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Chapter 1: Problem 91

Use a graphing utility to graph each function. Write a paragraph describing any similarities and differences you observe among the graphs. (a) \(y=x\) (b) \(y=x^{2}\) (c) \(y=x^{3}\) (d) \(y=x^{4}\) (e) \(y=x^{5}\) (f) \(y=x^{6}\)

Short Answer

Expert verified
The graphs of these functions are all divergent, where the graph of \(y=x\) is a line while the remaining are curves, which are symmetrical either around the y-axis or about the origin. The curves become steeper as the degree of the function increases, indicating a higher rate of change for higher exponents.

Step by step solution

01

Graphing

Use a graphing utility to graph each function, which are \(y=x\), \(y=x^{2}\), \(y=x^{3}\), \(y=x^{4}\), \(y=x^{5}\), and \(y=x^{6}\). Make sure all graphs are clearly labelled.
02

Analyzing the Graphs

Look at each graph individually and then together. Take note of key characteristics including the shape, slopes, symmetry, and roots. Compare these characteristics across all the graphs.
03

Describing Similarities

All functions pass through the origin (0,0) and are continuous for all real numbers. Functions with even degree (e.g. \(x^{2}\), \(x^{4}\), \(x^{6}\)) are symmetrical across the y-axis while functions with odd degree (e.g. \(x\), \(x^{3}\), \(x^{5}\)) are symmetrical with respect to the origin.
04

Describing Differences

The main differences between the functions are in their steepness and shape. The functions \(y=x\), \(y=x^{3}\), and \(y=x^{5}\) are linear, cubic and quintic and have a slope that changes from negative to positive as x increases, while the functions \(y=x^{2}\), \(y=x^{4}\), and \(y=x^{6}\) are quadratic, biquadratic and hexic and their slopes increase smoothly but always remain positive.

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