Question

A flexible cable always hangs in the shape of a catenary \(y = a\cosh \left( {\frac{x}{a}} \right)\), where \(c\) and \(a\) are constants and \(a > {\bf{0}}\) (see Figure 4 and Exercise 60). Graph several members of the family of functions \(y = a\cosh \left( {\frac{x}{a}} \right)\). How does the graph change as \(a\) varies?

Short answer

As the height of the rope at its lowest point is \(a\). As the value of \(a\) increases the height of rope at its lowest point also increases.

Step-by-step solution

Construct the graph of \(y = a\cosh \left( {\frac{x}{a}} \right)\) for the different values of \(a\)

Assume that \(a = 1,2,3\;{\rm{and}}\;4\).

Draw the graph of the function\(y = 1\cosh \left( {\frac{x}{1}} \right)\),\(y = 2\cosh \left( {\frac{x}{2}} \right)\), \(y = 3\cosh \left( {\frac{x}{3}} \right)\) and \(y = 4\cosh \left( {\frac{x}{4}} \right)\) by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(1\cosh \left( {\frac{x}{1}} \right)\)in the\({Y_1}\)tab.
2. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(2\cosh \left( {\frac{x}{2}} \right)\)in the\({Y_2}\)tab.
3. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(3\cosh \left( {\frac{x}{3}} \right)\)in the\({Y_3}\)tab.
4. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(4\cosh \left( {\frac{x}{4}} \right)\)in the\({Y_4}\)tab.
5. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graph of the function \(y = 1\cosh \left( {\frac{x}{1}} \right)\),\(y = 2\cosh \left( {\frac{x}{2}} \right)\), \(y = 3\cosh \left( {\frac{x}{3}} \right)\) and \(y = 4\cosh \left( {\frac{x}{4}} \right)\) is shown below:

Find the changes in graph

Consider the function,\(y = a\cosh \left( {\frac{x}{a}} \right)\).Since the minimum value of\(\cosh \left( {\frac{x}{a}} \right)\)is \(1\).

As the height of the rope at its lowest point is \(a\). As the value of \(a\) increases the height of rope at its lowest point also increases.