📚 5 Functions of ax + b | 五种 ax+b 型函数
In IB Mathematics, the expression ax + b acts as a fundamental building block that appears inside linear, quadratic, absolute value, reciprocal and square‑root functions. Mastering these five function families not only strengthens algebraic manipulation but also prepares you for modelling, calculus and graphical transformations. This article unpacks each function step by step, clarifies the role of parameters a and b, and links the concepts to typical IB exam questions.
在 IB 数学中，表达式 ax + b 是一个最基础的构建模块，它出现在线性、二次、绝对值、倒数和平方根函数内部。掌握这五个函数族不仅能强化代数运算能力，还能为建模、微积分和图像变换打下扎实基础。本文逐一拆解每个函数，厘清参数 a 和 b 的作用，并与典型的 IB 考题建立联系。
1. Understanding ax + b as a Building Block | 理解基础模块 ax + b
The linear expression ax + b, with a ≠ 0, is the simplest non‑constant function. It combines a stretch factor a and a vertical shift b when viewed as y = ax + b. However, when it is embedded inside other functions — for example as the argument of a square root — it controls horizontal transformations. IB students learn to interpret a and b as tools for translation and scaling, first along the x‑axis and then applying an outer rule.
线性表达式 ax + b（其中 a ≠ 0）是最简单的非常值函数。当写成 y = ax + b 时，它包含伸缩因子 a 和垂直平移量 b。但一旦它嵌入其他函数内部——例如作为平方根的自变量——它便控制着水平变换。IB 学生要学会将 a 和 b 视为平移与缩放的工具，先在 x 轴上操作，再施加外层的运算法则。
2. Function 1: Linear f(x) = ax + b | 函数一：线性函数 f(x) = ax + b
The parent function f(x) = x is transformed directly: a gives the slope (gradient) and b gives the y‑intercept. The graph is a straight line crossing the y‑axis at (0, b) and rising if a > 0 or falling if a < 0. The domain is all real numbers, ℝ, and the range is also ℝ. Zeros occur at x = −b/a. In IB exams, linear functions often set the stage for piecewise definitions or systems of equations.
母函数 f(x) = x 直接变换得到该形式：a 给出斜率（梯度），b 给出 y 轴截距。图像是一条直线，与 y 轴交于 (0, b)，当 a > 0 时上升，a < 0 时下降。定义域为全体实数 ℝ，值域也是 ℝ。零点为 x = −b/a。在 IB 考试中，线性函数常作为分段函数或方程组的基础出现。
3. Function 2: Quadratic f(x) = (ax + b)² | 函数二：二次函数 f(x) = (ax + b)²
Writing f(x) = (ax + b)² shows a horizontal transformation of the basic quadratic y = x². The vertex lies at x = −b/a, y = 0. Factorising gives f(x) = a² (x + b/a)², so the graph is a parabola with a vertical stretch of a² and a horizontal shift of −b/a units. The domain is ℝ is [0, ∞) because the square is never negative. The axis of symmetry is the vertical line x = −b/a.
将函数写成 f(x) = (ax + b)²，它表示基本二次函数 y = x² 的一个水平变换。顶点位于 x = −b/a，y = 0。因式分解得 f(x) = a² (x + b/a)²，因此图像是一条抛物线，垂直拉伸倍数为 a²，水平平移 −b/a 个单位。定义域为 ℝ；值域为 [0, ∞)，因为平方永不为负。对称轴是直线 x = −b/a。
4. Function 3: Absolute Value f(x) = |ax + b| | 函数三：绝对值函数 f(x) = |ax + b|
The graph of f(x) = |ax + b| is a V‑shape whose vertex is at x = −b/a. For x > −b/a the function behaves like y = ax + b; for x < −b/a it behaves like y = −(ax + b). Therefore the right branch has slope a and the left branch has slope −a. Domain is ℝ eliminates negative outputs, making this function ideal for modelling distances or errors.
f(x) = |ax + b| 的图像呈 V 形，顶点在 x = −b/a。当 x > −b/a 时，函数表现为 y = ax + b；当 x < −b/a 时，表现为 y = −(ax + b)。因此右侧分支斜率为 a，左侧分支斜率为 −a。定义域为 ℝ；值域为 [0, ∞)。绝对值消除了负输出，使该函数非常适合对距离或误差进行建模。
5. Function 4: Reciprocal f(x) = 1/(ax + b) | 函数四：倒数函数 f(x) = 1/(ax + b)
The reciprocal function f(x) = 1/(ax + b) is undefined where the denominator equals zero, i.e. at x = −b/a. Its graph is a rectangular hyperbola with a vertical asymptote x = −b/a and a horizontal asymptote y = 0. The domain is {x ∈ ℝ | x ≠ −b/a}; the range is {y ∈ ℝ | y ≠ 0}. The shape depends on a: if a > 0, the graph is in quadrants I and III of its asymptotes; if a < 0, it lies in quadrants II and IV.
倒数函数 f(x) = 1/(ax + b) 在分母为零处无定义，即 x = −b/a。其图像是一条等轴双曲线，具有垂直渐近线 x = −b/a 和水平渐近线 y = 0。定义域为 {x ∈ ℝ | x ≠ −b/a}；值域为 {y ∈ ℝ | y ≠ 0}。图像形态取决于 a：若 a > 0，图像位于渐近线坐标系的 I、III 象限；若 a < 0，则位于 II、IV 象限。
6. Function 5: Square Root f(x) = √ax + b | 函数五：平方根函数 f(x) = √ax + b
For f(x) = √ax + b to be real, the radicand must be non‑negative: ax + b ≥ 0. Solving the inequality gives the domain. If a > 0, the domain is [−b/a, ∞); if a < 0, the domain is (−∞, −b/a]. The graph is a half‑parabola lying on its side, starting at the endpoint x = −b/a. The range is [0, ∞). The basic shape is a transformation of y = √x under a horizontal stretch/compression and a horizontal shift.
为使 f(x) = √ax + b 为实函数，被开方数须非负：ax + b ≥ 0。解此不等式即得定义域：若 a > 0，定义域为 [−b/a, ∞)；若 a < 0，定义域为 (−∞, −b/a]。图像是一条侧卧的半抛物线，起点为 x = −b/a。值域为 [0, ∞)。其基本形状是 y = √x 经过水平伸缩和平移得到的变换。
7. The Common Thread: Zeros and Critical Points | 共同线索：零点与关键点
All five functions share a significant x‑value: x = −b/a. For the linear function it is the root; for the quadratic it is the vertex and double root; for the absolute value it is the vertex; for the reciprocal it is the vertical asymptote and excluded value; for the square root it is the endpoint of the domain. IB exam questions frequently ask for these features, so recognising x = −b/a as a 'critical coordinate' saves time and reduces errors.
所有五个函数共享一个重要的 x 值：x = −b/a。对线性函数而言，它是零点；对二次函数，它是顶点和二重根；对绝对值函数，它是顶点；对倒数函数，它是垂直渐近线和排除值；对平方根函数，它是定义域的端点。IB 试题经常考察这些特征，因此将 x = −b/a 视为“关键坐标”可以节省时间并减少错误。
8. Transformations: How a and b Affect Each Graph | 变换：a 和 b 如何影响图像
When ax + b appears as the inner function, the transformations are applied horizontally before the outer operation. Writing ax + b = a(x + b/a) reveals a horizontal shift by −b/a and a horizontal stretch by 1/|a|. The sign of a also triggers a reflection in the y‑axis if a < 0. Compare f(x) = √2x + 4 and f(x) = √−2x + 4 to see the flip clearly.
当 ax + b 作为内层函数出现时，变换先在水平方向进行，然后再执行外层运算。将 ax + b 改写为 a(x + b/a) 可以清晰看到水平平移 −b/a 以及水平伸缩因子 1/|a|。若 a 为负，还会引起关于 y 轴的反射。比较 f(x) = √2x + 4 与 f(x) = √−2x + 4 就可以清楚地观察到这种翻转。
9. Domain and Range in a Nutshell | 定义域与值域一览
The table below summarises the key properties for quick revision. Note that while a is always non‑zero, its sign determines the direction of inequalities. In IB exams, always state domains using set notation or interval notation, clearly linking them to the condition derived from the function.
下表总结了关键性质以便快速复习。注意 a 永远不为零，但其符号决定了不等号的方向。在 IB 考试中，务必用集合符号或区间符号写清定义域，并清楚地展示它是如何从函数条件推导出来的。
| Function / 函数 | Domain / 定义域 | Range / 值域 |
|---|---|---|
| f(x) = ax + b | ℝ | ℝ |
| f(x) = (ax + b)² | ℝ | [0, ∞) |
| f(x) = |ax + b| | ℝ | [0, ∞) |
| f(x) = 1/(ax + b) | ℝ \ {−b/a} | ℝ \ {0} |
| f(x) = √ax + b | [−b/a, ∞) if a > 0; (−∞, −b/a] if a < 0 | [0, ∞) |
10. Exam‑Style Connections and Problem Solving | 考试中的联系与解题思路
IB Mathematics frequently weaves these functions into composite problems: finding intersections of y = |2x − 3| and y = 1/(x + 1), or determining where √4 − x is greater than |x − 2|. You may also be asked to write piecewise definitions for absolute‑value functions, or to describe the sequence of transformations that maps y = 1/x onto y = 1/(2x − 5). Practise solving equations and inequalities analytically, and always support your answers with a quick sketch.
IB 数学常将这些函数编织成复合问题：求 y = |2x − 3| 与 y = 1/(x + 1) 的交点，或判断 √4 − x 何时大于 |x − 2|。考题也可能要求写出绝对值函数的分段定义，或者描述将 y = 1/x 映射到 y = 1/(2x − 5) 的变换序列。平时要多练习用代数方法解方程和不等式，并始终用快速草图来佐证答案。
11. Summary and Revision Tips | 总结与复习建议
The five functions of ax + b — linear, quadratic, absolute value, reciprocal and square root — form a logical family where the inner linear expression dictates horizontal behaviour. Remember that x = −b/a is almost always a key point. Master the domains and ranges, sketch graphs using transformation logic, and you will handle a large portion of IB function questions with confidence.
五种 ax + b 型函数——线性、二次、绝对值、倒数和平方根——构成一个逻辑紧密的家族，其中内层线性表达式支配着水平方向的性质。请记住 x = −b/a 几乎总是一个关键点。熟练掌握定义域和值域，运用变换逻辑绘制草图，你就能自信地应对 IB 函数试题中的一大部分。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图