📚 Linear Functions | 线性函数
A linear function is one of the most fundamental concepts in IB Mathematics. It models a constant rate of change and forms the basis for understanding more complex algebraic and geometric ideas. In this revision guide, we explore the definition, properties, graphical representation, and various applications of linear functions as required by the IB syllabus.
线性函数是 IB 数学中最基本的概念之一。它描述常量变化率，并构成理解更复杂代数与几何思想的基础。本复习指南将深入探讨线性函数的定义、性质、图形表示以及 IB 大纲要求的各类应用。
1. What is a Linear Function? | 什么是线性函数？
constants. The graph of a linear function is a straight line. The variable x is the independent variable, and f(x) is the dependent variable. The defining characteristic is that the rate of change of the function with respect to x is constant.
线性函数是形如 f(x) = mx + c 的函数，其中 m 与 c 为实常数。线性函数的图像是一条直线。变量 x 为自变量，f(x) 为因变量。其定义特征为该函数关于 x 的变化率是恒定的。
he domain is all real numbers unless context restricts it. In IB examinations, you will often encounter linear functions in modelling contexts, where the domain may be limited to positive values or a specific interval.
用集合记号表示，线性函数可写作 f: R → R，f(x) = mx + c。其定义域为所有实数，除非上下文有所限定。在 IB 考试中，你常会在建模情境下遇见线性函数，此时定义域可能局限于正值或一个特定区间。
2. Slope (Gradient) and Rate of Change | 斜率与变化率
horizontal change between any two distinct points on the line: m = (y₂ − y₁) / (x₂ − x₁).
斜率 m 衡量直线的倾斜程度与方向。其定义为直线上任意两个相异点之间垂直变化与水平变化之比：m = (y₂ − y₁) / (x₂ − x₁)。
undefined slope occurs for vertical lines x = k, where the denominator (x₂ − x₁) is zero.
斜率为正表示直线随 x 增加而上升；斜率为负表示直线随 x 增加而下降。斜率为零时得到水平线 y = c。对于竖直线 x = k，分母 (x₂ − x₁) 为零，斜率无定义。
In the context of applications, the slope represents a constant rate of change. For example, if a car travels at a constant speed, the distance-time graph is linear with slope equal to the speed. IB exam questions often ask you to interpret the slope in a real-world scenario.
在应用情境中，斜率表示恒定的变化率。例如，若汽车以恒定速度行驶，距离−时间图呈线性，斜率等于速度。IB 考题常要求你在实际场景中解释斜率的含义。
3. y-intercept and x-intercept | y 轴截距与 x 轴截距
x + c, the y-intercept is (0, c). Thus the constant term c directly gives the y-intercept.
y 轴截距是直线与 y 轴相交的点，此时 x = 0。对于函数 f(x) = mx + c，y 轴截距为 (0, c)。因此常数项 c 直接给出 y 轴截距。
where the line crosses the x-axis, found by setting f(x) = 0 and solving for x: mx + c = 0 ⇒ x = −c/m, provided m ≠ 0. This gives the intercept point (−c/m, 0).
x 轴截距是直线与 x 轴相交的点，可令 f(x) = 0 求解：mx + c = 0 ⇒ x = −c/m，其中 m ≠ 0。所得截距点为 (−c/m, 0)。
Understanding intercepts is essential for sketching graphs and for solving real-world problems, such as break-even analysis in business studies, where the x-intercept may represent the point of zero profit.
理解截距对于草图绘制和解决实际问题至关重要，例如商业研究中的盈亏平衡分析，此时 x 轴截距可表示利润为零的点。
4. Forms of Linear Equations | 线性方程的形式
the y-intercept. This form is convenient for graphing and for identifying slope directly.
线性方程可用多种等价形式表示。最常见的是斜截式：y = mx + c。此处 m 是斜率，c 是 y 轴截距。该形式便于作图及直接识别斜率。
), where m is the slope and (x₁, y₁) is a known point on the line. This form is particularly useful when you know a point and the slope but not the y-intercept.
点斜式为 y − y₁ = m(x − x₁)，其中 m 是斜率，(x₁, y₁) 是直线上已知点。当你已知一点和斜率但不知 y 轴截距时，该形式尤为有用。
for finding intersections and for linear programming in IB Mathematics: Applications and Interpretation.
一般式为 Ax + By + C = 0，其中 A, B, C 为整数，通常要求 A ≥ 0。该形式便于求交点，以及在 IB 数学：应用与解释中进行线性规划。
iven 2x − 3y + 6 = 0, rewrite as y = (2/3)x + 2 to identify slope and y-intercept quickly.
在这些形式之间进行转换是 IB 考试考查的关键技能。例如，给定 2x − 3y + 6 = 0，可改写为 y = (2/3)x + 2，以快速识别斜率和 y 轴截距。
5. Graphing Linear Functions | 绘制线性函数图像
To graph a linear function, you need either two points or one point and the slope. A common method is to plot the y-intercept and use the slope to find a second point: from (0, c), move right 1 unit and up m units if m > 0, or down |m| units if m < 0.
要绘制线性函数图像，你需要两点或一点与斜率。常用方法是标出 y 轴截距并利用斜率确定另一点：从 (0, c) 出发，若 m > 0 则向右 1 单位并向上 m 单位；若 m < 0 则向下 |m| 单位。
tively. Plot these two points and draw a line through them. This approach is very useful when the equation is in standard form.
另一种方法是分别令 y = 0 和 x = 0 求出 x 轴与 y 轴截距。标出这两点并过它们画出直线。当方程以一般式给出时，该方法十分有用。
When using a graphing calculator (required for IB), you can enter the function and view the graph. The calculator can also find intercepts and intersection points, which is helpful for checking your work.
使用图形计算器（IB 考试要求）时，你可输入函数并查看图像。计算器还能求截距和交点，有助于核对你的计算结果。
6. Parallel and Perpendicular Lines | 平行与垂直直线
= m₂. Their y-intercepts must be different, otherwise they are coincident (the same line).
两条直线平行当且仅当它们的斜率相等。即直线 L₁: y = m₁x + c₁ 与 L₂: y = m₂x + c₂ 平行时 m₁ = m₂。它们的 y 轴截距必须不同，否则为重合直线。
cular: a vertical line x = a has undefined slope, and a horizontal line y = b has slope 0.
两条直线垂直当且仅当它们的斜率之积为 −1。即 m₁ × m₂ = −1，这意味着 m₂ = −1/m₁。竖直线与水平线亦相互垂直：竖直线 x = a 的斜率无定义，水平线 y = b 的斜率为 0。
These relationships are frequently used in coordinate geometry problems and in constructing perpendicular bisectors (linking to Voronoi diagrams in the IB Applications course).
这些关系常用于坐标几何问题以及作垂直平分线（关联到 IB 应用课程中的 Voronoi 图）。
7. Finding Equations from Given Conditions | 根据条件求方程
IB questions often ask: find the equation of a line given two points; given a point and slope; or given a point and a relationship to another line (parallel or perpendicular). To find the equation, first determine the slope if not given, then use point-slope form and simplify.
IB 考题常要求：已知两点求直线方程；已知一点和斜率；或已知一点及与另一直线的关系（平行或垂直）求方程。若未直接给出斜率，先求出斜率，然后使用点斜式并化简。
5 = 2(x − 2), hence y = 2x + 1. Always check that both points satisfy the final equation.
例如，求过 A(2, 5) 和 B(4, 9) 的直线。斜率 m = (9 − 5)/(4 − 2) = 2。使用点 A：y − 5 = 2(x − 2)，因此 y = 2x + 1。务必检验最终方程是否满足两点坐标。
4), the required slope is 2 (since −½ × 2 = −1). Equation: y − 4 = 2(x − 1) → y = 2x + 2.
当直线需与 y = −½x + 3 垂直且经过点 (1, 4) 时，所求斜率为 2（因 −½ × 2 = −1）。方程为 y − 4 = 2(x − 1) → y = 2x + 2。
8. Solving Linear Equations Graphically | 线性方程的图像解法
of their intersection. For 2D linear systems, the intersection of two lines gives the simultaneous solution.
一元线性方程如 2x − 3 = 5 可用代数方法求解。在图像上，你可将等号两边视作函数：y = 2x − 3 与 y = 5，方程的解即为两者交点的 x 坐标。对于二元线性方程组，两直线交点给出联立解。
and c₁ ≠ c₂ (parallel lines), and infinitely many solutions if the lines are coincident (same line).
方程组 y = m₁x + c₁ 与 y = m₂x + c₂ 当 m₁ ≠ m₂ 时恰有一解（相交直线）；当 m₁ = m₂ 且 c₁ ≠ c₂ 时无解（平行线）；若两直线重合，则有无穷多解。
This links to the concept of linear independence and to solving equations using matrices in IB HL. The graphical interpretation reinforces the algebraic approach.
这与线性无关的概念相关，也联系到 IB 高水平课程中使用矩阵求解方程。图像解释能巩固代数方法。
9. Linear Functions in Modelling | 线性函数建模
le cost per unit and b is fixed cost; distance-time relations for constant speed; and temperature conversions.
线性模型在 IB 中被广泛用于刻画实际情境中变量间的关系。线性模型假定变化率恒定。例子包括：成本函数 C(x) = mx + b，其中 m 为单位可变成本，b 为固定成本；匀速运动的距离−时间关系；以及温度转换。
oefficient r that indicates how well the model fits. IB courses stress interpreting the parameters in context.
建模题可能给出数据表，要求使用 GDC 求出线性回归线。最佳拟合线 y = ax + b 配有相关系数 r，可指示模型拟合优度。IB 课程强调需在实际语境中解释参数。
Be cautious: linear models are valid only within the relevant domain. Extrapolation beyond the range of data can lead to unreliable predictions. Always discuss limitations.
需注意：线性模型仅在相关定义域内有效。超出数据范围的外推可能导致不可靠的预测。务必讨论其局限性。
10. Piecewise Linear Functions | 分段线性函数
A piecewise linear function is defined by different linear expressions on different intervals of the domain. For example, a tax bracket system or a shipping cost schedule can be modelled by piecewise linear functions.
分段线性函数由定义域不同区间上的不同线性表达式所定义。例如，税收阶梯制度或运输费用表可用分段线性函数建模。
Graphing such a function requires attention to endpoints: open circles for excluded values, closed circles for included values. Evaluating the function at a specific x involves determining which piece applies.
绘制此类函数图像时需注意端点：空心圆表示不含该值，实心圆表示包含该值。在特定 x 处求值需判断应使用哪一段表达式。
IB exam questions may ask for the equation of a piece of the function given a graph, or to graph a defined piecewise linear function, often within the topic of functions and modelling.
IB 考题可能根据图像求某段表达式，或要求绘制已定义的分段线性函数图像，常见于函数与建模主题中。
11. Linear Inequalities | 线性不等式
Solving linear inequalities follows similar rules to equations, but with one key exception: multiplying or dividing both sides by a negative number reverses the inequality sign. For example, −2x > 4 ⇒ x < −2.
解线性不等式遵循类似方程的法则，但有一个关键例外：两边同时乘或除以负数时，不等号方向改变。例如，−2x > 4 ⇒ x < −2。
below or on the horizontal line y = 7. This set can be expressed using interval notation.
从图像上看，不等式 3x + 1 ≤ 7 的解为使直线 y = 3x + 1 位于水平线 y = 7 下方或恰好在其上的所有 x 值集合。该集合可用区间表示法表达。
Systems of linear inequalities define regions in the coordinate plane. These are fundamental in linear programming, where you identify feasible regions and optimize a linear objective function, a key component of the IB Applications and Interpretation HL course.
线性不等式组定义坐标平面上的区域。这是线性规划的基础，需要识别可行域并优化线性目标函数，是 IB 应用与解释高水平课程的核心内容。
12. Exam Tips and Common Mistakes | 考试技巧与常见错误
ose marks unless the question specifies point-slope form. Convert to slope-intercept or general form as appropriate.
务必化简最终方程。若将答案留为 y − 3 = 2(x + 1) 可能失分，除非题目明确要求点斜式。应酌情转化为斜截式或一般式。
Check your slope calculation carefully, especially the sign: (y₂ − y₁)/(x₂ − x₁). A common error is switching numerator and denominator or misapplying points. Also verify that your line satisfies given conditions.
仔细检查斜率计算，尤其是符号：(y₂ − y₁)/(x₂ − x₁)。常见错误有颠倒分子分母或代错点。同时验证所求直线满足已知条件。
When a question gives a point and requires a perpendicular line, remember the negative reciprocal. Many students forget the negative sign. For horizontal and vertical lines, use the special cases directly.
当题目给出一点并要求一条垂线时，记住负倒数关系。许多学生忘记负号。遇到水平线与竖直线时，直接使用特殊情况。
In modelling, do not extrapolate without justification. State the domain clearly. If using a GDC to find a regression line, quote the equation with the correlation coefficient and comment on the fit.
在建模中，不要无根据地外推。应清晰陈述定义域。若用 GDC 求回归线，需给出方程及相关系数并评论拟合优度。
Finally, always label axes and scale when sketching graphs. Use a ruler for straight lines. Demonstrate understanding of intercepts and slope in your sketch.
最后，绘制草图时始终标注坐标轴与单位。画直线务必使用直尺。在草图中展示你对截距和斜率的理解。
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