Graphs of Hyperbolic Functions | 双曲函数的图像

📚 Graphs of Hyperbolic Functions | 双曲函数的图像

The hyperbolic functions cosh x, sinh x and tanh x are defined in terms of exponential functions. Understanding their graphs is essential for solving equations, sketching transformations and interpreting physical models. In this revision guide, we examine the key features of each graph, including symmetry, asymptotes, stationary points and range.

双曲函数 cosh x、sinh x 和 tanh x 是用指数函数定义的。理解它们的图像对于解方程、绘制变换图像以及解释物理模型至关重要。在本复习指南中，我们研究每类图像的关键特征，包括对称性、渐近线、驻点和值域。

1. Definitions and Basic Identities | 定义与基本恒等式

The hyperbolic functions are defined as:

sinh x = (eˣ − e⁻ˣ) / 2

cosh x = (eˣ + e⁻ˣ) / 2

tanh x = sinh x / cosh x = (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ)

These definitions immediately imply that cosh x is an even function and sinh x and tanh x are odd functions.

这些定义立即表明 cosh x 是偶函数，而 sinh x 和 tanh x 是奇函数。

2. Graph of y = sinh x | y = sinh x 的图像

The graph of sinh x passes through the origin. As x → ∞, sinh x → ∞; as x → −∞, sinh x → −∞. The curve is always increasing, and it has no stationary points or asymptotes.

sinh x 的图像经过原点。当 x → ∞ 时，sinh x → ∞；当 x → −∞ 时，sinh x → −∞。曲线始终递增，没有驻点，也没有渐近线。

For large positive x, sinh x behaves like eˣ/2; for large negative x, it behaves like −e⁻ˣ/2. The graph is rotationally symmetric about the origin because it is an odd function.

当 x 很大且为正时，sinh x 的行为类似于 eˣ/2；当 x 很大且为负时，它类似于 −e⁻ˣ/2。由于它是奇函数，图像关于原点旋转对称。

3. Graph of y = cosh x | y = cosh x 的图像

The graph of cosh x is a U-shaped curve, similar to a parabola but flatter at the bottom. Its minimum value is cosh 0 = 1, so the lowest point is (0, 1).

cosh x 的图像是一条 U 形曲线，类似于抛物线，但底部更平坦。其最小值为 cosh 0 = 1，因此最低点为 (0, 1)。

The function is even, so the graph is symmetric about the y-axis. As x → ±∞, cosh x → ∞. There are no asymptotes and no x-intercepts.

该函数是偶函数，因此图像关于 y 轴对称。当 x → ±∞ 时，cosh x → ∞。图像没有渐近线，也没有 x 轴截距。

4. Graph of y = tanh x | y = tanh x 的图像

The graph of tanh x passes through the origin and is increasing for all x. Its range is (−1, 1), and it has two horizontal asymptotes: y = 1 as x → ∞ and y = −1 as x → −∞.

tanh x 的图像经过原点，并且对所有 x 递增。其值域为 (−1, 1)，它有两条水平渐近线：当 x → ∞ 时 y = 1，当 x → −∞ 时 y = −1。

The curve approaches these asymptotes quickly but never crosses them. Like sinh x, it is an odd function and has rotational symmetry about the origin.

曲线迅速接近这些渐近线，但永远不会越过它们。与 sinh x 一样，它是奇函数，具有关于原点的旋转对称性。

5. Key Features Summary Table | 关键特征汇总表

Function	Domain	Range	Parity	Asymptotes
sinh x	ℝ	ℝ	Odd	None
cosh x	ℝ	[1, ∞)	Even	None
tanh x	ℝ	(−1, 1)	Odd	y = 1, y = −1

Comparing these features helps you identify which hyperbolic function is represented by a given sketch.

比较这些特征有助于你识别给定草图所代表的双曲函数。

6. Gradients and Stationary Points | 导数与驻点

The derivatives are:

d/dx (sinh x) = cosh x

d/dx (cosh x) = sinh x

d/dx (tanh x) = sech² x = 1 / cosh² x

Since cosh x > 0 for all x, the gradient of sinh x is always positive, so the graph is strictly increasing. The gradient of tanh x is always positive as well, but it tends to 0 as x → ±∞.

由于 cosh x 对所有 x 都大于 0，因此 sinh x 的导数始终为正，所以图像严格递增。tanh x 的导数也始终为正，但当 x → ±∞ 时趋向于 0。

For cosh x, the only stationary point is at x = 0, where cosh 0 = 1 is a global minimum.

对于 cosh x，唯一的驻点在 x = 0 处，此时 cosh 0 = 1 是全局最小值。

7. Reciprocal Hyperbolic Functions | 倒数双曲函数

The reciprocal functions are defined as:

cosech x = 1 / sinh x, sech x = 1 / cosh x, coth x = 1 / tanh x

The graph of sech x has a maximum point at (0, 1) and tends to 0 as x → ±∞. Since cosh x ≥ 1, we have 0 < sech x ≤ 1.

sech x 的图像在 (0, 1) 处有最大值，并且当 x → ±∞ 时趋向于 0。由于 cosh x ≥ 1，所以 0 < sech x ≤ 1。

The graph of cosech x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. Similarly, coth x has a vertical asymptote at x = 0 and horizontal asymptotes y = 1 and y = −1.

cosech x 的图像在 x = 0 处有垂直渐近线，在 y = 0 处有水平渐近线。类似地，coth x 在 x = 0 处有垂直渐近线，并有水平渐近线 y = 1 和 y = −1。

8. Transformations of Hyperbolic Graphs | 双曲图像的变换

All standard curve transformations apply to hyperbolic functions. For example, y = cosh x + 2 shifts the graph up by 2 units, giving a minimum at (0, 3).

所有标准曲线变换都适用于双曲函数。例如，y = cosh x + 2 将图像向上平移 2 个单位，得到最小值点 (0, 3)。

y = 2 sinh x is a vertical stretch by a factor of 2, while y = sinh(3x) is a horizontal compression by a factor of 1/3. Reflections across the axes can be applied using the parity properties.

y = 2 sinh x 是垂直方向上放大 2 倍，而 y = sinh(3x) 是水平方向上压缩到原来的 1/3。利用奇偶性可以进行关于坐标轴的反射。

9. Solving Equations Graphically | 利用图像解方程

The number of solutions to equations such as cosh x = k depends on k. If k < 1, there are no solutions; if k = 1, one solution x = 0; if k > 1, two solutions x = ±a.

形如 cosh x = k 的方程的解的个数取决于 k。如果 k < 1，则无解；如果 k = 1，有一个解 x = 0；如果 k > 1，有两个解 x = ±a。

For sinh x = k, there is always exactly one solution for any real k, because sinh x is a bijection from ℝ to ℝ. For tanh x = k, there is exactly one solution when |k| < 1 and no solution when |k| ≥ 1.

对于 sinh x = k，任意实数 k 都有且仅有一个解，因为 sinh x 是从 ℝ 到 ℝ 的双射。对于 tanh x = k，当 |k| < 1 时恰有一个解，当 |k| ≥ 1 时无解。

10. Examples of Sketching | 画图示例

Example 1: Sketch y = 2 cosh x − 3. The graph is a U-shape with minimum value 2 × 1 − 3 = −1 at x = 0. It is symmetric about the y-axis.

示例 1：画出 y = 2 cosh x − 3 的图像。该图像为 U 形，在 x = 0 处最小值为 2 × 1 − 3 = −1。它关于 y 轴对称。

Example 2: Sketch y = 1 − tanh x. The graph is a reflection of tanh x in the x-axis, then shifted up by 1. Thus it has horizontal asymptotes y = 0 as x → ∞ and y = 2 as x → −∞, and it passes through (0, 1).

示例 2：画出 y = 1 − tanh x 的图像。该图像是 tanh x 关于 x 轴反射后向上平移 1 个单位得到的。因此它有水平渐近线：当 x → ∞ 时 y = 0，当 x → −∞ 时 y = 2，并且经过 (0, 1)。

11. Connection with Circles and Trigonometric Graphs | 与圆和三角函数图像的联系

Hyperbolic functions are related to the rectangular hyperbola x² − y² = 1 through the parametrisation x = cosh t, y = sinh t. This is analogous to the unit circle parametrisation x = cos t, y = sin t.

双曲函数通过参数化 x = cosh t, y = sinh t 与直角双曲线 x² − y² = 1 相联系。这与单位圆的参数化 x = cos t, y = sin t 类似。

Unlike trigonometric graphs, sinh x and tanh x are not periodic. Their shapes reflect exponential growth and saturation, making them useful in modelling hanging cables, population growth and velocity with air resistance.

与三角函数图像不同，sinh x 和 tanh x 不是周期性的。它们的形状反映了指数增长和饱和效应，因此常用于模拟悬链线、人口增长以及考虑空气阻力时的速度。

12. Common Pitfalls in Exams | 考试常见易错点

Students often confuse the graph of cosh x with that of a quadratic. Remember that cosh x grows exponentially, so its arms rise much faster than a parabola.

学生常把 cosh x 的图像与二次函数图像混淆。请记住 cosh x 是指数增长的，因此其两臂上升得比抛物线快得多。

Another common mistake is forgetting that tanh x never reaches ±1. When sketching, always draw the asymptotes clearly and label them. Also, note that sinh 0 = 0 and cosh 0 = 1, not the reverse.

另一个常见错误是忘记 tanh x 永远不会达到 ±1。画图时，务必清楚地画出渐近线并标注。另外，注意 sinh 0 = 0，cosh 0 = 1，不要记反。

Finally, when solving equations like cosh x = 0, do not attempt to find real solutions — there are none. Always check the range of the function before writing solutions.

最后，在解如 cosh x = 0 的方程时，不要尝试寻找实数解——它们不存在。在写出解之前，务必检查函数的值域。

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Graphs of Hyperbolic Functions | 双曲函数的图像