📚 Reciprocal Graphs | 倒数函数图像
ere k is a constant. These graphs have distinct features such as asymptotes and symmetry, and they never touch the axes. Understanding reciprocal graphs is essential for solving equations graphically and modelling inverse relationships.
倒数函数图像是IGCSE数学中一类重要的非线性图像。它们来源于形如 y = k/x 或 y = k/x² 的函数，其中 k 为常数。这些图像具有渐近线和对称性等独特特征，且永远不会与坐标轴相交。理解倒数函数图像对于图解方程和建立反比关系模型至关重要。
1. Introduction to Reciprocal Functions | 倒数函数简介
Since division by zero is undefined, x cannot be zero, and consequently y never equals zero. These restrictions produce the asymptotes that define the graph's shape.
倒数函数是指变量 x 出现在分母中的函数。最简单形式为 f(x) = 1/x。由于除以零无定义，x 不能为零，因此 y 也永远不会等于零。这些限制产生了定义图像形状的渐近线。
The name 'reciprocal' comes from the idea that y is the reciprocal of x multiplied by a constant. These functions model inverse proportion: as x increases, y decreases, and their product remains constant. In IGCSE, you will learn to recognise, sketch and transform reciprocal graphs.
'倒数'这一名称源于 y 是 x 乘以一个常数的倒数。这些函数模拟反比关系：当 x 增大时，y 减小，且它们的乘积保持恒定。在IGCSE中，你将学习识别、绘制和变换倒数函数图像。
2. The Basic Reciprocal Function y = 1/x | 基本倒数函数 y = 1/x
arate curves called branches. For x > 0, y is positive; for x < 0, y is negative. The curve approaches both axes but never touches them. This shape is a rectangular hyperbola.
y = 1/x 的图像由两条分开的曲线（称为分支）组成。当 x > 0 时，y 为正；当 x < 0 时，y 为负。曲线趋近于两坐标轴但永不相交。这一形状是等轴双曲线。
Key points on the graph can be listed in a table:
图像上的关键点可列表如下：
| x | y = 1/x |
|---|---|
| -2 | -0.5 |
| -1 | -1 |
| -0.5 | -2 |
| 0.5 | 2 |
| 1 | 1 |
| 2 | 0.5 |
hen (-a, -b) also lies on the graph. This property comes from the fact that f(-x) = -f(x).
图像关于原点对称，即具有2阶旋转对称性。若 (a, b) 在图像上，则 (-a, -b) 也在图像上。这一性质来源于 f(-x) = -f(x)。
3. Asymptotes of y = 1/x | y = 1/x的渐近线
ends to positive or negative infinity, y approaches 0, so y = 0 is a horizontal asymptote.
当 x 从正方向趋近于 0 时，y 趋于正无穷大；当 x 从负方向趋近于 0 时，y 趋于负无穷大。直线 x = 0 是垂直渐近线。同样，当 x 趋于正无穷或负无穷时，y 趋近于 0，因此 y = 0 是水平渐近线。
ain. When sketching, these asymptotes should be drawn as dashed lines to guide the shape of the curve.
图像永远不会与坐标轴相交，因为没有有限的 x 值能使 y = 0，且 x = 0 被排除在定义域之外。绘图时，应用虚线画出这些渐近线，以引导曲线的形状。
4. Graph of y = k/x (k > 0) | y = k/x的图像 (k > 0)
the origin, while smaller values pull them closer. The asymptotes remain x = 0 and y = 0.
当 k 为正时，y = k/x 的图像位于第一和第三象限，与 y = 1/x 相似。较大的 k 值使分支远离原点拉伸，较小的 k 值则拉近分支。渐近线仍为 x = 0 和 y = 0。
is odd: f(-x) = -f(x). This means the graph has 180° rotational symmetry about the origin.
例如，y = 3/x 经过 (1, 3)、(3, 1)、(-1, -3) 和 (-3, -1)。该函数为奇函数：f(-x) = -f(x)，这意味着图像关于原点具有 180° 旋转对称性。
Key properties for any k > 0:
对于任意 k > 0 的关键性质：
- Domain: x ∈ ℝ, x ≠ 0 | 定义域：x ∈ ℝ, x ≠ 0
- Range: y ∈ ℝ, y ≠ 0 | 值域：y ∈ ℝ, y ≠ 0
- Asymptotes: x = 0, y = 0 | 渐近线：x = 0, y = 0
- Quadrants: 1st and 3rd | 象限：第一和第三象限
5. Graph of y = k/x (k < 0) | y = k/x的图像 (k < 0)
x, the graph appears in the second and fourth quadrants. For x > 0, y is negative; for x < 0, y is positive. This is a reflection of the positive k graph in the x-axis (or equivalently in the y-axis).
当 k 为负时，例如 y = -2/x，图像出现在第二和第四象限。当 x > 0 时，y 为负；当 x < 0 时，y 为正。这可以看作是 k 为正的图像关于 x 轴（或等效于关于 y 轴）的反射。
and the graph never crosses them. The function remains odd, so rotational symmetry about the origin is preserved. Points such as (1, -2) and (-1, 2) lie on the curve.
渐近线仍为 x = 0 和 y = 0，图像永不相交。该函数仍为奇函数，因此关于原点的旋转对称性保持不变。如 (1, -2) 和 (-1, 2) 等点位于曲线上。
Comparing the sign of k:
比较 k 的符号：
| Sign of k | k 符号 | Quadrants | 象限 | Example point | 示例点 |
|---|---|---|
| k > 0 | 1st and 3rd | 第一、三 | (1, k) |
| k < 0 | 2nd and 4th | 第二、四 | (1, k) with negative y |
6. Reciprocal Squared Function y = k/x² | 倒数平方函数 y = k/x²
rocal-type function. Its graph has two separate branches on either side of the y-axis, but both lie in the upper or lower half-plane depending on the sign of k. Since x² is always positive for x ≠ 0, y always has the same sign as k.
函数 y = k/x² 是另一种倒数型函数。其图像在 y 轴两侧各有一条分支，但二者均位于上半平面或下半平面，取决于 k 的符号。由于 x² 对于 x ≠ 0 恒为正，y 的符号始终与 k 相同。
is, and y → -∞ as x → 0. The asymptotes are still x = 0 (vertical) and y = 0 (horizontal).
当 k > 0 时，两分支均位于 x 轴上方。当 x 从任意一侧趋近于 0 时，y → +∞。当 k < 0 时，两分支均位于 x 轴下方，且当 x → 0 时 y → -∞。渐近线仍为 x = 0（垂直）和 y = 0（水平）。
function: f(-x) = f(x). This is a key difference from y = k/x, which has origin symmetry.
y = k/x² 的图像关于 y 轴对称，为偶函数：f(-x) = f(x)。这是与具有原点对称性的 y = k/x 的一个关键区别。
7. Transformations of Reciprocal Graphs | 倒数函数图像的变换
horizontal asymptote shifts to y = k. The constant a controls the stretch and reflection.
我们可以对倒数函数进行平移变换。对于一般函数 y = a/(x - h) + k，垂直渐近线移至 x = h，水平渐近线移至 y = k。常数 a 控制拉伸和反射。
x. If a is negative, the graph is reflected in the horizontal asymptote. The shape remains a rectangular hyperbola.
若 a 为正，分支相对于新渐近线的象限方位与 y = 1/x 相同。若 a 为负，图像关于水平渐近线反射。形状仍为等轴双曲线。
se lines instead of the axes. Plotting a few points helps to determine the exact position.
示例：y = 2/(x - 3) + 1 具有垂直渐近线 x = 3 和水平渐近线 y = 1。曲线将趋近于这些直线而非坐标轴。描出几个点有助于确定准确位置。
8. Sketching Reciprocal Graphs | 绘制倒数函数图像
/(x - h) + k they are x = h and y = k. Draw these as dashed lines on your coordinate grid.
绘制倒数函数图像时，首先要确定渐近线。对于 y = k/x，渐近线为 x = 0 和 y = 0；对于 y = a/(x - h) + k，渐近线为 x = h 和 y = k。在坐标网格上用虚线画出这些直线。
. For y = k/x, k > 0 means first and third quadrants; k < 0 means second and fourth.
接着，通过检查分子常数的符号，确定分支（相对于渐近线）所在的象限。对于 y = k/x，k > 0 意味着第一和第三象限；k < 0 意味着第二和第四象限。
Find a few key points by substituting chosen x-values, particularly where the curve will pass through integer coordinates. Draw smooth curves that approach but never cross the asymptotes. Label the graph clearly and never join the two branches across the asymptote.
通过代入选定的 x 值求出几个关键点，特别是曲线经过整数坐标的位置。画出光滑的曲线，使其趋近但永不超过渐近线。清晰地标注图像，切勿跨越渐近线连接两分支。
9. Intersection with Axes and Line | 与坐标轴及直线的交点
such as y = c. To find these intersections, solve k/x = c, giving x = k/c, provided c ≠ 0.
形如 y = k/x 的倒数图像不与 x 轴或 y 轴相交，因为 y 永不为零且 x 永不为零。但它们可能与水平直线如 y = c 相交。求交点需解方程 k/x = c，得 x = k/c，前提是 c ≠ 0。
ratic equation. It may have zero, one or two real solutions, representing the number of intersection points.
与斜线 y = mx + c 相交时，解方程 k/x = mx + c。两边同乘 x（因 x ≠ 0）得 k = mx² + cx，这是一个二次方程。它可能有零个、一个或两个实数解，代表交点的个数。
Graphically, you can simply read the intersection points from a plotted graph. This is especially useful in IGCSE questions where you are given a printed grid and asked to solve an equation by drawing a suitable straight line.
在图像上，你可以直接从已绘制的图形中读取交点。这在IGCSE题目中特别有用，题目往往给出一个印刷好的网格，要求通过画一条合适的直线来解方程。
10. Solving Equations Using Reciprocal Graphs | 利用倒数图像解方程
me axes. The x-coordinates of their intersection points are the solutions of the equation.
倒数图像为解方程提供了一种直观方法。例如，要解 2/x = x + 1，可在同一坐标系中画出 y = 2/x 和 y = x + 1，其交点的横坐标即为方程的解。
Sometimes rearranging the equation into the form
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