The estimated revenues r (in millions of dollars) from sales...

The estimated revenues $r$ (in millions of dollars) from sales of in-vehicle technologies in the United States from 2003 through 2008 can be approximated by the model $$r=157.30 t^{2}-397.4 t+6114, \quad 3 \leq t \leq 8$$ where $t$ represents the year, with $t=3$ corresponding to $2003 .$ (Source: Consumer Electronics Association) (a) Use a graphing utility to graph the model. (b) Find the average rate of change of the model from 2003 through 2008 . Interpret your answer in the context of the problem.

The estimated revenues $r$ (in millions of dollars) from sales of in-vehicle technologies in the United States from 2003 through 2008 can be approximated by the model
$$r=157.30 t^{2}-397.4 t+6114, \quad 3 \leq t \leq 8$$
where $t$ represents the year, with $t=3$ corresponding to $2003 .$ (Source: Consumer Electronics Association)
(a) Use a graphing utility to graph the model.
(b) Find the average rate of change of the model from 2003 through 2008 . Interpret your answer in the context of the problem.

Key Concepts

Graphical Representation

Graphical representation involves using plotting tools or graphing utilities to visually display the behavior of functions. Plotting a function helps in understanding its overall trend, identifying key features such as turning points, and communicating the behavior of the model clearly, which is essential for interpreting real-world data.

Average Rate of Change

The average rate of change is a calculation that measures the change in the function's value relative to the change in the independent variable over a specific interval. It is computed as the difference in the function values at the end points divided by the difference in the independent variable values, essentially providing the slope of the secant line connecting these points.

Mathematical Modeling

Mathematical modeling involves formulating real-world phenomena as mathematical expressions or equations. In this context, the revenue over time is represented by a quadratic function, which allows for analysis and prediction of trends using mathematical methods.

Quadratic Functions

A quadratic function is a second-degree polynomial that typically takes the form ax² + bx + c. These functions model situations where the rate of change itself changes linearly. Their graphs are parabolic, and key features such as the vertex and axis of symmetry help characterize their behavior over a domain.

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