The table shows the numbers of cellular phone subscribers...

The table shows the numbers of cellular phone subscribers (in millions) in the United States for selected years. $$\begin{array}{|l|c|c|c|c|c|c|}\hline \text { Year } & 1995 & 1998 & 2001 & 2004 & 2007 & 2010 \\\hline \text { Number } & 34 & 69 & 128 &182 & 255 & 303 \\\hline\end{array}$$ (a) Use the regression capabilities of a graphing utility to find a mathematical model of the form $y=a t^{2}+b t+c$ for the data. In the model, $y$ represents the number of subscribers (in millions) and $t$ represents the year, with $t=5$ corresponding to 1995 (b) Use a graphing utility to plot the data and graph the model. Compare the data with the model. (c) Use the model to predict the number of cellular phone subscribers in the United States in the year $2020 .$ PICTURE CANT COPY

The table shows the numbers of cellular phone subscribers (in millions) in the United States for selected years.
$$\begin{array}{|l|c|c|c|c|c|c|}\hline \text { Year } & 1995 & 1998 & 2001 & 2004 & 2007 & 2010 \\\hline \text { Number } & 34 & 69 & 128 &182 & 255 & 303 \\\hline\end{array}$$
(a) Use the regression capabilities of a graphing utility to find a mathematical model of the form $y=a t^{2}+b t+c$ for the data. In the model, $y$ represents the number of subscribers (in millions) and $t$ represents the year, with $t=5$ corresponding to 1995
(b) Use a graphing utility to plot the data and graph the model. Compare the data with the
model.
(c) Use the model to predict the number of cellular phone subscribers in the United States in the year $2020 .$
PICTURE CANT COPY

Key Concepts

Quadratic Regression

This is a statistical method used to fit a quadratic model (a polynomial of degree 2) to a set of data points. It involves finding the coefficients of the quadratic equation y = at² + bt + c that best describe the relationship between the independent and dependent variables, typically by minimizing the sum of the squared errors between the observed and predicted values.

Mathematical Modeling

Mathematical modeling is the process of representing real-world phenomena using mathematical constructs. In this context, it involves using a quadratic equation to capture the trend in the number of cellular phone subscribers over time, allowing for both analysis of past behavior and prediction of future outcomes.

Data Visualization

Data visualization involves graphing both the actual data points and the mathematical model to provide a visual representation of how well the model fits the data. This step is crucial in assessing the accuracy of the model and identifying any discrepancies between the model's predictions and the observed values.

Prediction and Extrapolation

Once a mathematical model is established, it can be used to predict future values outside the range of the original data. Extrapolation involves extending the model to make forecasts about future events, such as estimating the number of subscribers in a year not included in the data set.

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