1.14 Function Model Construction

Overview of Jeff Bezos's Net Worth

• Jeff Bezos is a prominent figure in business, known for his significant wealth and impact on the tech industry.

• Comparison of Bezos's net worth at different years, highlighting significant changes and trends in his financial growth.

Overview of Net Worth Graphing

• Initial Value: 1994 - Bezos started with no billion dollars.

• 1998: Reached $12 billion.

• 2002: Dropped to $10 billion.

• 2008: Experienced a major decrease to $2 billion due to market fluctuations.

• Post-2008: Resumed a dramatic increase in wealth, reaching unprecedented levels.

Data Analysis and Modeling

• Shape of Data: The net worth data resembles a cubic function overall due to its rise and fall pattern.

• Transition to Calculations:

  • Years since 1994 are used for X-axis (e.g., 0 for 1994, 4 for 1998, etc.).

  • Data points plotted for accurate graphical representation.

Calculator Usage

• Entering Data: Steps provided to input x-values (years) and y-values (net worth) into a calculator.

• Stat Plot Configuration: Instructions on enabling the scatter plot to visualize data.

• Window Settings: Adjusting the view to capture all relevant data points for clarity.

Performing Regression Analysis

• Choosing Regression Type: Opting for cubic regression to fit the model to the plotted data.

• Data Entry Verification: Ensuring data is correctly inputted for accurate model calculations.

• Regression Output: The calculator provides coefficients for the cubic equation in the form ax^3 + bx^2 + cx + d.

• Truncating Values: Rules for rounding or truncating coefficients to maintain precision in results.

Predictive Analysis

• Average Rate of Change Calculation:

  • Example calculation for change in net worth from 2008 to 2022.

  • Calculating Values: Specific values substituted from regression function into the x-variables for years 14 and 28.

  • Final Outcome: Deriving a significant average rate of change of $13.484 billion per year from 2008 to 2022.

Regression Types Overview

• Types of Regression:

  • Linear: Straight line fit.

  • Quadratic: Parabolic shape, indicating acceleration or deceleration.

  • Cubic: N-shaped curve for more complex data relationships.

  • Quartic: W-shaped curve, allowing for multiple turning points.

  • Exponential: Rapid growth models.

  • Logarithmic: Inverse growth trends.

  • Logistic: Population trend with a plateau.

  • Sine and Cosine: Periodic functions, applicable in various contexts.

Inversely Proportional Functions

• Formula: y = k/x, where k is the constant of proportionality.

• Example Problem: Analyzing a scenario with magnetic force inversely related to distance squared.

• Application: Solving for k given a specific force and distance, then using it to answer follow-up questions about different forces.

Piecewise Functions

• Context: Analyzing snow accumulation over time using piecewise function definitions.

• Breaking Down Time Intervals:

  • Quadratic growth in the first period (0 to 4 hours).

  • Constant depth from 4 to 6 hours.

  • Linear growth between 6 to 9 hours.

• Calculating Average Rate of Change:

  • Finding values at specific intervals and assessing the depth change over time, demonstrating the importance of deriving accurate outputs for discontinuous functions.

  • Result for the depth of snow accumulation after specific hours analyzed.

Conclusion

• Reflecting on practices and calculations using regression models and piecewise functions solidifies understanding of mathematical modeling.

• Encouragement for further study and mastery of the concepts presented.