📚 Reciprocal Graphs | 倒数函数图像
Reciprocal functions are a fundamental family of curves in A-Level mathematics, appearing in pure mathematics and required for sketching, transformations, and solving equations and inequalities. Understanding their asymptotes, shape, and key features is essential for success in the Edexcel specification.
倒数函数是A-Level数学中一类基本的曲线，出现在纯数学中，需要掌握其图像绘制、变换以及解方程和不等式。理解其渐近线、形状和关键特征对于在Edexcel考试中取得成功至关重要。
1. Understanding Reciprocal Functions | 理解倒数函数
k/x or a transformation of it, where k is a non-zero constant. The variable x appears in the denominator, which introduces restrictions on the domain and leads to characteristic hyperbolic shapes.
倒数函数是可以写成 f(x) = k/x 或其变换形式的函数，其中 k 为非零常数。变量 x 出现在分母中，这就对定义域产生了限制，并导致典型的双曲线形状。
= a/(bx + c) + d. Recognising the parent function y = 1/x and applying transformations is key.
在Edexcel A-Level课程中，你需要处理基本的倒数图像，也要处理分母为线性表达式的有理函数，例如 y = a/(bx + c) + d。识别出母函数 y = 1/x 并施加变换是关键。
2. The Basic Reciprocal Graph y = 1/x | 基本倒数图像 y = 1/x
rate branches: one in the first quadrant and one in the third quadrant. The curve never touches the x-axis or the y-axis, but approaches them arbitrarily closely.
最简单的倒数函数是 y = 1/x。它的图像由两个独立的分支组成：一个在第一象限，另一个在第三象限。曲线永远不会碰到 x 轴或 y 轴，但会无限接近它们。
y = 1/x
y = 1/x
from the positive side, y tends to positive infinity. On the negative side, as x → 0⁻, y → –∞, and as x → – → 0⁻. The graph is symmetrical about the line y = x, and it is an odd function because f(–x) = –f(x).
对于大的正 x 值，y 变得非常小且为正；当 x 从正方向趋近于 0 时，y 趋向正无穷大。在负数一侧，当 x→0⁻，y→–∞，当 x→–∞ 时，y→0⁻。该图像关于直线 y = x 对称，并且是一个奇函数，因为 f(–x) = –f(x)。
3. Asymptotes and Their Significance | 渐近线及其意义
is) and the horizontal asymptote y = 0 (the x-axis). These occur because division by zero is undefined, and as |x| increases without bound, 1/x approaches zero.
渐近线是一条曲线无限接近但永不相交的直线。对于 y = 1/x，有两条渐近线：垂直渐近线 x = 0（y 轴）和水平渐近线 y = 0（x 轴）。出现这些渐近线是因为除以零未定义，并且当 |x| 无限增大时，1/x 趋近于零。
Vertical asymptote: x = 0
垂直渐近线：x = 0
Horizontal asymptote: y = 0
水平渐近线：y = 0
Asymptotes are crucial when sketching because they divide the plane into regions where the curve can exist. Ignoring them leads to incorrect graph shapes. In exam questions, you must indicate asymptotes as dashed lines and label them with their equations.
渐近线在画草图时至关重要，因为它们将平面划分为曲线可以存在的区域。忽略它们会导致图像形状错误。在考试题目中，你必须用虚线画出渐近线，并标注其方程。
4. Effect of the Constant of Proportionality: y = a/x | 比例常数的影响：y = a/x
curve. For a > 0, the two branches remain in the first and third quadrants, and the graph stretches vertically away from the origin as |a| raph is reflected in the x-axis, moving the branches into the second and fourth quadrants.
在 y = a/x 中引入常数乘数 a 会改变曲线的陡峭程度。当 a > 0 时，两个分支依旧位于第一和第三象限，随着 |a| 增大，图像在垂直方向上远离原点伸展。当 a < 0 时，图像关于 x 轴反射，分支转移到第二和第四象限。
y = 2/x, y = 1/x, y = 0.5/x (all with same asymptotes)
y = 2/x, y = 1/x, y = 0.5/x（渐近线相同）
he constant of proportionality, and the product xy = a is constant for any point on the curve.
无论 a 取何值，只要未施加平移，垂直渐近线和水平渐近线仍分别为 x = 0 和 y = 0。常数 a 有时被称为比例常数，且对于曲线上任意点，乘积 xy = a 是常数。
5. Horizontal and Vertical Translations: y = a/(x - h) + k | 水平和垂直平移：y = a/(x - h) + k
The general transformation of the reciprocal function is given by y = a/(x – till controls the scale and reflection; h and k determine the new centre of the hyperbola.
倒数函数的一般变换由 y = a/(x – h) + k 给出。这将垂直渐近线从 x = 0 移至 x = h，水平渐近线从 y = 0 移至 y = k。参数 a 仍然控制缩放和反射；h 与 k 决定了双曲线的新中心。
Asymptotes: x = h, y = k
渐近线：x = h, y = k
For example, y = 3/(x + 2) – 1 has a vertical asymptote at x = –2 and a horizontal asymptote at y = –1. The graph is centred at (–2, – branches. Remember that the shape remains the same as y = 1/x; only the position changes.
例如，y = 3/(x + 2) – 1 的垂直渐近线为 x = –2，水平渐近线为 y = –1。图像中心位于 (–2, –1)。画草图时，先画出渐近虚线，然后标出几个关键点以确定分支的方向。记住，形状与 y = 1/x 相同，变化的只是位置。
6. Determining Asymptotes from Equation | 由方程确定渐近线
ve for x. For y = a/(bx + c) + d, the denominator is bx + c; the vertical asymptote is x = –c/b. The horizontal asymptote is found by letting |x| become very large: the term a/(bx + c) approaches 0, so the horizontal asymptote is y = d.
要找到倒数函数的垂直渐近线，令分母等于零并求解 x。对于 y = a/(bx + c) + d，分母为 bx + c；垂直渐近线为 x = –c/b。水平渐近线则通过令 |x| 变得非常大来求得：项 a/(bx + c) 趋近于 0，因此水平渐近线为 y = d。
If the function is given as a rational expression like y = (2x + 1)/(x – 3), perform algebraic division or rewrite it in the form p + q/(x – 3). This reveals the horizontal asymptote y = p and the vertical asymptote x = 3.
如果函数以有理表达式给出，例如 y = (2x + 1)/(x – 3)，执行代数除法或将其重写为 p + q/(x – 3) 的形式。这就能显示出水平渐近线 y = p 和垂直渐近线 x = 3。
y = (2x + 1)/(x – 3) → y = 2 + 7/(x – 3), asymptotes: x = 3, y = 2
y = (2x + 1)/(x – 3) → y = 2 + 7/(x – 3)，渐近线：x = 3, y = 2
7. Sketching Reciprocal Graphs | 绘制倒数函数图像草图
mptote and examine end behaviour for the horizontal asymptote. Next, determine the sign of a to know whether the graph lies in the first/third quadrants relative to the centre, or second/fourth.
一个可靠的画图流程首先确定渐近线：解分母 = 0 求得垂直渐近线，并考察末端行为以获得水平渐近线。接着，判断 a 的符号，以了解图像相对于中心是位于第一/第三象限，还是第二/第四象限。
to find the x-intercept. Then choose a couple of additional x-values on either side of the vertical asymptote to calculate coordinates. Plot these points, draw the asymptotes as dashed lines, and smoothly sketch the two branches.
求截距：令 x = 0 求 y 截距（如果存在），令 y = 0 求 x 截距。然后在垂直渐近线两侧选取若干 x 值计算坐标。标出这些点，画出渐近虚线，并平滑地绘出两条分支。
| Step | Action |
| 1 | Find and draw asymptotes |
| 2 | Check sign of a (orientation) |
| 3 | Calculate and plot intercepts |
| 4 | Plot extra points either side of vertical asymptote |
| 5 | Draw smooth hyperbolic branches |
| 步骤 | 操作 |
| 1 | 求出并画出渐近线 |
| 2 | 检查 a 的符号（方向） |
| 3 | 计算并标出截距 |
| 4 | 在垂直渐近线两侧标出额外点 |
| 5 | 画出平滑的双曲线分支 |
8. Reflections and Stretches | 反射与伸缩
Reciprocal graphs obey the same transformation rules as other functions. Replacing y with –y reflects the graph in the x-axis; replacing x with – se y = 1/x is odd, a reflection in the y-axis is equivalent to a reflection in the x-axis.
倒数图像遵循与其他函数相同的变换规则。用 –y 替换 y 将使图像关于 x 轴反射；用 –x 替换 x 则关于 y 轴反射。由于 y = 1/x 是奇函数，关于 y 轴的反射等同于关于 x 轴的反射。
he denominator by a constant, as in y = 1/(bx), is a horizontal stretch by scale factor 1/|b| s stretched. A coefficient inside the denominator also moves the vertical asymptote to x = –c/b.
用常数乘以分母，例如 y = 1/(bx)，是通过尺度因子 1/|b| 进行的水平伸缩。若 b > 1，图像在水平方向压缩；若 0 < b < 1，则会拉伸。分母内的系数也会将垂直渐近线移至 x = –c/b。
y = 1/(2x) → vertical asymptote still x = 0, but steeper
y = 1/(2x) → 垂直渐近线仍为 x = 0，但更陡
9. Intercepts and Domain/Range | 截距、定义域与值域
und by substituting x = 0, provided that x = 0 is not the vertical asymptote. For y = a/(x – h) + k, the y-intercept exists only if h ≠ 0 and is given by y = a/(–h) + k. The x-intercept occurs when y = 0; solving a/(x – h) + k = 0 gives x = h – a/k, provided k ≠ 0.
y 截距通过代入 x = 0 求得，前提是 x = 0 不是垂直渐近线。对于 y = a/(x – h) + k，仅当 h ≠ 0 时 y 截距存在，且为 y = a/(–h) + k。x 截距发生在 y = 0 时；求解 a/(x – h) + k = 0 得到 x = h – a/k，前提是 k ≠ 0。
The domain of a reciprocal function excludes any x that makes the denominator zero, so domain: x ∈ ℝ, x ≠ h. The range excludes the value of the horizontal asymptote, giving range: y ∈ ℝ, y ≠ k. Always state domain and range in set notation or using inequalities.
倒数函数的定义域排除任何使分母为零的 x，因此定义域：x ∈ ℝ，x ≠ h。值域排除水平渐近线的值，即值域：y ∈ ℝ，y ≠ k。始终用集合符号或不等式表示定义域与值域。
Domain: {x ∈ ℝ : x ≠ h} Range: {y ∈ ℝ : y ≠ k}
定义域：{x ∈ ℝ : x ≠ h} 值域：{y ∈ ℝ : y ≠ k}
10. Using Reciprocal Graphs to Solve Equations | 利用倒数图像解方程
Equations involving reciprocal expressions, such as 2/(x – 1) = x + 3, can be solved graphically by considering two separate functions: y = 2/(x – helpful when algebraic rearrangement leads to a quadratic that is difficult to factorise.
涉及倒数表达式的方程，例如 2/(x – 1) = x + 3，可以通过考虑两个单独的函数 y = 2/(x – 1) 和 y = x + 3 来图解求解。它们的交点给出解。当代数变换导致难以因式分解的二次方程时，这种方法尤其有用。
You must sketch both graphs accurately, label asymptotes and intercepts, and indicate the intersection points clearly. The x-coordinates of these intersections are the solutions to the original equation. Always check if any solutions lie on a vertical asymptote, as they would be invalid.
你必须准确地画出两个函数的草图，标注渐近线和截距，并清晰地标出交点。这些交点的 x 坐标即是原方程的解。始终检查是否有解落在垂直渐近线上，因为它们将是无效的。
Solve 1/x = 2x + 1 → sketch y = 1/x and y = 2x + 1
解 1/x = 2x + 1 → 画 y = 1/x 和 y = 2x + 1 的草图
11. Reciprocal Graphs and Inequalities | 倒数图像与不等式
consists of the x-intervals where the reciprocal graph lies above the line. Be careful when solving algebraically because multiplying by x can change the inequality direction depending on the sign of x.
像 1/x > 2 这样的倒数不等式可以通过图像求解。画出 y = 1/x 和直线 y = 2。解集由倒数图像位于直线上方的 x 区间组成。在使用代数方法求解时要小心，因为乘以 x 会根据 x 的符号改变不等号的方向。
From the graph, 1/x > 2 is satisfied for 0 < x < 1/2. The region x < 0 never works because 1/x is negative. Graphical interpretation avoids sign errors and is often preferred in exam mark schemes when a sketch is explicitly requested.
从图像上看，当 0 < x < 1/2 时满足 1/x > 2。x < 0 的区域永远不行，因为 1/x 为负。图像解释可以避免符号错误，当题目明确要求画草图时，考试评分方案通常也偏好这种方法。
Solution: 0 < x < ½
解：0 < x < ½
12. Common Mistakes and Exam Tips | 常见错误与考试技巧
ing the vertical asymptote when the denominator contains a coefficient of x. For y = 1/(2x – 4), students often set x = 4 instead of solving 2x – 4 = 0 to get x = 2. Always solve the linear equation completely.
一个典型错误是分母含有 x 的系数时，错判了垂直渐近线。对于 y = 1/(2x – 4)，学生常常设 x = 4，而不是解 2x – 4 = 0 得 x = 2。务必完整求解线性方程。
Another mistake is drawing the branches touching the asymptotes. The curve must approach asymptotes but never cross them. Use dashed lines and keep the curve distinct. Also, when finding the range, forgetting to exclude the value k leads to an incomplete answer.
另一个错误是画出与渐近线接触的分支。曲线必须趋近渐近线，但绝不能穿过它们。使用虚线并保持曲线分离开。此外，在求值域时，忘记排除 k 值会导致答案不完整。
ew asymptotes after each transformation. A well-drawn sketch with labelled asymptotes, intercepts and coordinates of key points will secure full marks on graph questions.
在变换问题中，从 y = 1/x 逐步构建。在每次变换后说明新的渐近线。一幅绘制精良、标有渐近线、截距和关键点坐标的草图将确保在图像题上拿到满分。
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