Solve each problem. Selected values of the stopping distance...

Solve each problem. Selected values of the stopping distance $y$ in feet of a car traveling $x$ mph are given in the table. $$\begin{array}{|c|c|}\hline \begin{array}{c} \text { Speed } \\ \text { (in mph) } \end{array} & \begin{array}{c} \text { Stopping Distance } \\ \text { (in feet) } \end{array} \\ \hline 20 & 46 \\ 30 & 87 \\ 40 & 140 \\ 50 & 240 \\ 60 & 282 \\ 70 & 371 \\ \hline \end{array}$$ (a) Plot the data. (b) The quadratic function $$f(x)=0.056057 x^{2}+1.06657 x$$ is one model for the data. Find and interpret $f(45)$ (c) Graph the function in the same window as the data to determine how well $f$ models the stopping distance.

Solve each problem.
Selected values of the stopping distance $y$ in feet of a car traveling $x$ mph are given in the table.
$$\begin{array}{|c|c|}\hline
\begin{array}{c}
\text { Speed } \\
\text { (in mph) }
\end{array} & \begin{array}{c}
\text { Stopping Distance } \\
\text { (in feet) }
\end{array} \\
\hline 20 & 46 \\
30 & 87 \\
40 & 140 \\
50 & 240 \\
60 & 282 \\
70 & 371 \\ \hline
\end{array}$$
(a) Plot the data.
(b) The quadratic function
$$f(x)=0.056057 x^{2}+1.06657 x$$
is one model for the data. Find and interpret $f(45)$
(c) Graph the function in the same window as the data to determine how well $f$ models the stopping distance.

Key Concepts

Interpreting Model Outputs

Interpreting a model output involves understanding the significance of the numerical result within the context of the problem. It requires converting an abstract output from a mathematical function into meaningful information about the real-world scenario being modeled.

Function Evaluation

Function evaluation is the process of substituting a specific input into a function to determine the corresponding output. This process is essential in understanding how changes in the independent variable affect the outcome, and it helps in interpreting the behavior of the model under different conditions.

Graph Comparison of Models and Data

This concept involves overlaying the model's graph with the actual data points on the same coordinate system to visually assess the fit of the model. A good model will closely follow the trend of the data, while discrepancies might indicate regions where the model could be improved or where outliers exist.

Graphing Empirical Data

This concept involves plotting a set of observed data points on a coordinate plane so as to visually represent the relationship between two variables. Graphing data is crucial for understanding trends, patterns, or anomalies that might not be immediately evident from a table of numbers.

Quadratic Functions as Models

Quadratic functions are often used to model real-world phenomena that exhibit a parabolic behavior. In many cases, especially when the relationship between variables is non-linear, a quadratic model can approximate the underlying trend in the data, allowing for estimation and prediction.

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