Gas mileage is tested for a car under different driving...

Gas mileage is tested for a car under different driving conditions. At lower speeds, the car is driven in stop and go traffic. At higher speeds, the car must overcome more wind resistance. The variable $x$ given in the table represents the speed (in mph) for a compact car, and $m(x)$ represents the gas mileage (in $\mathrm{mpg}$ ). $$\begin{array}{|c|c|c|c|c|c|} \hline \boldsymbol{x} & 25 & 30 & 35 & 40 & 45 \\ \hline \boldsymbol{m}(\boldsymbol{x}) & 22.7 & 25.1 & 27.9 & 30.8 & 31.9 \\ \hline \end{array}$$ $$\begin{array}{|c|c|c|c|c|} \hline \boldsymbol{x} & 50 & 55 & 60 & 65 \\ \hline \boldsymbol{m}(\boldsymbol{x}) & 30.9 & 28.4 & 24.2 & 21.9 \\ \hline \end{array}$$ a. Use regression to find a quadratic function to model the data. b. At what speed is the gas mileage the greatest? Round to the nearest mile per hour. c. What is the maximum gas mileage? Round to the nearest mile per gallon.

Gas mileage is tested for a car under different driving conditions. At lower speeds, the car is driven in stop and go traffic. At higher speeds, the car must overcome more wind resistance. The variable $x$ given in the table represents the speed (in mph) for a compact car, and $m(x)$ represents the gas mileage (in $\mathrm{mpg}$ ).
$$\begin{array}{|c|c|c|c|c|c|}
\hline \boldsymbol{x} & 25 & 30 & 35 & 40 & 45 \\
\hline \boldsymbol{m}(\boldsymbol{x}) & 22.7 & 25.1 & 27.9 & 30.8 & 31.9 \\
\hline
\end{array}$$
$$\begin{array}{|c|c|c|c|c|}
\hline \boldsymbol{x} & 50 & 55 & 60 & 65 \\
\hline \boldsymbol{m}(\boldsymbol{x}) & 30.9 & 28.4 & 24.2 & 21.9 \\
\hline
\end{array}$$
a. Use regression to find a quadratic function to model the data.
b. At what speed is the gas mileage the greatest? Round to the nearest mile per hour.
c. What is the maximum gas mileage? Round to the nearest mile per gallon.

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Key Concepts

Regression Analysis

Regression analysis involves using statistical techniques to model and approximate the relationship between two or more variables by fitting a curve to the data. In this context, quadratic regression is used to identify a quadratic function that best represents the underlying relationship between the independent variable (speed) and the dependent variable (mileage) based on the collected data.

Quadratic Functions

A quadratic function is a second-degree polynomial typically expressed in the form ax² + bx + c. These functions exhibit parabolic behavior, meaning their graphs are U-shaped or inverted U-shaped depending on the sign of the coefficient a. They are commonly used in modeling scenarios where the relationship between variables shows a turning point or optimum.

Vertex of a Parabola

The vertex of a parabola is the point where the function reaches its maximum or minimum value, depending on whether the parabola opens downward or upward. When modeling real-world scenarios with quadratic functions, finding the vertex is essential for determining the optimal value of the dependent variable (in this case, the highest gas mileage) and the corresponding independent variable value (the speed at which that optimum occurs).

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