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Let's analyze the data and the problem statement from the image. The table shows the relationship between the number of billboard advertisements (independent variable \(x\)) and kite sales (dependent variable \(y\)).

Here’s a breakdown of the dataLet's analyze the data and the problem statement from the image. The table shows the relationship between the number of billboard advertisements (independent variable \(x\)) and kite sales (dependent variable \(y\)).

Here’s a breakdown of the 
- \(x = 1\), \(y = 120\)
- \(x = 2\), \(y = 229\)
- \(x = 3\), \(y = 344\)
- \(x = 4\), \(y = 463\)
- \(x = 5\), \(y = 588\)
- \(x = 6\), \(y = 717\)

The problem asks for a quadratic function to model this data, and we are given four possible quadratic models:
1. \( y = 2.5x^2 + 15.4x \)
2. \( y = 0.15x^2 - 10.6x + 119 \)
3. \( y = 2.5x^2 + 102x + 15.4 \)
4. \( y = 0.15x^2 + 119 \)

To find the correct model, we need to check which one fits the data best. This can be done by substituting the values of \(x\) (number of billboard ads) and checking if the resulting \(y\) (kite sales) is close to the given values.

I will test these equations with the first and last data points. Let's start by evaluating the four functions for \(x = 1\) and \(x = 6\).Here are the results for each model when evaluated at \(x = 1\) and \(x = 6\):

- **Model 1**: \(y(1) = 17.9\), \(y(6) = 182.4\)
- **Model 2**: \(y(1) = 108.55\), \(y(6) = 60.8\)
- **Model 3**: \(y(1) = 119.9\), \(y(6) = 717.4\)
- **Model 4**: \(y(1) = 119.15\), \(y(6) = 124.4\)

Comparing these values to the actual kite sales from the table:
- \(y(1) = 120\)
- \(y(6) = 717\)

**Model 3** fits the data best, as its predicted values (\(y(1) = 119.9\) and \(y(6) = 717.4\)) are very close to the actual kite sales.

Thus, the correct model is **Model 3: \( y = 2.5x^2 + 102x + 15.4 \)**.

Would you like more details, or have any questions?

Here are some related questions to expand on this topic:
1. How do you derive a quadratic equation from data points?
2. What are the key features of a quadratic function (e.g., vertex, axis of symmetry)?
3. How can we use quadratic regression to find a model?
4. What other types of functions can model real-world data?
5. How do you determine the best model for a set of data?

**Tip:** When modeling data with a quadratic function, checking values at the extremes of the data set often helps to quickly eliminate incorrect models.

Kai's Kites decided to advertise on billboards around town and track the relationship between the number of billboard advertisements they posted and the number of kites they sold. This table gives the total number of billboard advertisements they bought compared to the total number of kites they sold. The data can be modeled by a quadratic function. What function best models the data?

Analyze how a quadratic function models the relationship between the number of billboard advertisements and kite sales for Kai's Kites. The problem compares data points and matches them to a quadratic equation. Suitable for students in Grades 9-12.

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