The table below shows the results from a study that compared speed (in miles per hour) and average fuel economy (in miler per gallon) for cars.  Find a quadratic model for the data.

Speed (mph)	Fuel Economy (mpg)
15	22.3
20	25.5
25	27.5
30	29.0
35	28.8
40	30
45	29.9
50	30.2
55	30.4
60	28.8
65	27.4
70	25.3

The table below shows the results from a study that compared speed (in miles per hour) and average fuel economy (in miler per gallon) for cars.  Find a quadratic model for the data.

Speed (mph)	Fuel Economy (mpg)
15	22.3
20	25.5
25	27.5
30	29.0
35	28.8
40	30
45	29.9
50	30.2
55	30.4
60	28.8
65	27.4
70	25.3

To find a quadratic model for the given data, we'll use a quadratic regression approach, which fits a quadratic equation of the form:

\[
y = ax^2 + bx + c
\]

Where:
- \( y \) is the fuel economy (in miles per gallon),
- \( x \) is the speed (in miles per hour),
- \( a \), \( b \), and \( c \) are the coefficients that we need to determine.

### Step 1: Organize the Data
Given:
- \( x = \) Speed (mph): \( 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70 \)
- \( y = \) Fuel Economy (mpg): \( 22.3, 25.5, 27.5, 29.0, 28.8, 30.0, 29.9, 30.2, 30.4, 28.8, 27.4, 25.3 \)

### Step 2: Perform Quadratic Regression
We can find the coefficients \( a \), \( b \), and \( c \) by using a quadratic regression method, which typically involves solving a system of linear equations derived from the least squares method. However, using computational tools simplifies the process.

Let's calculate the coefficients for the quadratic model.The quadratic model for the given data is:

\[
y = -0.0082x^2 + 0.7459x + 13.4722
\]

Where:
- \( a = -0.0082 \)
- \( b = 0.7459 \)
- \( c = 13.4722 \)

This equation describes the relationship between speed (in miles per hour) and fuel economy (in miles per gallon) for the cars in the study.

Would you like further details or have any questions?

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Here are some related questions:

1. How would you interpret the coefficients in this quadratic model?
2. What is the significance of the sign of the \(a\) coefficient in the quadratic model?
3. How would you use this quadratic model to predict the fuel economy at 75 mph?
4. Can a quadratic model always be used to represent the relationship between speed and fuel economy?
5. How would the quadratic model change if the dataset included a wider range of speeds?
6. How does the quadratic model compare to a linear model for the same data?
7. What are the limitations of using a quadratic model for this type of data?
8. How might the model be validated using additional data points?

**Tip:** When interpreting a quadratic model, remember that the sign and magnitude of the coefficients can indicate whether the curve opens upwards or downwards, and how steep the curve is.

Explore the quadratic model derived from a study comparing speed and fuel economy in cars. Learn how to find coefficients using quadratic regression and interpret their significance. Suitable for advanced high school students.

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