📚 Graphs of Hyperbolic Functions | 双曲函数图像
Hyperbolic functions are defined using exponential functions, but their graphs display a striking beauty and symmetry that mirrors their trigonometric cousins. In this article, we explore the shapes, key features, and transformations of the six standard hyperbolic functions, along with their inverses and exam-relevant techniques.
双曲函数由指数函数定义，但它们的图像展现出一种与三角函数惊人相似的优美与对称性。本文将深入探讨六个标准双曲函数的形状、关键特征与变换，同时涵盖反函数及与考试相关的解题技巧。
1. Definitions from Exponential Functions | 由指数函数给出的定义
Before sketching any graph, we must recall the formal definitions. The two primary hyperbolic functions are:
在绘制任何图像之前，我们必须回顾其形式定义。两个最基本的双曲函数为：
sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2
From these, four more functions are derived: tanh x = sinh x / cosh x, coth x = cosh x / sinh x, sech x = 1 / cosh x, and cosech x = 1 / sinh x. Each of these six functions has a distinct graph that you must be able to sketch from memory.
由这两个函数可推导出另外四个函数：tanh x = sinh x / cosh x，coth x = cosh x / sinh x，sech x = 1 / cosh x，cosech x = 1 / sinh x。这六个函数各具独特的图像，你需要能够凭记忆画出草图。
2. The Graph of y = sinh x | y = sinh x 的图像
The graph of y = sinh x passes through the origin and is an odd function, meaning sinh(−x) = −sinh x. As x → +∞, sinh x → +∞, and as x → −∞, sinh x → −∞. There are no asymptotes; the curve grows exponentially in both directions but is steeper than a simple exponential because it is the difference of two exponentials.
y = sinh x 的图像经过原点，是一个奇函数，即 sinh(−x) = −sinh x。当 x → +∞ 时，sinh x → +∞；当 x → −∞ 时，sinh x → −∞。图像没有渐近线，曲线在两个方向均呈现指数增长，但由于它是两个指数函数的差，因此比简单指数函数更陡峭。
The gradient at the origin is exactly 1, and the curve is monotonically increasing everywhere. Locally, near x = 0, sinh x behaves like x (its linear approximation), which is why the tangent line at the origin is y = x.
在原点的梯度恰为 1，曲线在整个定义域上单调递增。在 x = 0 附近，sinh x 近似等于 x（即其线性逼近），因此原点处的切线为 y = x。
3. The Graph of y = cosh x | y = cosh x 的图像
The graph of y = cosh x is an even function, symmetric about the y-axis. It has a global minimum at (0, 1), since cosh 0 = (1 + 1)/2 = 1. As |x| → ∞, cosh x → ∞. The curve resembles a parabola but grows much faster; it is often called a catenary when describing hanging cables.
y = cosh x 的图像是偶函数，关于 y 轴对称。它在 (0, 1) 处取得全局最小值，因为 cosh 0 = (1 + 1)/2 = 1。当 |x| → ∞ 时，cosh x → ∞。该曲线类似抛物线，但增长速度快得多；描述悬索时它常被称为悬链线。
There are no asymptotes, and the curve never dips below y = 1. A useful approximation for large x is cosh x ≈ eˣ/2 (from the right) or e⁻ˣ/2 (from the left).
曲线没有渐近线，且始终不低于 y = 1。一个有用的近似是：当 x 很大时，从右侧看 cosh x ≈ eˣ/2，从左侧看则约为 e⁻ˣ/2。
4. The Graph of y = tanh x | y = tanh x 的图像
y = tanh x is odd, passes through the origin, and has two horizontal asymptotes: y = 1 as x → +∞, and y = −1 as x → −∞. The function is monotonically increasing and always lies between −1 and 1.
y = tanh x 是奇函数，经过原点，有两条水平渐近线：当 x → +∞ 时 y = 1，当 x → −∞ 时 y = −1。该函数单调递增，且始终介于 −1 与 1 之间。
The graph approaches each asymptote rapidly but never touches them. Its point of inflection is at the origin, where the graph changes from concave down to concave up. The gradient at the origin is 1, identical to that of sinh x and the line y = x.
图像迅速靠近每条渐近线，但永远不会触及。其拐点在原点处，图像由下凹变为上凸。原点处的梯度为 1，与 sinh x 及直线 y = x 相同。
5. The Graphs of y = coth x, sech x, cosech x | coth x、sech x、cosech x 的图像
y = coth x is odd and has a vertical asymptote at x = 0. It has horizontal asymptotes y = 1 and y = −1. As x → 0⁺, coth x → +∞; as x → 0⁻, coth x → −∞. In the first quadrant, the branch looks like the portion of tanh x but with y > 1.
y = coth x 是奇函数，在 x = 0 处有垂直渐近线。它有水平渐近线 y = 1 和 y = −1。当 x → 0⁺ 时，coth x → +∞；当 x → 0⁻ 时，coth x → −∞。在第一象限，其分支类似于 tanh x 的形状，但 y 值大于 1。
y = sech x is even, has a maximum value of 1 at x = 0, and has y = 0 as a horizontal asymptote. It is always positive, resembling a smooth bell curve. y = cosech x is odd, has a vertical asymptote at x = 0, and tends to 0 as |x| → ∞, but it is negative for x < 0 and positive for x > 0.
y = sech x 是偶函数，在 x = 0 处取最大值 1，以 y = 0 为水平渐近线。它始终为正，形似光滑的钟形曲线。y = cosech x 是奇函数，在 x = 0 处有垂直渐近线，当 |x| → ∞ 时趋近于 0，但 x < 0 时为负，x > 0 时为正。
6. Key Features: Symmetry, Asymptotes, and Intercepts | 关键特征：对称性、渐近线与截距
For all six graph sketches, memorise the following table of key features:
对于这六个图像草图，请牢记以下关键特征表：
| Function | Symmetry | Intercepts | Asymptotes |
| sinh x | Odd | (0,0) | None |
| cosh x | Even | (0,1) | None |
| tanh x | Odd | (0,0) | y = ±1 |
| coth x | Odd | None | x = 0, y = ±1 |
| sech x | Even | (0,1) | y = 0 |
| cosech x | Odd | None | x = 0, y = 0 |
This table is essentially a checklist: when sketching, always mark the asymptotes and intercepts first.
该表格实际上是一个检查清单：绘图时，始终先标出渐近线和截距。
7. Hyperbolic vs Trigonometric Graphs | 双曲图像与三角函数图像的对比
There is a useful correspondence: for any real x, the point (cos x, sin x) lies on the unit circle x² + y² = 1, while the point (cosh x, sinh x) lies on the hyperbola x² − y² = 1. This is why they are called hyperbolic functions.
这里有一个非常有用的对应关系：对于任意实数 x，点 (cos x, sin x) 位于单位圆 x² + y² = 1 上，而点 (cosh x, sinh x) 位于双曲线 x² − y² = 1 上。这正是“双曲函数”名称的由来。
Visually, sin x oscillates between −1 and 1 forever, whereas sinh x rises monotonically without bound. Similarly, cos x oscillates, but cosh x has no upper bound and only increases as |x| grows. The tangent-like functions tan x has vertical asymptotes at π/2 + kπ, while tanh x is smooth with horizontal asymptotes. Recognising these differences prevents common sketching errors.
从图像上看，sin x 永远在 −1 与 1 之间振荡，而 sinh x 则单调上升、无界增长。类似地，cos x 振荡，而 cosh x 无上界，随 |x| 增大而增大。tan x 在 π/2 + kπ 处有垂直渐近线，而 tanh x 光滑且具有水平渐近线。认清这些差异可以避免常见的绘图错误。
8. Transformations of Hyperbolic Graphs | 双曲函数图像的变换
Standard transformations apply exactly as they do to any function. For example, y = 2 cosh(x − 1) is a horizontal shift right by 1 unit and a vertical stretch by factor 2. The minimum point moves from (0,1) to (1,2). Such questions frequently appear in exams.
标准变换规则对双曲函数同样适用。例如，y = 2 cosh(x − 1) 是将图像向右平移 1 个单位，并沿垂直方向拉伸 2 倍。最小值点从 (0,1) 移动到 (1,2)。这类题目在考试中频繁出现。
Also remember the effect of a vertical reflection: y = −tanh x is the mirror image of y = tanh x in the x-axis. A horizontal reflection, y = tanh(−x), is equivalent to −tanh x because tanh is odd. For even functions like cosh x, we have cosh(−x) = cosh x, so the graph is unchanged by reflection in the y-axis.
同时记住垂直反射的效果：y = −tanh x 是 y = tanh x 关于 x 轴的镜像。由于 tanh 是奇函数，水平反射 y = tanh(−x) 等价于 −tanh x。对于 cosh x 这类偶函数，cosh(−x) = cosh x，因此图像关于 y 轴的反射不会改变形状。
9. Graphs of Inverse Hyperbolic Functions | 反双曲函数的图像
The inverse hyperbolic functions are defined using logarithms:
反双曲函数通过对数定义：
arsinh x = ln(x + √(x² + 1)), arcosh x = ln(x + √(x² − 1)), artanh x = ½ ln((1 + x)/(1 − x))
The graph of y = arsinh x is the reflection of y = sinh x in the line y = x. It is odd, passes through the origin, and increases logarithmically for large x. y = arcosh x has domain x ≥ 1 and is the upper branch of a graph increasing from the point (1,0). Since cosh x is not one-to-one, its inverse only exists for x ≥ 0, giving arcosh a restricted domain of x ≥ 1.
y = arsinh x 的图像是 y = sinh x 关于直线 y = x 的反射。它是奇函数，经过原点，对于大的 x 呈对数增长。y = arcosh x 的定义域为 x ≥ 1，是从点 (1,0) 开始上升的上方分支。由于 cosh x 不是一一映射，其反函数仅对 x ≥ 0 存在，因此 arcosh 的定义域限制为 x ≥ 1。
y = artanh x has domain −1 < x < 1, with vertical asymptotes at x = 1 and x = −1. It passes through the origin and is odd. When sketching these inverses, always indicate the restricted domain and the asymptotes clearly.
y = artanh x 的定义域为 −1 < x < 1，在 x = 1 与 x = −1 处有垂直渐近线。它经过原点且为奇函数。绘制这些反函数时，务必清楚标出受限定义域和渐近线。
10. Solving Equations and Inequalities Using Graphs | 利用图像解方程与不等式
A common exam question asks you to determine the number of solutions to an equation involving hyperbolic functions. For instance, the equation 2 cosh x = 3 has exactly two solutions because the horizontal line y = 3 intersects the graph of y = 2 cosh x twice (once for x > 0 and once for x < 0). However, 2 cosh x = 1 has no solutions, since the minimum value of 2 cosh x is 2.
一个常见的考试题是确定包含双曲函数的方程的解的个数。例如，方程 2 cosh x = 3 恰有两个解，因为水平线 y = 3 与 y = 2 cosh x 的图像相交两次（一次在 x > 0，一次在 x < 0）。然而，2 cosh x = 1 无解，因为 2 cosh x 的最小值为 2。
For inequalities, consider tanh x > 0.5. Since tanh is increasing, the solution is x > artanh 0.5, and the graph visually shows that all points to the right of the intersection satisfy the inequality. Similarly, sech x < 0.8 means the curve dips below the horizontal line, giving two intervals: one negative and one positive.
对于不等式，以 tanh x > 0.5 为例。因为 tanh 单调递增，解为 x > artanh 0.5，图像直观显示交点右侧的所有点都满足不等式。类似地，sech x < 0.8 意味着曲线低于水平线 0.8，解集包含两个区间：一个在负半轴，一个在正半轴。
11. Applications in Real-World Contexts | 双曲图像的实际应用
The graph of cosh x, known as a catenary, models the shape of a hanging chain or cable under gravity. The equation y = a cosh(x/a) describes the curve perfectly, and A-Level questions may ask you to interpret this shape visually or algebraically.
cosh x 的图像被称为悬链线，用于模拟重力作用下悬挂的链条或电缆的形状。方程 y = a cosh(x/a) 精确描述该曲线，A-Level 题目可能会要求你从图像或代数角度进行解读。
The graph of tanh x appears in mathematical models of velocity in certain physical systems, where a quantity approaches a limiting value. For example, the voltage across a charging capacitor in an RC circuit follows a curve similar to tanh x, rising steeply then flattening out. Recognising this saturated-growth pattern helps you connect mathematical graphs to practical problems in mechanics and electronics.
tanh x 的图像出现在某些物理系统中速度的数学模型里，其中某个量趋近于极限值。例如，RC 电路中电容充电时两端的电压遵循类似 tanh x 的曲线：先急剧上升然后趋于平坦。识别这种饱和增长模式，可以帮助你将数学图像与力学及电子学中的实际问题联系起来。
12. Common Mistakes and Exam Tips | 常见错误与考试建议
A frequent error is drawing cosh x as a parabola. While they look similar near the origin, the cosh curve rises exponentially, so it will quickly outpace a parabola. Always check a few points, such as cosh 0 = 1, cosh 1 ≈ 1.543, and cosh 2 ≈ 3.762, to confirm the rate of growth.
一个常见错误是将 cosh x 画成抛物线。虽然它们在原点附近看起来很相似，但 cosh 曲线呈指数上升，会迅速超过抛物线。务必检查几个点，例如 cosh 0 = 1、cosh 1 ≈ 1.543、cosh 2 ≈ 3.762，以确认增长速度。
Another mistake is forgetting that sinh x is an odd function, so its graph must show rotational symmetry about the origin. Also, when asked to sketch the inverse, many students draw the entire reflection of cosh x, which is incorrect because the inverse only exists for x ≥ 0 on the original function. In exams, explicitly state the domain of any inverse function you sketch.
另一个错误是忘记 sinh x 是奇函数，因此其图像必须关于原点呈旋转对称性。此外，当要求绘制反函数时，许多学生画出整个 cosh x 的反射图像，这是错误的，因为反函数只对原函数 x ≥ 0 的部分存在。考试中，请明确写出你所绘制的任何反函数的定义域。
Finally, always label asymptotes with their equations, mark the origin intercepts, and use a smooth continuous curve. For marks to be awarded, the general shape is more important than exact precision, but the key features must be accurate.
最后，始终用方程标记渐近线、标出原点处的截距，并画平滑连续曲线。想要获得分数，整体形状比精确度更重要，但关键特征必须准确。
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