📚 GCSE Maths: Graph Formulas Summary and Compilation | GCSE 数学：图像公式归纳与汇总
In GCSE Mathematics, understanding the shapes and equations of different graphs is essential for both Foundation and Higher tier papers. This article summarises the key graph formulas you need to recognise, sketch, and transform, including linear, quadratic, cubic, reciprocal, exponential, trigonometric and circle graphs, as well as transformations of functions. Being confident with these forms will help you interpret questions on drawing graphs, solving equations graphically and identifying curves from their equations.
在 GCSE 数学中，掌握各类图像的形状和方程是基础与高等级试卷的共同核心。本文归纳了需要辨认、绘制和变换的关键图像公式，涵盖直线、二次、三次、反比例、指数、三角函数和圆的图像，以及函数变换。熟悉这些标准形式，有助于解答画图题、图像法解方程和根据方程识别曲线类型。
1. Linear Graphs | 直线图像
A linear equation produces a straight line. The general form is y = mx + c, where m is the gradient and c is the y‑intercept.
线性方程对应一条直线。一般式为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。
y = mx + c
gative m slopes downwards, and m = 0 gives a horizontal line y = c. The line crosses the y‑axis at (0, c). To find the x‑intercept, set y = 0 and solve for x, giving (−c/m, 0).
斜率 m 描述倾斜程度：m > 0 直线向上倾斜，m < 0 向下倾斜，m = 0 时为水平线 y = c。直线与 y 轴交于点 (0, c)。求 x 截距可令 y = 0，解出 x，得交点 (−c/m, 0)。
- Gradient m = rise / run
- Parallel lines have equal gradients
- Perpendicular lines: m₁ × m₂ = −1
- 斜率 m = 纵向变化 / 横向变化
- 平行直线斜率相等
- 互相垂直的直线满足 m₁ × m₂ = −1
ee linear graphs for vertical lines x = k (undefined gradient) and horizontal lines y = c.
也会出现垂直线 x = k（斜率无定义）和水平线 y = c 的图像。
2. Quadratic Graphs | 二次函数图像
Quadratic graphs are parabolas. The standard form is y = ax² + bx + c, where a ≠ 0. The sign of a determines whether the parabola opens upwards (a > 0) or downwards (a < 0).
二次图像是抛物线。标准式为 y = ax² + bx + c（a ≠ 0）。a 的正负决定开口方向：a > 0 开口向上，a < 0 开口向下。
y = ax² + bx + c
The turning point (vertex) can be found by completing the square to give the form y = a(x − p)² + q, where (p, q) is the vertex. The axis of symmetry is x = p. The y‑intercept is at (0, c). Quadratic graphs may have 0, 1 or 2 x‑ olving ax² + bx + c = 0 using factorising, the quadratic formula or completing the square.
顶点可通过配方化为 y = a(x − p)² + q 求得，此时顶点为 (p, q)，对称轴为 x = p。y 轴截距为 (0, c)。二次图像与 x 轴可有 0 个、1 个或 2 个交点，通过因式分解、求根公式或配方法解 ax² + bx + c = 0 可得。
- Positive quadratic (a > 0): ∪ shape, minimum point
- Negative quadratic (a < 0): ∩ shape, maximum point
- Line of symmetry x = −b/(2a)
- 正二次 (a > 0)：∪ 形，有最小值点
- 负二次 (a < 0)：∩ 形，有最大值点
- 对称轴 x = −b/(2a)
3. Cubic Graphs | 三次函数图像
Cubic functions have the general form y = ax³ + bx² + cx + d, with a ≠ 0. Typical GCSE cubic graphs show a distinctive ‘wiggle’ and can cross the x‑axis up to three times.
三次函数的一般式为 y = ax³ + bx² + cx + d（a ≠ 0）。GCSE 常见的三次图像呈现优雅的弯曲，并可与 x 轴相交多达三次。
y = ax³ + bx² + cx + d
When a > 0, the graph generally goes from bottom‑left to top‑right; when a < 0, it goes from top‑left to bottom‑ epeated roots in the factorised form, such as y = (x − r)²(x − s), create a touch at the x‑axis.
当 a > 0 时，曲线总体从左下到右上延伸；a < 0 时从左到右下。简单三次图像如 y = x³ 经过原点并有一个拐点。因式分解形式中出现重根时，例如 y = (x − r)²(x − s)，图像会在 x 轴处相切。
- y = x³: basic cubic, passes through (0,0)
- y = (x − 1)(x + 2)(x − 3): crosses x‑axis at 1, −2, 3
- y = (x + 1)²(x − 2): touches at x = −1, crosses at x = 2
- y = x³：基本三次函数，过原点
- y = (x − 1)(x + 2)(x − 3)：在 x = 1, −2, 3 处穿过 x 轴
- y = (x + 1)²(x − 2)：在 x = −1 处相切，在 x = 2 处穿过 x 轴
4. Reciprocal Graphs | 反比例函数图像
Reciprocal graphs have the form y = k/x, where k is a constant and x ≠ 0. The graph consists of two separate branches forming a hyperbola.
反比例函数图像的形式为 y = k/x，其中 k 为常数且 x ≠ 0。图像由两条分支构成双曲线。
y = k/x
0, they are in the second and fourth quadrants. The graph never touches the axes, so the x‑axis and y‑axis are asymptotes. As x approaches ±∞, y tends to 0; as x approaches 0, y tends to ±∞.
若 k > 0，分支位于第一和第三象限；若 k < 0，则位于第二和第四象限。图像不与坐标轴相交，x 轴和 y 轴是渐近线。当 x → ±∞ 时 y → 0；当 x → 0 时 y → ±∞。
- y = 1/x: standard reciprocal, k = 1
- y = −2/x: reflection in y‑axis or x‑axis
- Translated reciprocal: y = k/(x − a) + b
- y = 1/x：标准反比例，k = 1
- y = −2/x：关于坐标轴反射的双曲线
- 平移反比例：y = k/(x − a) + b
5. Exponential Graphs | 指数函数图像
Exponential graphs show rapid growth or decay. The basic form is y = aˣ, where a > 0 and a ≠ 1. If a > 1, the graph models exponential growth; if 0 < a < 1, it models exponential decay.
指数图像呈现快速增长或衰减。基本形式为 y = aˣ（a > 0，a ≠ 1）。若 a > 1，图像表示指数增长；若 0 < a < 1，则为指数衰减。
y = aˣ
All exponential graphs pass through (0,1) because a⁰ = 1. The x‑ apidly for positive x; for decay, it decreases towards zero. Common examples include y = 2ˣ and y = (½)ˣ.
所有指数图像都经过 (0,1)，因为 a⁰ = 1。x 轴为水平渐近线 (y = 0)。在增长型中，当 x > 0 时曲线快速上升；在衰减型中，曲线随 x 增大而趋近于零。常见例子有 y = 2ˣ 和 y = (½)ˣ。
- y = 2ˣ: goes through (1,2), steep rise
- y = 3ˣ: steeper growth
- y = (0.5)ˣ: decay, goes through (1,0.5)
- y = 2ˣ：经过 (1,2)，陡峭上升
- y = 3ˣ：增长更陡
- y = (0.5)ˣ：衰减，经过 (1,0.5)
6. Trigonometric Graphs | 三角函数图像
= sin x, y = cos x and y = tan x, for angles usually measured in degrees from 0° to 360°.
GCSE 中三个关键的三角函数图像是 y = sin x、y = cos x 和 y = tan x，角度通常以度为单位，区间 0° 到 360°。
y = sin x, y = cos x, y = tan x
ps to 0 at 90°, −1 at 180°, 0 at 270°, and 1 at 360°. Both sine and cosine have amplitude 1 and range −1 ≤ y ≤ 1. y = tan x has asymptotes at x = 90° and 270°, with a period of 180° and range all real numbers.
y = sin x 从 0 开始，在 90° 达到最大值 1，180° 回到 0，270° 为 −1，360° 回到 0，周期为 360°。y = cos x 从 1 开始，90° 为 0，180° 为 −1，270° 为 0，360° 回到 1。正弦和余弦的振幅均为 1，值域为 [−1, 1]。y = tan x 在 x = 90° 和 270° 处有渐近线，周期为 180°，值域为所有实数。
- Sine graph: wave starting at origin
- Cosine graph: wave starting at maximum
- Tangent graph: repeated every 180° with vertical asymptotes
- 正弦图像：从原点出发的波形
- 余弦图像：从最大值出发的波形
- 正切图像：每 180° 重复一次，有垂直渐近线
7. Circle Equations and Graphs | 圆的方程与图像
A circle with centre (0,0) and radius r has the equation x² + y² = r². It is not a function (fails the vertical line test), but its graph is a key locus in GCSE Mathematics.
以原点为圆心、半径为 r 的圆方程为 x² + y² = r²。它不是函数（不满足垂线检验），但其图像是 GCSE 数学中的重要轨迹。
x² + y² = r²
ch the circle, locate the centre and plot points at a distance r along the axes. The value r must be positive, and the graph is symmetric about both axes.
若圆心在 (a, b)，方程变为 (x − a)² + (y − b)² = r²。画图时先标出圆心，再沿坐标轴方向截取距离为 r 的点。r 必须为正，图像关于两坐标轴对称。
- Centre (0,0): x² + y² = 25 gives radius 5
- Centre (3, −2): (x − 3)² + (y + 2)² = 16
- Radius found by √r²; ensure r > 0
- 圆心 (0,0)：x² + y² = 25 表示半径 5
- 圆心 (3, −2)：(x − 3)² + (y + 2)² = 16
- 半径 = √r²，确保 r > 0
8. Graph Transformations | 图像变换
tch related graphs quickly. The mapping below shows the standard transformations for GCSE.
理解如何对 y = f(x) 进行变换，有助于快速绘制相关图像。下表展示了 GCSE 常见变换。
| Transformation | Equation | Effect on graph |
|---|---|---|
| Vertical translation | y = f(x) + a | Shifts up by a if a > 0, down if a < 0 |
| Horizontal translation | y = f(x + a) | Shifts left by a if a > 0, right if a < 0 |
| Vertical stretch | y = a f(x) | Stretch vertically by scale factor a |
| Horizontal stretch | y = f(ax) | Stretch horizontally by scale factor 1/a |
| Reflection in x-axis | y = −f(x) | Every y-coordinate changes sign |
| Reflection in y-axis | y = f(−x) | Every x-coordinate changes sign |
For trigonometric graphs, a vertical stretch y = A sin x changes the amplitude to |A|, while y = sin Bx changes the period to 360°/|B|.
对于三角函数图像，纵向拉伸 y = A sin x 将振幅变为 |A|；y = sin Bx 将周期变为 360°/|B|。
9. Identifying Graphs from Equations | 从方程识别图像
Exam questions often ask you to match equations to their graphs. Key clues are the highest power, the sign of coefficients, and the constant term.
考试中常要求将方程与图像配对。关键线索是最高次数项、系数的正负以及常数项。
- Highest power 1: linear straight line
- Highest power 2: quadratic parabola
- Highest power 3: cubic curve
- Equation contains k/x: reciprocal hyperbola
- Equation contains aˣ: exponential curve
- Equation has x² + y²: circle
- 最高次数为 1：直线
- 最高次数为 2：二次抛物线
- 最高次数为 3：三次曲线
- 含有 k/x：反比例双曲线
- 含有 aˣ：指数曲线
- 出现 x² + y²：圆
Always check the y‑intercept, roots and whether the graph is positive or negative for large x. For quadratics, look at the sign of a; for cubics, check the leading coefficient sign and the number of turning points.
始终检查 y 截距、根以及 x 很大时图像的走向。对二次函数，看 a 的符号；对三次函数，检查首项系数的符号与拐点数量。
10. Intersection Points and Solving Equations Graphically | 图像交点与图解方程
intersection points. For instance, to solve f(x) = g(x), draw y = f(x) and y = g(x); the x‑coordinates of the intersections are the solutions.
可以通过绘制两条曲线并找出交点来解方程。例如，解 f(x) = g(x) 时，画出 y = f(x) 和 y = g(x)，交点的横坐标即为方程的解。
and read off approximate solutions. This method also works for simultaneous equations and when finding the roots of a cubic by intersecting with the x‑axis.
GCSE 常见题型是画出二次函数与直线，如 y = x² − 3 和 y = 2x，然后读取近似解。此方法同样适用于解方程组，或通过令曲线与 x 轴相交求三次方程的根。
Always label your graphs clearly and use the given axes. When an approximate solution is read from a graph, it is acceptable to give values to one or two decimal places.
绘图时务必标注清晰并使用给定坐标系。从图像读取近似解时，保留一位或两位小数即可。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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