Question

35. Graph several members of the family of functions

\(f\left( x \right) = \frac{a}{2}\left( {{e^{x/a}} + {e^{ - x/a}}} \right)\)

where \(a > 0\). How does the graph change as \(a\) increases?

Short answer

The given function \(f\left( x \right)\) for \(a = 1\), 2 and 5 is shown below:

As \(a\) it increases, the corresponding graph is shifted upwards, and its width along the \(y\) axis is also increased.

Step-by-step solution

Analyze the given function

The given function is \(f\left( x \right) = \frac{a}{2}\left( {{e^{x/a}} + {e^{ - x/a}}} \right)\).

At \(x = 0\), the function is defined as \(f\left( 0 \right) = a\). So, the \(y\) intercept of this function is \(a\).

As the value of \(a\) increases, the value of \(y\) the intercept of this function will also increase.

Thus, it will be in an upward direction and vice-versa.

Find the function \(f\left( x \right)\) for different values of \(a\)

Let the values of \(a\) be 1, 2, and 5. Substitute these values for \(a\) into \(f\left( x \right)\) and simplify to get \(f\left( x \right)\) as \(f\left( x \right) = \frac{1}{2}\left( {{e^x} + {e^{ - x}}} \right)\), \(f\left( x \right) = {e^{x/2}} + {e^{ - x/2}}\) and \(\frac{5}{2}\left( {{e^{x/5}} + {e^{ - x/5}}} \right)\) respectively.

Draw the graph

Plot the functions\(f\left( x \right) = \frac{1}{2}\left( {{e^x} + {e^{ - x}}} \right)\), \(f\left( x \right) = {e^{x/2}} + {e^{ - x/2}}\) and \(\frac{5}{2}\left( {{e^{x/5}} + {e^{ - x/5}}} \right)\) on the same window, by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(0.5\left( {{e^x} + {e^{ - x}}} \right)\)in the\({Y_1}\)tab,\({e^{x/2}} + {e^{ - x/2}}\)in the\({Y_2}\)tab,\(2.5\left( {{e^{x/5}} + {e^{ - x/5}}} \right)\)in the\({Y_3}\)tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graph of function functions\(f\left( x \right) = \frac{1}{2}\left( {{e^x} + {e^{ - x}}} \right)\), \(f\left( x \right) = {e^{x/2}} + {e^{ - x/2}}\) and \(\frac{5}{2}\left( {{e^{x/5}} + {e^{ - x/5}}} \right)\) is shown below:

On observing the graph carefully, it is observed that as \(a\) it increases, the corresponding graph is shifted upwards, and its width along the \(y\) axis is also increased.