The percent pn of United States adults who met federal...

The percent $p_{n}$ of United States adults who met federal physical activity guidelines from 2007 through 2014 can be approximated by $$p_{n}=0.0061 n^{3}-0.419 n^{2}+7.85 n+4.9$ $n=7,8, \dots14$$ where $n$ is the year, with $n=7$ corresponding to $2007 .$ (Source: National Center for Health Statistics) (a) Write the terms of this finite sequence. Use a graphing utility to construct a bar graph that represents the sequence. (b) What can you conclude from the bar graph in part (a)?

The percent $p_{n}$ of United States adults who met federal physical activity guidelines from 2007 through 2014 can be approximated by $$p_{n}=0.0061 n^{3}-0.419 n^{2}+7.85 n+4.9$
$n=7,8, \dots14$$
where $n$ is the year, with $n=7$ corresponding to $2007 .$ (Source:
National Center for Health Statistics)
(a) Write the terms
of this finite sequence. Use a graphing utility to construct a bar graph that represents the sequence.
(b) What can you conclude from the bar graph in part (a)?

Key Concepts

Trend Analysis

Trend analysis involves examining the pattern or direction in which data points are moving over time. When applied to sequences depicted in graphs, it allows one to identify increases, decreases, or irregular patterns, thereby facilitating an understanding of the underlying behavior of the system being studied.

Bar Graph Representation

A bar graph is a visual tool that represents data with rectangular bars, where each bar’s length corresponds to the value of the data in its category. This type of graph is especially useful for comparing the magnitudes of finite sequence terms side by side, allowing for clear visual comparison.

Graphing Utility

A graphing utility refers to software or a calculator that is capable of plotting equations and sequences. It is essential for visualizing data, as it helps in quickly generating graphs that illustrate the behavior of mathematical functions, making it easier to analyze and interpret trends.

Polynomial Function

A polynomial function is composed of variables raised to various nonnegative integer powers with corresponding coefficients. In many real-world applications, polynomials are used to model smooth changes and trends across data sets, providing a framework to approximate complex relationships.

Finite Sequence

A finite sequence is an ordered list of numbers that has a fixed number of terms. In mathematical modeling, sequences are often generated by applying a rule or formula to a set of indices, which helps in understanding discrete behaviors over a limited interval.

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