Problem 87 Use a graphing utility to graph ... [FREE SOLUTION]

Chapter 3: Problem 87

Use a graphing utility to graph each pair of functions in the same viewing window. Describe any similarities and differences in the graphs. (a) \(y_{1}=2^{x}, y_{2}=x^{2}\) (b) \(y_{1}=3^{x}, y_{2}=x^{3}\)

Short Answer

Expert verified

Both pairs share the similarity of increasing graphs. For the first pair, \(2^{x}\) is always positive and increases more steeply than \(x^{2}\), which also takes on negative values for negative x. Similarly, for the second pair, \(3^{x}\) is also always positive and increases, albeit less steeply than \(x^{3}\), which takes on both positive and negative values based on the value of x.

Step by step solution

Graph the functions

Using a graphing utility, input each pair of functions and graph them. This produces two graphs for each pair: one for \(y_{1}=2^{x}\) and \(y_{2}=x^{2}\), and another for \(y_{1}=3^{x}\) and \(y_{2}=x^{3}\).

Observe the behavior of the graphs

Once the graphs are plotted, observe their behavior. For the first pair, the graphs of \(x^{2}\) starts at the origin and goes up in both directions while \(2^{x}\) never gets below the x-axis and increases more steeply than \(x^{2}\) as x increases. For the second pair, the graph of \(x^{3}\) passes through the origin, and increases more rapidly for positive x-values, and decreases more rapidly for negative x-values. \(3^{x}\) also never gets below the x-axis and is increasing, but less steeply than \(x^{3}\) for large positive values of x.

Describe the differences and similarities between the graphs

Compare the behaviors of both graphs in each pair. The main similarity in each pair is that the graphs are increasing in the positive direction of x. The differences lie in their growth rates and the fact that the exponential functions are always positive, while the polynomial functions can take positive and negative values.

Conclude the analysis

Based on these observations, we can conclude that while both function types increase as x increases, exponential functions grow at a faster rate. Additionally, exponential functions are always positive, while polynomial functions can take on negative values for negative x-values.

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91影视

Use a graphing utility to graph each pair of functions in the same viewing window. Describe any similarities and differences in the graphs. (a) \(y_{1}=2^{x}, y_{2}=x^{2}\) (b) \(y_{1}=3^{x}, y_{2}=x^{3}\)

Short Answer

Step by step solution

Graph the functions

Observe the behavior of the graphs

Describe the differences and similarities between the graphs

Conclude the analysis

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