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### Problem Summary:
- We are given a quadratic regression equation for predicting the profit $ \hat{y} $ of a company based on the number of years $ x $:
  \[
  \hat{y} = -0.34x^2 + 4.43x + 3.46
  \]
- We need to find the predicted profit for **year 4**.

### Calculation:
Plugging $ x = 4 $ into the equation:

\[
\hat{y} = -0.34(4)^2 + 4.43(4) + 3.46
\]

I'll perform the calculation.The predicted profit for year 4 using the given quadratic regression equation is approximately **$15.74 million**.

The correct answer is **Option B: $15.74 million**.

Would you like more details or have any other questions? Here are some related questions you might find interesting:

1. How do you derive a quadratic regression equation from data points?
2. What are the key components of a quadratic equation?
3. How does the coefficient of $ x^2 $ influence the shape of a quadratic graph?
4. What is the importance of using regression analysis in business forecasting?
5. How can residuals help in assessing the accuracy of a regression model?

**Tip:** Always check the fit of the regression model by analyzing residuals and ensuring no systematic patterns remain unmodeled.

The quadratic regression equation that models these data is: ŷ = -0.34x^2 + 4.43x + 3.46. Using this quadratic regression equation, what was the predicted profit in year 4?

Learn how to use a quadratic regression equation to predict company profits. In this example, the equation ŷ = -0.34x^2 + 4.43x + 3.46 is used to calculate the predicted profit for year 4. This lesson covers quadratic equations and regression analysis, suitable for students in Grades 10-12.

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