📚 IGCSE Maths Chapter G7: Graphs of Linear and Quadratic Functions | IGCSE数学第G7章：一次与二次函数图像
Graphs are a central part of the IGCSE Mathematics syllabus. They allow us to represent relationships between variables visually and to solve equations in a geometric way. This article focuses on Chapter G7 of the student book, covering linear and quadratic functions, their graphs, key features and common applications. We begin with the coordinate plane, build up the algebra of straight lines, then move to parabolas, their turning points and roots, and finish with graph transformations, modelling questions and exam technique.
图像是 IGCSE 数学大纲的核心内容。它让我们直观地表示变量之间的关系，并用几何方法解方程。本文围绕学生用书第 G7 章，重点讲解一次函数与二次函数、它们的图像、关键特征以及常见应用。我们会从坐标平面讲起，逐步建立直线的代数表达，再进入抛物线、顶点与根，最后讨论图像变换、建模题与应试技巧。
1. The Coordinate Plane and Plotting Points | 坐标平面与描点
Every graph starts on the coordinate plane. The Cartesian plane consists of a horizontal x-axis and a vertical y-axis. A point is described by an ordered pair (x, y). The first number is the horizontal distance from the origin, and the second is the vertical distance. Points are plotted by moving along the x-axis first, then along the y-axis. The origin is the point (0, 0) where the two axes cross, and the axes divide the plane into four quadrants.
所有图像都从坐标平面开始。笛卡尔平面由水平的 x 轴和竖直的 y 轴组成。一个点用有序数对 (x, y) 表示。第一个数是距离原点的水平距离，第二个数是竖直距离。描点时先沿 x 轴移动，再沿 y 轴移动。原点是两轴交点 (0, 0)，两坐标轴把平面分成四个象限。
The signs of the coordinates tell you which quadrant a point lies in. This matters in exam questions where you must decide whether an intersection point is a valid solution in context, for example when a negative time or a negative length is meaningless.
坐标的正负号决定点落在哪个象限。这在考试中很重要：当你需要判断交点在实际情境中是否有效时，例如负数的时间或负数长度没有意义，就要用到象限信息。
| Quadrant 象限 | x | y | Example 例子 |
| First 第一 | + | + | (3, 2) |
| Second 第二 | − | + | (−3, 2) |
| Third 第三 | − | − | (−3, −2) |
| Fourth 第四 | + | − | (3, −2) |
Choosing a sensible scale is a skill in itself. If x runs from 0 to 20, using one square per unit wastes space and makes the curve look flat; using one square per five units keeps the shape readable. Always label the axes and the scale, because untidy axes lose method marks even when the curve is correct.
选择合适的刻度本身就是一项技能。如果 x 从 0 到 20，每格 1 个单位会浪费空间，让曲线看起来太平；每格 5 个单位则能让形状更清晰。一定要标注坐标轴和刻度，因为即使曲线画对了，坐标轴不规范也会丢分。
2. Linear Functions and y = mx + c | 一次函数与 y = mx + c
, where m is the gradient and c is the y-intercept. Its graph is always a straight line, so two points are enough to draw it, but a third point is a useful check. A table of values is the standard starting point: choose simple x values such as −2, −1, 0, 1, 2, substitute each one, and record the resulting y values.
一次函数的形式是 y = mx + c，其中 m 是斜率，c 是 y 轴截距。它的图像永远是一条直线，所以两点就能确定它，但第三个点可以用来检验。通常先列数值表：选取简单的 x 值，例如 −2、−1、0、1、2，逐个代入并记录对应的 y 值。
epts are often enough to sketch the line quickly. Remember that a line such as y = 3 is horizontal with gradient 0, while a line such as x = 3 is vertical with an undefined gradient and is not written in the form y = mx + c.
x 轴截距通过令 y = 0 解出 x 得到，y 轴截距通过令 x = 0 得到，直接就是 y = c。有了这两个截距，通常就能快速画出直线。要记住，像 y = 3 这样的直线是水平的，斜率为 0；而像 x = 3 这样的直线是竖直的，斜率不存在，不能写成 y = mx + c 的形式。
ich shows the regular pattern of a linear function: every time x increases by 1, y increases by 2.
下面是 y = 2x − 1 的典型数值表，它体现了一次函数的规律：x 每增加 1，y 就增加 2。
| x | −2 | −1 | 0 | 1 | 2 |
| y = 2x − 1 | −5 | −3 | −1 | 1 | 3 |
3. Gradient and Intercept: Interpreting m and c | 斜率与截距：解读 m 与 c
The gradient measures steepness and direction. For two points on a line, the gradient is calculated with the standard formula below. The symbol Δ means "change in", so the gradient is the change in y divided by the change in x.
斜率衡量的是陡峭程度与方向。对于直线上的两点，斜率可用下面的标准公式计算。符号 Δ 表示"变化量"，所以斜率等于 y 的变化量除以 x 的变化量。
m = (y₂ − y₁) ÷ (x₂ − x₁) = Δy ÷ Δx
er the line, the larger the numerical value of m, regardless of sign. In real contexts the gradient is a rate: cost per item, speed in metres per second, or population change per year. The intercept c is the starting value, the quantity present when x = 0, such as a fixed call-out charge or an initial height.
斜率为正表示直线从左到右上升，为负表示下降，为 0 则是水平线。直线越陡，m 的数值越大（不论正负）。在实际情境中，斜率就是变化率：每件商品的成本、以米每秒计的速度、每年的人口变化。截距 c 是初始值，即 x = 0 时的数量，例如固定的上门费或初始高度。
Two useful checks in exams: if the line goes down as you read left to right, your answer must be negative; and if two points have the same y value, the gradient is 0, not undefined. Reversing the order of subtraction in both the numerator and denominator is fine, but reversing only one of them flips the sign and loses the mark.
考试中两个实用的检查：如果从左向右看直线是下降的，答案必须为负；如果两点的 y 值相同，斜率是 0，而不是不存在。分子分母同时交换顺序没问题，但只交换其中一个就会改变符号，从而失分。
4. Finding the Equation of a Line | 求直线方程
There are three common routes to the equation of a straight line, and you should be able to switch between them confidently. The first is to read the gradient and intercept directly from a graph. The second is to use two given points. The third is to use one point plus the gradient.
求直线方程通常有三条路径，你需要能熟练地在它们之间切换。第一，直接从图像上读斜率和截距；第二，利用已知的两点；第三，利用一点加斜率。
From a graph: pick two points on the line that sit exactly on grid intersections, count the vertical rise and the horizontal run, then read c where the line crosses the y-axis.
从图像：选取直线上正好落在网格交点的两点，数出竖直上升量与水平位移量，再读出直线与 y 轴交点处的 c。
rst find m with the gradient formula, then substitute one point into y = mx + c to find c.
已知两点：先用斜率公式求 m，再把其中一点代入 y = mx + c 求 c。
nd a gradient: substitute the point and gradient straight into y = mx + c and solve for c.
已知一点与斜率：直接把点和斜率代入 y = mx + c，解出 c。
7 = 6 + c, so c = 1. The equation is y = 3x + 1, which can be verified by substituting the second point: 3 × 5 + 1 = 16.
例题：求经过 (2, 7) 与 (5, 16) 的直线方程。斜率是 (16 − 7) ÷ (5 − 2) = 9 ÷ 3 = 3。把 (2, 7) 代入 y = 3x + c 得 7 = 6 + c，所以 c = 1。方程为 y = 3x + 1，可用第二点验证：3 × 5 + 1 = 16。
5. Parallel and Perpendicular Lines | 平行线与垂直直线
tercepts are also equal the lines coincide, meaning they have infinitely many intersection points rather than none.
平行线斜率相同但截距不同，条件是 m₁ = m₂ 且 c₁ ≠ c₂。如果截距也相同，两直线重合，意味着它们有无穷多个交点，而不是没有交点。
Perpendicular lines meet at 90°, and their gradients multiply to −1. Equivalently, the perpendicular gradient is the negative reciprocal of the original. If a line has gradient 3, a perpendicular line has gradient −1 ÷ 3; if the gradient is −2 ÷ 5, the perpendicular gradient is 5 ÷ 2. A horizontal line is perpendicular to a vertical line, which is the one case where the rule cannot be applied numerically.
垂直直线相交成 90°，它们的斜率乘积为 −1。也就是说，垂直直线的斜率是原斜率的负倒数。若一条直线的斜率是 3，则垂直直线的斜率是 −1 ÷ 3；若斜率是 −2 ÷ 5，则垂直斜率是 5 ÷ 2。水平线与竖直线互相垂直，这是唯一无法用数值套用该法则的情形。
m₁ × m₂ = −1 ⇔ m₂ = −1 ÷ m₁
This rule is frequently used in coordinate geometry questions asking for the perpendicular bisector of a line segment. Find the midpoint with the average of the coordinates, find the negative reciprocal of the segment's gradient, then form the new equation. A quick sketch always helps you spot a sign error before it costs marks.
这条法则常用于坐标几何题中求线段的垂直平分线。先用坐标的平均值求中点，再求原线段斜率的负倒数，最后写出新方程。画一个简图总能在失分之前帮你发现符号错误。
6. Quadratic Functions and Their Parabolas | 二次函数与抛物线
x² + bx + c with a ≠ 0, and its graph is a parabola. If a is positive the parabola opens upwards and has a minimum point; if a is negative it opens downwards and has a maximum point. The larger the value of |a|, the narrower the curve, while a small |a| produces a wide, flatter shape.
二次函数的形式是 y = ax² + bx + c，其中 a ≠ 0，其图像是抛物线。若 a 为正，抛物线开口向上，有最低点；若 a 为负，则开口向下，有最高点。|a| 越大，曲线越窄；|a| 越小，曲线越宽越平。
tituting x = 0 leaves y = c. Quadratic graphs can have two, one or zero x-intercepts, and this number is controlled by the discriminant.
每条抛物线都关于过其顶点的竖直线对称，这条线称为对称轴。y 轴截距就是 c，因为令 x = 0 时 y = c。二次函数图像可能有 2 个、1 个或 0 个 x 轴交点，其个数由判别式决定。
Δ = b² − 4ac；Δ > 0 两个根，Δ = 0 一个根，Δ < 0 无实根
s of symmetry. Because of the symmetry, you never need to calculate both sides from scratch; whatever happens at x = h + d is mirrored at x = h − d.
作图时至少描五个点：顶点、尽可能两个交点，以及对称轴两侧互为镜像的一两个附加点。由于对称性，你不必从零计算两侧：在 x = h + d 处的取值与 x = h − d 处完全相同。
7. Key Features of Quadratic Graphs | 二次函数图像的关键特征
epts), the y-intercept, the vertex or turning point, and the axis of symmetry. The roots are the solutions of ax² + bx + c = 0 and can be found by factorising, completing the square, or using the quadratic formula.
考试很少只让你画抛物线，而是要求你从图像中提取信息。关键特征包括：根（x 轴交点）、y 轴截距、顶点（转折点）以及对称轴。根就是方程 ax² + bx + c = 0 的解，可用因式分解、配方法或求根公式得到。
x = (−b ± √b² − 4ac) ÷ (2a)
The axis of symmetry always passes through the vertex, and its equation is given by the formula below. Once you know this x value, substituting it back into the function gives the y coordinate of the turning point. This is often faster than completing the square when the question only asks for the coordinates of the vertex.
对称轴总是经过顶点，其方程由下面的公式给出。求出这个 x 值后，再代回函数即可得到顶点的 y 坐标。当题目只要求顶点坐标时，这往往比配方法更快。
x = −b ÷ (2a)
x² − 6x + 5, the y-intercept is 5 while the minimum is at (3, −4); these are completely different points. Another frequent mistake is to forget that a parabola with a positive a opens upwards, so it cannot have a maximum value, no matter where the vertex lies.
常见错误是把 y 轴截距与顶点混为一谈。对于 y = x² − 6x + 5，y 轴截距是 5，而最小值点在 (3, −4)，这是两个完全不同的点。另一个常见错误是忘记 a 为正的抛物线开口向上，因此无论顶点在哪里，它都不可能有最大值。
8. Completing the Square and the Turning Point | 配方法与顶点
s. When a = 1, halve the coefficient of x to find h, then subtract the square of that halved value to keep the expression balanced.
把二次函数写成配方式 a(x − h)² + k，可以立刻看出顶点 (h, k)，最小值或最大值也一目了然。当 a = 1 时，把 x 的系数除以 2 得到 h，再减去这个半数之平方，以保持表达式平衡。
so subtract 4: x² − 6x + 5 = (x − 3)² − 4. The vertex is therefore (3, −4) and the minimum value of the function is −4, occurring when x = 3.
例题：把 x² − 6x + 5 化为配方式。−6 的一半是 −3，所以先写 (x − 3)²。展开得 x² − 6x + 9，比原式多了 4，因此减去 4：x² − 6x + 5 = (x − 3)² − 4。所以顶点是 (3, −4)，函数的最小值为 −4，在 x = 3 时取得。
s two roots, which is exactly why students who forget it lose half the answer. For quadratics with a ≠ 1, first factor a out of the first two terms before completing the square inside the bracket.
令配方式等于 0 也能干净地解方程：(x − 3)² = 4，得 x − 3 = ±2，所以 x = 5 或 x = 1。注意 ± 号产生了两个根，这正是忘记它就会丢掉一半答案的原因。当 a ≠ 1 时，先从前两项中提出 a，再在括号内配方。
9. Solving Equations Graphically | 用图像解方程
his is why solving a linear and a quadratic equation simultaneously can produce two solutions: a straight line can cut a parabola at two points, touch it at one point, or miss it entirely.
图像把代数变成几何。方程 f(x) = g(x) 的解，就是 y = f(x) 与 y = g(x) 两条图像交点的 x 坐标。这就是为什么联立一次方程与二次方程可能有两个解：一条直线可以与抛物线相交于两点、相切于一点，或完全不相交。
x values where it meets the curve. If the curve does not quite reach that height, there is no real solution, which links directly to the discriminant being negative. Always give answers to the accuracy stated in the question, usually one or two decimal places, and show your reading lines on the grid.
用图像解 f(x) = k 时，画出水平线 y = k，读出它与曲线相交处的 x 值。如果曲线达不到这个高度，就没有实数解，这正好对应判别式为负的情形。答案要按题目要求的精度给出，通常保留一位或两位小数，并在网格上画出读取的辅助线。
Graphical methods are also how you estimate solutions that cannot be factorised. If a question says "use your graph to solve", an algebraic answer alone may not earn full marks, because the examiner is testing whether you can read a curve accurately. Conversely, if the question says "show that" or "find exactly", an answer read from a sketch is not acceptable.
对于无法因式分解的方程，图像法也是估算解的手段。如果题目说"利用你的图像求解"，只给代数答案可能拿不到满分，因为考官考查的正是你读图的准确性。反之，如果题目说"证明"或"精确求值"，从草图上读出的答案就不被接受。
10. Transformations of Graphs | 图像变换
Transformations describe how a known graph changes when its equation is altered. Knowing the standard rules lets you sketch a modified curve in seconds without a table of values, which saves valuable time in the exam.
图像变换描述的是当方程改变时，已知图像如何变化。掌握标准法则后，你可以在几秒内画出变形后的曲线，而不必列表，这在考试中非常省时。
y = f(x) + a translates the graph upwards by a units; y = f(x) − a moves it down.
y = f(x) + a 把图像向上平移 a 个单位；y = f(x) − a 则向下平移。
a units, and y = f(x − a) to the right. The direction feels reversed, so check carefully.
y = f(x + a) 把图像向左平移 a 个单位，y = f(x − a) 向右平移。方向感似乎相反，务必仔细。
aph vertically by scale factor a; values with a > 1 stretch, values with 0 < a < 1 squash.
y = af(x) 把图像沿竖直方向按比例 a 拉伸；a > 1 拉伸，0 < a < 1 压缩。
y = −f(x) reflects the graph in the x-axis, while y = f(−x) reflects it in the y-axis.
y = −f(x) 关于 x 轴翻折，y = f(−x) 关于 y 轴翻折。
e, y = 2x + 1 shifted up by 4 becomes y = 2x + 5, with the same gradient but a different intercept. Describing these changes in words is a common exam requirement.
以抛物线为例，y = (x − 2)² + 3 是把基本曲线 y = x² 向右移 2、向上移 3，顶点从 (0, 0) 移到 (2, 3)。以直线为例，y = 2x + 1 向上平移 4 个单位后变为 y = 2x + 5，斜率不变而截距改变。用文字描述这些变化是常见的考试要求。
11. Modelling and Real-Life Applications | 建模与实际应用
Linear models describe constant rates of change: a taxi fare with a fixed charge plus a rate per kilometre, the total cost of n identical items, or a temperature falling steadily. The gradient is the rate and the intercept is the fixed part. Always state clearly what the variables represent and include units in your final answer.
一次模型描述恒定的变化率：出租车起步价加每公里费用、n 件相同商品的总价，或匀速下降的温度。斜率是变化率，截距是固定部分。一定要说明变量代表什么，并在最终答案中带上单位。
peed 20 m/s, a simplified height model is h = 20t − 5t², where h is in metres and t in seconds. The roots t = 0 and t = 4 tell you when the object is at ground level, and the vertex at t = 2 gives the maximum height of 20 m.
二次模型描述变化率本身在改变的情形，最典型的是抛体运动。若物体以 20 m/s 的初速度向上抛出，简化后的高度模型为 h = 20t − 5t²，其中 h 以米为单位，t 以秒为单位。两根 t = 0 与 t = 4 表示物体位于地面的时刻，顶点 t = 2 则给出最大高度 20 米。
h = 20t − 5t²；顶点 t = 2 s，最大高度 h = 20 m
Other standard contexts include profit maximisation, where revenue minus cost gives a downward-opening parabola, and breaking distance, which grows roughly with the square of speed. In such questions the sensible domain matters: negative time and negative distance are discarded even if the algebra offers them, and answers are usually rounded to a sensible degree of accuracy.
其他常见情境包括利润最大化：收入减成本得到一条开口向下的抛物线
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