The numbers an (in thousands) of AIDS cases reported from...

The numbers $a_{n}$ (in thousands) of AIDS cases reported from 2003 through 2010 can be approximated by $a_{n}=-0.0126 n^{3}+0.391 n^{2}-4.21 n+48.5$ $n=3,4, \ldots, 10$ where $n$ is the year, with $n=3$ corresponding to $2003 .$ (Source: U.S. Centers for Disease Control and Prevention) (a) Write the terms of this finite sequence. Use a graphing utility to construct a bar graph that represents the sequence. (b) What does the graph in part (a) say about reported cases of AIDS?

The numbers $a_{n}$ (in thousands) of AIDS cases reported
from 2003 through 2010 can be approximated by $a_{n}=-0.0126 n^{3}+0.391 n^{2}-4.21 n+48.5$
$n=3,4, \ldots, 10$
where $n$ is the
year, with $n=3$
corresponding to
$2003 .$ (Source:
U.S. Centers for
Disease Control
and Prevention)
(a) Write the terms of this finite sequence. Use a
graphing utility to construct a bar graph that
represents the sequence.
(b) What does the graph in part (a) say about reported
cases of AIDS?

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Key Concepts

Finite Sequence

A finite sequence is an ordered list of numbers that has a limited number of elements. In many real-world applications, finite sequences are used to represent discrete data points collected over a specific period or within a defined category, allowing for targeted analysis and comparison.

Polynomial Modeling

Polynomial functions are mathematical expressions composed of terms in which variables are raised to whole number exponents and multiplied by coefficients. They are often used in modeling data because their continuous and smooth nature can approximate trends and behaviors in various datasets, making them useful for prediction and analysis.

Bar Graph Representation

A bar graph is a type of chart that represents data with rectangular bars, where the length or height of each bar is proportional to the value it represents. Bar graphs are ideal for visualizing discrete data points from a finite sequence, enabling easy comparison between different categories or time intervals.

Data Trend Analysis

Data trend analysis involves examining and interpreting data over time or across categories to identify patterns, such as increases, decreases, or cyclic behavior. This process is critical in many fields, as it helps to translate raw data into insights that can inform decision-making and forecast future behavior.

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