https://file.aiquickdraw.com/mathbot/file/17272051362059nrxfb1h.jpg

The image contains a dataset showing the number of hours of daylight over time, where:

- \( m \) is the number of months since January 1st.
- \( h(m) \) represents the hours of daylight observed at the start of each month.

We are asked to generate a quadratic function using regression to model this data set.

The data is:
- \( m = 0 \), \( h(0) = 11.57 \)
- \( m = 1 \), \( h(1) = 12.5 \)
- \( m = 2 \), \( h(2) = 13.3 \)
- \( m = 3 \), \( h(3) = 13.8 \)
- \( m = 4 \), \( h(4) = 13.9 \)
- \( m = 5 \), \( h(5) = 13.53 \)
- \( m = 6 \), \( h(6) = 11.85 \)

To find the quadratic regression equation \( y = ax^2 + bx + c \), I will perform the regression on this dataset. Let's compute the coefficients.The quadratic function that models the given data set is:

\[
h(m) = -0.2345m^2 + 1.5321m + 11.3738
\]

This equation can be used to estimate the number of daylight hours based on the number of months since January 1st.

Would you like more details or have any questions? Here are some related questions for further exploration:

1. How does the quadratic function change if we include more data points beyond 6 months?
2. Can this model predict daylight hours accurately for months beyond the given data?
3. What does the negative coefficient of \( m^2 \) tell us about the pattern of daylight hours?
4. How would a linear regression compare to this quadratic regression in terms of accuracy?
5. How does the quadratic function behave for negative values of \( m \) (before January)?

**Tip:** Quadratic functions are often useful for modeling patterns with an initial increase followed by a decrease, as seen in cyclical or seasonal data like daylight hours.

On January 1st, Ray noticed that there were 11.57 hours of daylight in Houston. He recorded the hours of daylight on the first of each month for the next six months. His data is: (m, h(m)) = [(0, 11.57), (1, 12.5), (2, 13.3), (3, 13.8), (4, 13.9), (5, 13.53), (6, 11.85)]. Use regression to generate a quadratic function that models this dataset.

Discover how to generate a quadratic function to model the daylight hours in Houston over the course of several months. Data points include hours of daylight from January to June, ranging from 11.57 to 13.9 hours. Suitable for high school students learning quadratic equations and regression techniques.

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