Problem 86 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Use a graphing utility to graph \(y=2^{-x}, y=3^{-x}\), \(y=5^{-x}\), and \(y=(0.5)^{-x}\) on the same set of axes. How does a change in base affect the graph of the exponential function? (Suggestion: Use the graphing window \([-3,3] 1\) by \([-3,3] 1\).)

Step-by-Step Solution

Answer

Larger bases result in steeper declines in the graph. Smaller bases decay slower.

Step 1 - Understand the Function

Understand that you need to graph exponential functions with different bases: \(y=2^{-x}\), \(y=3^{-x}\), \(y=5^{-x}\), and \(y=(0.5)^{-x}\). Each function is of the form \(y=b^{-x}\).

Step 2 - Set Up the Graphing Window

Set up the graphing utility with the suggested window limits: \([-3,3] 1\) for the x-axis and \([-3,3] 1\) for the y-axis.

Step 3 - Input the Functions

Input the four functions into the graphing utility: \(y=2^{-x}\), \(y=3^{-x}\), \(y=5^{-x}\), and \(y=(0.5)^{-x}\).

Step 4 - Graph Each Function

Graph each function on the same set of axes. Observe how each curve behaves.

Step 5 - Analyze the Graphs

Notice that all functions \(y=b^{-x}\) are decreasing since the base is greater than 1 for 2, 3, and 5, and less than 1 for 0.5. Larger bases result in graphs that decay faster.

Step 6 - Conclusion

The change in base affects how quickly the exponential function decreases. A larger base results in a steeper decline, whereas a smaller base results in a slower decline.

Key Concepts

graphing utility

A graphing utility is an invaluable tool for visualizing mathematical functions.
With a graphing utility, you can quickly and accurately plot functions and observe their behaviors.
In this exercise, you need to graph four exponential functions: \(y=2^{-x}\), \(y=3^{-x}\), \(y=5^{-x}\), and \(y=(0.5)^{-x}\).
Each function is of the form \(y=b^{-x}\), where ‘b’ is the base.
When using a graphing utility, it's essential to set the correct window limits to get a clear view of the graph.
For this exercise, set your window to \([-3, 3] 1\) for both x and y-axes.
This setting means that both the x and y values will range from -3 to 3, which will provide a good view to compare the graphs.
Once you input the functions into the graphing utility and plot them, you'll see how they behave on the same set of axes.

base change effect

Changing the base in an exponential function has a pronounced effect on the shape of its graph.
In this exercise, you are asked to compare the graphs of four functions with different bases: 2, 3, 5, and 0.5.
Notice how the base affects the steepness of the decline.
For example, the function \(y=5^{-x}\) decreases much more rapidly than \(y=2^{-x}\) because the base 5 is larger, causing the function to decay faster.
Conversely, when the base is less than 1, like in \(y=(0.5)^{-x}\), the function decreases more slowly.
This is because raising a fraction to a negative exponent effectively increases the function value.
Observing these differences graphically allows you to understand how exponential functions expand and decay based on their bases.

function behavior analysis

Performing a behavior analysis on exponential functions involves observing how they change under different conditions.
In this case, the key observation is how these functions behave as they decrease.
All the functions \(y=2^{-x}\), \(y=3^{-x}\), \(y=5^{-x}\), and \(y=(0.5)^{-x}\) are decreasing functions.
This is because the exponent in each function is negative.
However, the rate of decrease – or how steeply they decline – varies.
With a larger base, the function decreases faster.
For instance, \(y=5^{-x}\) decreases very sharply compared to \(y=2^{-x}\).
Conversely, \(y=(0.5)^{-x}\) decreases very slowly because its base is less than 1.
Understanding these behaviors is crucial for interpreting and predicting how exponential functions behave in different scenarios.