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

Key Concepts

Data Visualization

Data visualization involves graphing discrete data points to observe trends, patterns, or relationships in the data. In this problem, plotting the stopping distances versus speeds allows for a visual interpretation of how increasing speed correlates with increasing stopping distance, and it provides a basis for comparing actual data points with the predictions of a mathematical model.

Quadratic Function Modeling

Quadratic function modeling uses a polynomial of degree 2 to represent relationships between variables. This type of model is particularly useful when the relationship between the independent and dependent variables is not linear, as is often the case in real-world phenomena like stopping distances, which may increase at an accelerating rate with speed.

Function Evaluation and Interpretation

Function evaluation is the process of substituting a specific value into a function to obtain an output, which in this context means plugging in a given speed into the quadratic model to predict the stopping distance. The interpretation of this value helps in understanding and explaining the behavior of the modeled scenario, such as estimating the stopping distance at an intermediate speed.

Graphical Analysis for Model Fit

Graphical analysis for model fit involves overlaying the graph of a mathematical model onto the plotted data points to assess how well the model captures the underlying trend. This comparison provides visual insight into the accuracy and appropriateness of the model, highlighting areas where the model may either overestimate or underestimate the actual values.

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