The video discusses selecting mathematical models based on word problems, data, and graphs. It covers identifying linear, quadratic, cubic, and absolute value functions. The key points are:
- Linear models have constant rate of change.
- Quadratic models have constant second differences.
- Cubic models have constant third differences.
The video also explores how to apply these models to real-world scenarios, such as calculating perimeter, area, and volume, and discusses the importance of understanding the restricted domain and range of each model. It concludes by showing how to find specific values and average rates of change within these models using a graphing calculator.
1. Mr. Bean's stamp collection has a constant rate of change of 5 stamps per day, indicating a linear function.
2. Mr. Kelly's stamp collection has a second constant rate of change of 2, suggesting a quadratic function.
3. Mr. Sullivan's stamp collection has a third constant rate of change of 6, indicating a cubic function.
4. Mr. Breast's stamp collection shows a constant rate of change of 6 stamps every two days, representing a linear function.
5. The discussion shifts to applying these functions to geometry problems, where perimeter problems lead to linear functions, area problems to quadratic functions, and volume problems to cubic functions.
6. The example of a bottle rocket's height over time is explored using a quadratic function.
7. The height of the rocket at 2 seconds is 67 feet.
8. The restricted domain is [0, 4.046] seconds, reflecting the time the rocket is in the air.
9. The restricted range is [0, 67] feet, representing the height above ground level.
10. The average rate of change on the interval [0, 4.046] seconds is used to analyze the rocket's motion.