IB Mathematics: Linear Functions Graphs and Properties | IB数学：线性函数图像与性质

📚 IB Mathematics: Linear Functions Graphs and Properties | IB数学：线性函数图像与性质

A linear function is one of the most fundamental concepts in IB Mathematics. It appears across all levels, from Analysis and Approaches (AA) to Applications and Interpretation (AI). Understanding its graph and properties is essential for solving problems involving rates of change, intercepts, and systems of equations.

线性函数是IB数学中最基础的概念之一，贯穿分析与方法（AA）和应用与解释（AI）全部级别。掌握其图像与性质，对于解决涉及变化率、截距以及方程组的问题至关重要。

1. Definition and General Form | 定义与一般形式

A linear function can be written in the form \( f(x) = mx + c \), where \( m \) is the gradient (slope) and \( c \) is the y-intercept. This is called the slope-intercept form. In IB questions, you may also see the standard form \( ax + by = d \) and the point-slope form \( y – y_1 = m(x – x_1) \).

线性函数可以写成 \( f(x) = mx + c \) 的形式，其中 \( m \) 是斜率（梯度），\( c \) 是 y 截距。这称为斜截式。在IB题目中，你还会看到标准式 \( ax + by = d \) 以及点斜式 \( y – y_1 = m(x – x_1) \)。

f(x) = mx + c

The domain and range of a non-vertical linear function are both all real numbers. The graph is a straight line with constant gradient.

非垂直线性函数的定义域和值域均为全体实数。其图像是一条斜率恒定的直线。

2. The Gradient (Slope) | 梯度（斜率）

The gradient measures the steepness and direction of a line. It is defined as the change in y divided by the change in x:

梯度衡量一条线的陡峭程度和方向，定义为 y 的变化量除以 x 的变化量：

m = (y₂ − y₁) / (x₂ − x₁) = Δy / Δx

If m > 0, the line rises from left to right.

如果 m > 0，直线从左到右上升。
If m < 0, the line falls from left to right.

如果 m < 0，直线从左到右下降。
If m = 0, the line is horizontal.

如果 m = 0，直线是水平的。
A vertical line has undefined gradient.

垂直线的斜率不存在。

Parallel lines have equal gradients. Perpendicular lines have gradients that multiply to −1, i.e. \( m_1 \times m_2 = -1 \).

平行线的斜率相等。垂直线的斜率乘积为 −1，即 \( m_1 \times m_2 = -1 \)。

3. Intercepts | 截距

The y-intercept is the point where the line crosses the y-axis, found by setting x = 0. In the form y = mx + c, the y-intercept is simply c. The x-intercept is found by setting y = 0 and solving for x.

y 截距是直线与 y 轴的交点，令 x = 0 即可求得。在 y = mx + c 中，y 截距就是 c。x 截距则令 y = 0 并解出 x。

x-intercept = −c / m (provided m ≠ 0)

Example: For y = 2x − 4, the y-intercept is (0, −4) and the x-intercept is (2, 0).

例如：对于 y = 2x − 4，y 截距为 (0, −4)，x 截距为 (2, 0)。

4. Graphing a Linear Function | 绘制线性函数图像

To graph a linear function, you need at least two points. The easiest method is to find both intercepts and draw the line through them. Alternatively, start at the y-intercept and use the gradient as rise over run.

绘制线性函数至少需要两个点。最简单的方法是求出两个截距并连线。另一种方法是从 y 截距出发，利用斜率“上升量除以水平距离”移动。

Example: Graph y = ½ x + 1

y-intercept: (0, 1)

y 截距：(0, 1)
Gradient = ½, so from (0,1) move up 1 unit and right 2 units to (2,2).

斜率 = ½，从 (0,1) 上升 1 个单位、右移 2 个单位到达 (2,2)。
Draw a straight line through both points.

过这两点画一条直线。

Always label axes and use a ruler. In IB exams, graph sketching may require intercepts, gradient, or specific points.

画图时务必标注坐标轴并使用直尺。在IB考试中，画草图可能要求标出截距、斜率或特定点。

5. Finding the Equation from a Graph | 从图像求方程

If you are given a graph, determine the gradient by selecting two clear points and calculating m. Then identify the y-intercept c directly, or use a point to substitute into y = mx + c.

若已知图像，选取两个清晰点计算 m 得到斜率，然后直接读出 y 截距 c，或代入一个点的坐标求解 c。

y − y₁ = m(x − x₁)

Example: A line passes through (2,5) and (4,9). Gradient m = (9−5)/(4−2) = 2. Using point-slope form: y − 5 = 2(x − 2), so y = 2x + 1.

例：一条线经过 (2,5) 和 (4,9)。斜率 m = (9−5)/(4−2) = 2。使用点斜式：y − 5 = 2(x − 2)，化简得 y = 2x + 1。

6. Special Cases: Horizontal and Vertical Lines | 特殊情况：水平线与垂直线

A horizontal line has equation y = k. Its gradient is 0. A vertical line has equation x = h. Its gradient is undefined and it is not a function unless restricted.

水平线方程为 y = k，斜率为 0。垂直线方程为 x = h，斜率不存在，除非限制定义域，否则它不是函数。

Line	Equation	Gradient
Horizontal	y = k	0
Vertical	x = h	Undefined

Remember that only non-vertical lines are linear functions. A vertical line fails the vertical line test.

注意只有非垂直线才是线性函数。垂直线无法通过垂直线检验。

7. Linear Functions in Real-World Contexts | 线性函数在实际情境中的应用

Linear functions model constant rate situations. For example, distance = speed × time, or cost = fixed charge + variable charge per unit. In IB AI, you may be asked to interpret the gradient and intercept in context.

线性函数用于建模恒定速率的情景。例如，距离 = 速度 × 时间，或成本 = 固定费用 + 每单位变动费用。在IB AI 中，你可能需要结合情境解释斜率和截距的实际含义。

Example: A taxi fare is given by C = 2.5 + 1.2x

The gradient 1.2 represents cost per kilometre.

斜率 1.2 表示每公里的费用。
The y-intercept 2.5 represents the flag fall (initial charge).

y 截距 2.5 表示起步价（初始费用）。

Always include units when interpreting slopes in word problems.

在应用题中解释斜率时，务必带上单位。

8. Intersection of Two Lines | 两条直线的交点

Two lines intersect at a point that satisfies both equations. Solve the system either by substitution or elimination. The solution is the coordinate of intersection.

两条直线相交于同时满足两个方程的点。通过代入法或消元法解方程组，解即为交点坐标。

Example: y = 2x + 1 and y = −x + 4

Set 2x + 1 = −x + 4 → 3x = 3 → x = 1. Then y = 2(1)+1 = 3. Intersection: (1,3).

令 2x + 1 = −x + 4 → 3x = 3 → x = 1。代入得 y = 2(1)+1 = 3。交点：(1,3)。

If lines are parallel (m equal), they never intersect. If they are the same line, they intersect infinitely many times.

如果两直线平行（m 相等），则永不相交。如果两直线重合，则有无数个交点。

9. Transformations of Linear Functions | 线性函数的变换

Changing the parameters m and c changes the graph. Increasing m makes the line steeper; decreasing m makes it flatter. Changing c shifts the line vertically. Horizontal shifts occur when x is replaced by (x − h).

改变参数 m 和 c 会改变图像。增大 m 使直线更陡，减小 m 使直线更平。改变 c 会使直线上下平移。将 x 替换为 (x − h) 可实现水平平移。

Vertical translation: y = mx + c + k

垂直平移：y = mx + c + k
Horizontal translation: y = m(x − h) + c

水平平移：y = m(x − h) + c
Reflection in y-axis: y = m(−x) + c = −mx + c

关于 y 轴对称：y = m(−x) + c = −mx + c

These transformations are useful in function-based IB questions.

这些变换在IB函数类题目中非常有用。

10. Common IB Exam Tips | 常见IB考试技巧

When solving linear function problems, always check the form requested by the question. Some questions require y = mx + c, while others accept ax + by = d. Use your GDC to verify intercepts when allowed.

解决线性函数问题时，务必注意题目要求的形式。有些题要求 y = mx + c，有些则接受 ax + by = d。允许使用图形计算器（GDC）时，可用其验证截距。

Read the gradient carefully: it may be given as a fraction, decimal, or percentage.

仔细理解斜率：可能以分数、小数或百分比给出。
Do not confuse x-intercept with y-intercept.

不要混淆 x 截距和 y 截距。
When finding the equation from two points, compute gradient first.

由两点求方程时，先算斜率。
For perpendicular lines, remember the product of gradients is −1.

对于垂直线，记住斜率乘积为 −1。

Practise sketching lines quickly by identifying intercepts and gradient. Mastery of linear functions builds a strong foundation for quadratic, exponential, and rational functions in later topics.

通过识别截距和斜率快速画图多加练习。掌握线性函数将为后续二次函数、指数函数和有理函数打下坚实基础。

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IB Mathematics: Linear Functions Graphs and Properties | IB数学：线性函数图像与性质