Question

Graph \(\quad\) (1) \(\mathrm{y}=\sin \mathrm{h} \mathrm{x} \quad\) (2) \(\mathrm{y}=\cosh \mathrm{x}\) (3) \(\mathrm{y}=\tan \mathrm{h} \mathrm{x}\)

Short answer

To graph the given hyperbolic functions, use their properties and plot key points: 1) For \(y = \sinh(x)\), the graph is symmetric with respect to the origin. Plot points for every multiple of π, compute their sinh values, and plot the points for negative x-values symmetrically. Connect the points with a smooth curve. 2) For \(y = \cosh(x)\), the graph is symmetric with respect to the y-axis. Plot points for every multiple of π, compute their cosh values, and plot the points for negative x-values symmetrically. Connect the points with a smooth curve. The graph has a minimum value of 1 when x = 0. 3) For \(y = \tanh(x)\), the graph is symmetric with respect to the origin and has horizontal asymptotes at y=-1 and y=1. Plot points for every multiple of π, compute their tanh values, and plot the points for negative x-values symmetrically. Connect the points with a smooth curve and draw horizontal dashed lines at y = -1 and y = 1, where the graph approaches but never reaches these lines.

Step-by-step solution

Understand the properties of the hyperbolic functions

The hyperbolic functions have some basic properties that are useful when graphing them: - sinh(x): The function is an odd function, meaning that sinh(-x) = -sinh(x). This property makes the graph of sinh(x) symmetric with respect to the origin. - cosh(x): The function is an even function, meaning that cosh(-x) = cosh(x). This property makes the graph of cosh(x) symmetric with respect to the y-axis. - tanh(x): The function is an odd function, meaning that tanh(-x) = -tanh(x). This property makes the tanh(x) graph symmetric with respect to the origin. Additionally, the function has horizontal asymptotes at y = -1 and y = 1.

Graph y = sinh(x)

The graph of y = sinh(x) can be drawn by plotting some key points on the x-axis and using the symmetry property of the function. - Start by plotting points on the x-axis for every multiple of π. For example, x = 0, π, 2π, 3π, and so on. - Compute the sinh values for these x-values and plot the corresponding points on the y-axis. - Recall that sinh(-x) = -sinh(x), so the graph is symmetric with respect to the origin. Plot the points for the negative x-values as well. - Connect the points with a smooth curve.

Graph y = cosh(x)

To draw the graph of y = cosh(x), again start by plotting some key points on the x-axis and use the symmetry property of the function. - Plot points on the x-axis for every multiple of π, e.g., x = 0, π, 2π, 3π, etc. - Compute the cosh values for these x-values and plot the corresponding points on the y-axis. - Recall that cosh(-x) = cosh(x), so the graph is symmetric with respect to the y-axis. Plot the points for the negative x-values as well, making sure that they are symmetrical to the positive x-values. - Connect the points with a smooth curve. Note that the graph has a minimum value of 1 when x = 0.

Graph y = tanh(x)

Finally, we will draw the graph of y = tanh(x) by plotting key points and using its properties. - Plot points on the x-axis for every multiple of π. - Compute the tanh values for these x-values and plot the corresponding points on the y-axis. - Recall that tanh(-x) = -tanh(x), so the graph is symmetric with respect to the origin. Plot the points for the negative x-values as well. - Connect the points with a smooth curve. - Remember that the function has horizontal asymptotes at y = -1 and y = 1, so draw two horizontal dashed lines at these y-values. The graph of the function will approach these lines but never reach them.

Key concepts

Properties of Hyperbolic Functions

Understanding the properties of hyperbolic functions is essential to accurately graph these mathematical expressions. Hyperbolic functions, which include sinh (hyperbolic sine), cosh (hyperbolic cosine), and tanh (hyperbolic tangent), exhibit unique behaviors that distinguish them from their trigonometric counterparts. For instance, sinh(x) and tanh(x) are odd functions, which exhibit symmetry about the origin. This means that if you take a point on the graph (x, y), the point (-x, -y) will also be on the graph. On the other hand, cosh(x) is an even function and shows symmetry about the y-axis, meaning points (x, y) and (-x, y) mirror each other.
Each hyperbolic function also exhibits characteristic shapes and features on their graphs. The sinh function displays unbounded growth in both directions, while the cosh function shows exponential growth in the positive direction and approaches a minimum at the origin. Conversely, the tanh function increases gradually to a horizontal asymptote, marking a limit that the curve approaches but never reaches. These aspects are crucial for plotting accurate graphs of hyperbolic functions.

Odd and Even Functions

When graphing functions, it is particularly useful to determine whether they are odd or even. Odd functions have rotational symmetry around the origin, essentially meaning that flipping the function 180 degrees around the origin leaves the graph unchanged. Mathematically, a function f(x) is odd if it satisfies the condition f(-x) = -f(x) for all x in its domain. In the context of our exercise, sinh(x) and tanh(x) are odd functions, so their graphs are symmetric with respect to the origin.

Even Functions

Even functions, on the contrary, have reflective symmetry across the y-axis. This is to say, folding the graph along the y-axis would show that both halves line up exactly. A function f(x) is even if f(-x) = f(x) for all x in the domain. The hyperbolic function cosh(x) is an even function, as its graph exhibits this kind of symmetry. Knowing whether a hyperbolic function is odd or even allows us to more efficiently plot the function, as we only need to calculate values for either the positive or the negative side of the x-axis, and then mirror those points accordingly.

Horizontal Asymptotes

Horizontal asymptotes are straight, horizontal lines that a graph approaches as the input, or x-values, move towards infinity in either direction. These lines represent values that the function will never actually reach no matter how large or small x becomes. In the graphing exercise for tanh(x), we note horizontal asymptotes at y = -1 and y = 1, indicating that as x goes to positive or negative infinity, the value of tanh(x) gets closer and closer to 1 and -1, respectively, without ever intersecting these lines.
Identifying horizontal asymptotes is crucial for graphing because it provides you with a boundary condition. It tells you the ultimate behavior of the function at the extremes of the domain. To illustrate these asymptotes, you usually draw dotted lines at the y-values where these asymptotes occur, which helps to visually reinforce the concept that the function's output will tend towards these values but not surpass them. This property is not just fundamental in graphing hyperbolic functions but also in understanding the limits and behavior of many other types of functions.