Question

How does the graph of the catenary \(y=a \cosh \frac{x}{a}\) change as \(a>0\) increases?

Short answer

As the value of \(a > 0\) increases in the catenary equation \(y = a\cosh\frac{x}{a}\), the graph shifts upwards, becomes more spread out horizontally, and is less steep near the center. These changes occur as the minimum point shifts up, the curve takes longer to approach its vertical asymptotes, and the steepness decreases near the origin.

Step-by-step solution

1. Understanding the Catenary Equation

The given catenary equation is \(y=a \cosh \frac{x}{a}\), where \(a\) is a positive constant. The hyperbolic cosine function can be defined as \(\cosh{x} = \frac{e^x + e^{-x}}{2}\), which leads to positive and non-negative values as its output. So, the catenary reaches its minimum value when \(x=0\), and it grows asymptotically as \(x\) moves towards positive and negative infinity.

2. Observe the Graph at x=0

At \(x=0\), we can observe the behavior of the graph. Plugging in \(x=0\) into the equation, we get $$ y = a \cosh \frac{0}{a} = a \cosh {0} = a \cdot 1 = a. $$ Regardless of the value of \(a\), the graph of the catenary will always pass through the point \((0,a)\). However, as the value of \(a\) increases, the graph will simply shift upward.

3. Observe the Graph for Large x

As \(x\) approaches infinity, \(\cosh \frac{x}{a}\) will grow exponentially. As a result, the graph will become more vertical. To see this, consider the limit $$ \lim_{x \to \infty} a \cosh \frac{x}{a}. $$ No matter the value of \(a\), this limit will be infinity. However, the larger the value of \(a\), the slower the graph will approach the vertical asymptotes at \(x=\pm \infty\). This means that for larger values of \(a\), the graph will be more spread out and less steep near the center.

4. Analyzing the Changes in the Graph

As \(a\) increases, the following changes occur in the graph of the catenary: - The minimum point of the graph shifts upward, since the graph passes through \((0,a)\). - The graph becomes less steep near the center and more spread out horizontally. - The graph still becomes asymptotically vertical as \(x\) approaches infinity, but it takes longer for larger values of \(a\). Therefore, the graph of the catenary \(y = a \cosh \frac{x}{a}\) changes as \(a>0\) increases by shifting upwards, spreading out horizontally, and becoming less steep.

Key concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change (differential calculus) and accumulation of quantities (integral calculus). It allows us to study the properties of functions and their graphs in detail.

When examining the catenary graph, calculus is a critical tool in understanding its behavior. For instance, the first derivative of the catenary function tells us about the slope of the graph at any given point, indicating how steep the curve is. In the catenary graph, we notice that as the value of 'a' increases, the slope of the curve becomes less steep around the origin.

Furthermore, second derivatives are used in calculus to analyze concavity and points of inflection. In the case of the catenary, the second derivative would remain positive, implying the graph is always concave upwards, resembling a hanging chain. Calculus, therefore, helps us comprehend how the shape of the catenary graph changes with different values of 'a'.

Hyperbolic Functions

Hyperbolic functions, which include hyperbolic sine (sinh) and hyperbolic cosine (cosh), are analogs of the trigonometric functions but for a hyperbola rather than a circle. To understand the catenary graph, one needs a firm grasp of these functions.

Hyperbolic cosine, defined as \(\cosh{x} = \frac{e^x + e^{-x}}{2}\), determines the shape of the catenary curve. As 'a' increases in the catenary equation \(y=a \cosh \frac{x}{a}\), the hyperbolic cosine function dictates that the curve becomes less steep. This function is always positive, leading to the catenary having no real asymptotes and being reflected symmetrically across the y-axis.

Understanding hyperbolic functions is essential to analyzing the nature of the graph and its response to changes in the parameter 'a'. These functions have unique properties, such as their derivatives and integrals, playing a pivotal role in the analysis of the shape and behavior of the catenary curve.

Asymptotes

An asymptote is a line that a curve approaches as it heads towards infinity. Studying asymptotes is part of understanding the end behavior of a function's graph. While the catenary curve \(y=a \cosh \frac{x}{a}\) does not possess real asymptotes due to the nature of the hyperbolic cosine function, its behavior at extreme values of 'x' is still worth exploring.

The catenary graph grows exponentially as 'x' moves towards positive and negative infinity, resembling vertical asymptotes at \(x=\pm \infty\). The speed at which the catenary approaches these invisible vertical bounds is related to the value of 'a': a larger 'a' means a slower approach to the vertical, indicating a more horizontal spread of the curve near the origin.

Understanding asymptotic behavior is crucial for grasp\ing the overall shape of the catenary curve and predicting how changes in 'a' will affect the curve's steepness and spread. This concept helps students visualize the extent of the graph even where it stretches beyond practical plotting.