Question

Sketch the graphs of $y=\cosh x,y=\sinh x,$ and $y=\tanh x$ (include asymptotes), and state whether each function is even, odd, or neither.

Short answer

Answer: The properties of the hyperbolic functions are as follows: - $y=\cosh x$ is an even function, and its values are always positive. - $y=\sinh x$ is an odd function, crosses the origin, and its values increase asymptotically. - $y=\tanh x$ is an odd function, crosses the origin, and its values have horizontal asymptotes at $y=1$ and $y=-1$.

Step-by-step solution

Sketch the graph of $y=\cosh x$

To sketch the graph of $y=\cosh x$, we can use the definition formula: $\cosh x=\frac{e^x+e^{-x}}{2}$. We notice that for all values of $x$, both $e^x$ and $e^{-x}$ are strictly positive, and their sum will also be positive. Therefore, the $\cosh x$ function is always positive. Let's plot the function using some points: $x=-2,y=\cosh (-2)\approx 3.76$ $x=-1,y=\cosh (-1)\approx 1.54$ $x=0,y=\cosh (0)=1$ $x=1,y=\cosh (1)\approx 1.54$ $x=2,y=\cosh (2)\approx 3.76$ Now, check if $\cosh x$ is even, odd, or neither. We substitute $-x$ for $x$ and see if we get $\cosh x$ or $-\cosh x$: $\cosh (-x)=\frac{e^{-x}+e^x}{2}=\cosh x$, which means $\cosh x$ is an even function.

Sketch the graph of $y=\sinh x$

To sketch the graph of $y=\sinh x$, we can use the definition formula: $\sinh x=\frac{e^x-e^{-x}}{2}$. Let's plot the function using some points: $x=-2,y=\sinh (-2)\approx -3.63$ $x=-1,y=\sinh (-1)\approx -1.18$ $x=0,y=\sinh (0)=0$ $x=1,y=\sinh (1)\approx 1.18$ $x=2,y=\sinh (2)\approx 3.63$ Now, check if $\sinh x$ is even, odd, or neither. We substitute $-x$ for $x$ and see if we get $\sinh x$ or $-\sinh x$: $\sinh (-x)=\frac{e^{-x}-e^x}{2}=-\sinh x$, which means $\sinh x$ is an odd function.

Sketch the graph of $y=\tanh x$

To sketch the graph of $y=\tanh x$, we can use the definition formula: $\tanh x=\frac{\sinh x}{\cosh x}$. Let's plot the function using some points: $x=-2,y=\tanh (-2)\approx -0.96$ $x=-1,y=\tanh (-1)\approx -0.76$ $x=0,y=\tanh (0)=0$ $x=1,y=\tanh (1)\approx 0.76$ $x=2,y=\tanh (2)\approx 0.96$ Notice that as $x$ increases, $\tanh x$ approaches 1, and as $x$ decreases, $\tanh x$ approaches -1. Therefore, the asymptotes are $y=1$ (upper) and $y=-1$ (lower). Check if $\tanh x$ is even, odd, or neither. We substitute $-x$ for $x$ and see if we get $\tanh x$ or $-\tanh x$: $\tanh (-x)=\frac{\sinh (-x)}{\cosh (-x)}=-\tanh x$, which means $\tanh x$ is an odd function. In summary, we have the following properties for each function: • $\cosh x$: even function, always positive • $\sinh x$: odd function, crosses the origin, asymptotically increasing • $\tanh x$: odd function, crosses the origin, asymptotes at $y=1$ and $y=-1$

Key concepts

cosh x

The hyperbolic cosine function, denoted as $\cosh x$, is a fundamental component in the study of hyperbolic functions. It is defined mathematically by the formula $\cosh x=\frac{e^x+e^{-x}}{2}$.

Characterized by its even symmetry, the graph of $\cosh x$ is shaped like a symmetric arch that stretches into infinity on the y-axis as $x$ moves further away from zero. This shape is a reflection of the fact that $\cosh x$ is an even function and produces the same value for $x$ and $-x$. Additionally, the graph never touches or crosses the x-axis, indicating it has no real roots and that the function is always positive.\

In practical terms, understanding the $\cosh x$ function is crucial for various applications, such as in the field of physics where it describes the shape of a hanging cable, known as a catenary.

sinh x

The hyperbolic sine function, represented as $\sinh x$, is equally important and is defined by the equation $\sinh x=\frac{e^x-e^{-x}}{2}$. Unlike $\cosh x$, the $\sinh x$ function features an odd symmetry, meaning that it is symmetrical with respect to the origin and that $\sinh (-x)=-\sinh (x)$.

The graph of the $\sinh x$ function exhibits a distinct 'S' shape which passes through the origin (0,0), highlighting that it is an odd function. It shows exponential growth in the positive direction and exponential decay in the negative direction. Students must note that the $\sinh x$ function does not have any asymptotes, as it continues to increase or decrease without approaching any fixed lines.

The $\sinh x$ function's behavior makes it relevant in fields like engineering where it can describe certain types of wave phenomena.

tanh x

The hyperbolic tangent function, denoted by $\tanh x$, is another core hyperbolic function given by the quotient $\tanh x=\frac{\sinh x}{\cosh x}$. Similar to $\sinh x$ in terms of symmetry, $\tanh x$ is an odd function with $\tanh (-x)=-\tanh (x)$.

The distinctive feature of the $\tanh x$ graph is its approach to the horizontal asymptotes at $y=1$ and $y=-1$ as $x$ approaches positive and negative infinity, respectively. This behavior indicates that the output of the $\tanh x$ function is bounded, never exceeding these horizontal lines, which is a useful property when modeling systems where growth is limited.

Students should recognize the practicality of the $\tanh x$ graph in real-life applications such as electrical engineering, where it can illustrate the voltage-current relationship for certain types of diodes.

graph sketching

Graph sketching is a crucial technique for visualizing the behavior of functions. It allows one to quickly assess important characteristics such as intercepts, symmetry, and asymptotic behavior.

For hyperbolic functions, sketching the graph involves plotting several key points and observing the symmetry to complete the shape of the curve. An even function like $\cosh x$ will be mirror-imaged across the y-axis, while an odd function like $\sinh x$ and $\tanh x$ will have rotational symmetry around the origin.

When sketching hyperbolic function graphs, students should start by identifying points where the function crosses the axes and look for any asymptotic behavior, as seen in $\tanh x$, which can guide the drawing of the curve. By connecting critical points and following the asymptotes, students can accurately sketch the functions, gaining a visual understanding of their properties.

even and odd functions

The concepts of even and odd functions are fundamental in understanding symmetry in mathematical functions. An even function is one that satisfies $f(x)=f(-x)$, meaning the graph of the function is symmetrical about the y-axis. Conversely, an odd function fulfills the condition $f(-x)=-f(x)$ and shows symmetry with respect to the origin.

For hyperbolic functions, $\cosh x$ is an even function, mirroring along the y-axis, and both $\sinh x$ and $\tanh x$ are odd functions, exhibiting rotational symmetry. Recognizing these properties not only aids in graph sketching but also has implications in advanced mathematics and physics, such as in Fourier series where functions are expressed as sums of sines and cosines, and in signal processing.

asymptotes

Asymptotes are lines that a graph approaches but never actually reaches. They can be vertical, horizontal, or slanted and are essential for understanding the long-term behavior of functions. For hyperbolic functions like $\tanh x$, the horizontal asymptotes at $y=1$ and $y=-1$ signify that the function values approach these lines but do not cross them as $x$ gets large in magnitude.

No asymptotes exist for $\cosh x$ and $\sinh x$ as these functions continue to increase or decrease without bound. Recognizing asymptotes is particularly important when dealing with limits and infinity in calculus. It represents the behavior of a function outside the constraints of the graphing window, and thus, becomes a critical concept when studying the end behavior of functions.