Problem 84 - Precalculus

Question

Plot the functions \(y=2^{x}, y=e^{x},\) and \(y=3^{x}\) in the same viewing screen. Explain why \(y=e^{x}\) lies between the other two graphs.

Step-by-Step Solution

Answer

The graph of \(y=e^{x}\) is between \(y=2^{x}\) and \(y=3^{x}\) because \(e\) is between 2 and 3.

Step 1: Identify the Functions

We must analyze and compare the three exponential functions: \(y=2^{x}\), \(y=e^{x}\), and \(y=3^{x}\). Each function represents a different exponential base. Here, \(e\) is approximately 2.718, which is greater than 2 but less than 3.

Step 2: Plot the Function Graphs

Plot the functions on the same graph using a graphing tool or calculator. Make sure to mark each curve clearly for \(y=2^{x}\), \(y=e^{x}\), and \(y=3^{x}\). Observe how the graphs behave as \(x\) increases or decreases.

Step 3: Analyze the Graphs

Notice that as \(x\) increases, \(y=2^{x}\) grows slower than \(y=e^{x}\), and \(y=e^{x}\) grows slower than \(y=3^{x}\). This results from the hierarchy of the bases: 2 is less than \(e\), which is less than 3. Hence, \(y=e^{x}\) is between \(y=2^{x}\) and \(y=3^{x}\).

Step 4: Conclusion

Since the base of \(y=e^{x}\) is between the bases of the other two functions, its graph naturally lies between \(y=2^{x}\) and \(y=3^{x}\). This pattern holds true for all positive values of \(x\) and is an implicit property of exponential functions with different bases.

Key Concepts

Function Graphs

Understanding function graphs is essential when studying exponential functions. Exponential function graphs have a few unique properties that make them interesting to analyze. For an exponential function like \(y=a^{x}\), the graph shows rapid growth (or decay) based on the value of \(a\).
The base \(a\) dictates the 'steepness' of the graph. If \(a > 1\), the function represents exponential growth, and the graph rises smoothly with increasing \(x\). Conversely, if \(0 < a < 1\), it indicates exponential decay, and the graph falls as \(x\) increases.

• The graph of \(y=2^x\) starts lower compared to \(y=3^x\) because 2 is smaller than 3.
• \(y=e^x\) holds a middle position since \(e\) (approximately 2.718) lies between 2 and 3.

The graphs generally pass through the point (0,1), since any non-zero number raised to the power of 0 will always equal 1. This creates a unique snapshot of exponential growth across different bases.

Exponential Growth

Exponential growth occurs when the rate of growth of a function is proportional to its current value. The classic form is expressed as \(y = a^x\), where \(a > 1\). As \(x\) increases, \(y\) grows more rapidly, characterizing the explosive nature of exponential growth.
With the functions \(y=2^x\), \(y=e^x\), and \(y=3^x\), all exhibit exponential growth, however, the speed or steepness of this growth varies:

• \(y=2^x\) grows steadily, but at a slower pace compared to the others.
• \(y=e^x\) grows faster than \(y=2^x\) due to its larger base.
• \(y=3^x\) shows the most rapid growth, reflecting the effect of the largest base.

This pattern highlights an essential aspect of exponential functions: small changes in the base can lead to substantial differences in growth behavior.

Base Comparisons

When comparing different exponential functions, the base plays a crucial role in determining their relative growth rates. The base of an exponential function such as \(y=a^x\) influences how quickly the function's value increases or decreases.

• In our comparison, \(y=2^x\) has the smallest base, resulting in slower growth.
• \(y=e^x\), with a base of approximately 2.718, grows faster than \(y=2^x\) but slower than \(y=3^x\).
• \(y=3^x\) boasts the largest base and therefore grows the fastest of the three functions.

These base comparisons help explain why \(y=e^x\) lies between \(y=2^x\) and \(y=3^x\) on the graph.
Exponential functions with larger bases will always grow more rapidly than those with smaller bases, given the same exponent \(x\). This is an intrinsic property of exponential behavior.