Therefore, the y-intercept is simply (0, c). Plotting this point is an essential first step when sketching a parabola, as it provides a concrete reference point on the graph.

常数项 ‘c’ 唯一地确定了 y 轴截距。这是图像与 y 轴交叉的点，其 x 坐标始终为零。因此，y轴截距就是 (0, c)。在画抛物线草图时，标出这个点是至关重要的第一步，因为它提供了一个具体的参考点。

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3. Axis of Symmetry and Vertex | 对称轴与顶点

Every parabola is symmetric about a vertical line known as the axis of symmetry. This line passes directly through the vertex, effectively dividing the parabola into two mirror images.

每条抛物线都关于一条称为对称轴的垂直线对称。该直线直接穿过顶点，将抛物线完美地分成镜像的两半。

Axis of Symmetry: x = -b / (2a)

对称轴方程：x = -b / (2a)

Since the vertex lies on the axis of symmetry, its x-coordinate is also -b/(2a). To find the full coordinates of the vertex, substitute this x-value back into the original quadratic equation to solve for the y-coordinate. The vertex represents the turning point of the function, and its coordinates are vital for determining both the range of the function and its maximum or minimum value in optimization contexts.

由于顶点位于对称轴上，其 x 坐标同样为 -b/(2a)。要得到顶点的完整坐标，需要将这个 x 值代回原二次方程，以解出 y 坐标。顶点代表了函数的转折点，其坐标对于确定函数的值域以及解决优化问题中的最大值或最小值至关重要。

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4. The Discriminant and X-Intercepts | 判别式与x轴交点

The discriminant, denoted by the symbol Δ (Delta), is a powerful algebraic tool derived from the coefficients of the quadratic equation ax² + bx + c = 0. It allows us to determine the number and nature of the roots (which correspond to the x-intercepts of the graph) without actually solving the equation.

判别式，用符号 Δ（Delta）表示，是由一元二次方程 ax² + bx + c = 0 的系数推导出的一个强大代数工具。它使我们无需实际解方程，就能确定根（对应图像与 x 轴的交点）的数量与性质。

Discriminant: Δ = b² – 4ac

判别式：Δ = b² – 4ac

• If Δ > 0: The equation has two distinct real roots. The parabola crosses the x-axis at two distinct points.

  如果 Δ > 0：方程有两个不同的实数根，抛物线与 x 轴相交于两个不同的点。

• If Δ = 0: The equation has exactly one real root (a repeated root). The parabola touches the x-axis at exactly one point, meaning the vertex lies on the x-axis.

  如果 Δ = 0：方程有一个实数根（重根），抛物线在唯一点与 x 轴相切，即顶点落在 x 轴上。

• If Δ < 0: The equation has no real roots. The parabola does not intersect the x-axis at all (it lies entirely above or below the x-axis).

  如果 Δ < 0：方程没有实数根，抛物线与 x 轴完全没有交点（它完全位于 x 轴上方或下方）。

Additionally, when roots do exist, Vieta’s formulas provide a direct relationship between the roots and the coefficients. If the roots are x₁ and x₂, then x₁ + x₂ = -b/a and x₁ × x₂ = c/a. These relationships are frequently tested in IB papers, often as a shortcut to solving for unknown parameters.

此外，当根存在时，韦达定理直接揭示了根与系数之间的关系。若两根为 x₁ 和 x₂，则 x₁ + x₂ = -b/a，x₁ × x₂ = c/a。这些关系式在 IB 考试中经常出现，通常作为求解未知参数的捷径。

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5. Range and Optimal Value | 值域与最值

The vertex of a quadratic function represents its absolute maximum or absolute minimum value, often referred to as the ‘optimal value’ in application problems (e.g., maximizing profit or minimizing area). This optimal value determines the overall range of the function.

二次函数的顶点代表了其绝对最大值或绝对最小值，在应用题中常被称为”最值”（例如，最大化利润或最小化面积）。这个最值决定了函数的整体值域。

If a > 0: Range = [

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