The table shows the retail market share of passenger cars...

The table shows the retail market share of passenger cars from Ford Motor Company as a percentage of the U.S. market. $$ \begin{array}{cc} \text { Year } & \text { Market Share } \\ 1980 & 17.2 \% \\ 1985 & 18.8 \% \\ 1990 & 20.0 \% \\ 1995 & 20.7 \% \\ 2000 & 20.2 \% \\ 2005 & 17.4 \% \\ \hline 2005 & 17.4 \% \\ \hline \end{array} $$ A mathematical model for this data is given by $$ f(x)=-0.0206 x^{2}+0.548 x+16.9 $$ where $x=0$ corresponds to 1980 . (A) Complete the following table. Round values of $f(x)$ to one decimal place. (B) Sketch the graph of $f$ and the market share data in the same coordinate system. (C) Use values of the modeling function $f$ to estimate Ford's market share in 2015 and in 2020 . (D) Write a brief verbal description of Ford's market share from 1980 to $2005 .$

The table shows the retail market share of passenger cars from Ford Motor Company as a percentage of the U.S. market.
$$
\begin{array}{cc}
\text { Year } & \text { Market Share } \\
1980 & 17.2 \% \\
1985 & 18.8 \% \\
1990 & 20.0 \% \\
1995 & 20.7 \% \\
2000 & 20.2 \% \\
2005 & 17.4 \% \\
\hline 2005 & 17.4 \% \\
\hline
\end{array}
$$
A mathematical model for this data is given by
$$
f(x)=-0.0206 x^{2}+0.548 x+16.9
$$
where $x=0$ corresponds to 1980 .
(A) Complete the following table. Round values of $f(x)$ to one decimal place.
(B) Sketch the graph of $f$ and the market share data in the same coordinate system.
(C) Use values of the modeling function $f$ to estimate Ford's market share in 2015 and in 2020 .
(D) Write a brief verbal description of Ford's market share from 1980 to $2005 .$

Key Concepts

Function Evaluation

Evaluating a function involves substituting given input values to calculate the corresponding outputs. This process is essential for predicting data points, filling in tables with estimated values, and verifying if the model accurately captures the trends of the observed data.

Trend Analysis

This involves analyzing the behavior of data over a period to understand patterns such as increases, decreases, or plateaus. In the context of market share, trend analysis assists in drawing conclusions about changes over time and in providing insights into market dynamics.

Graphing Functions and Data

Graphing involves plotting both the model (function) and the actual data points on the same coordinate system. This visual representation helps in comparing the fit of the model to the data, and in understanding the overall behavior and reliability of the model.

Mathematical Modeling

This concept involves creating functions that represent real-world data. In this context, a quadratic function is used to model the market share over time, allowing for predictions and analysis of trends. It provides a systematic way to understand and forecast behavior based on observed data.

Quadratic Functions

Quadratic functions are mathematical expressions of the form ax² + bx + c. They are used in modeling when data trends exhibit curvature, such as increasing then decreasing behavior, which helps to closely approximate real-life phenomena. Understanding their features helps in analyzing turning points and overall trend behavior.

Interpolation and Extrapolation

Interpolation is the process of estimating data points within the range of known data, while extrapolation involves predicting values outside that range. Using a mathematical model for extrapolation can enable predictions about future trends even when no data has been recorded yet.

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