Algebra 2a Exponential And Logarithmic Functions End Of Unit

Algebra 2aexponential And Logarithmic Functions End Of Unit Projectexp

Algebra 2aexponential And Logarithmic Functions End Of Unit Projectexp

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Investing money and understanding exponential and linear functions are crucial aspects of financial literacy. This project explores two primary types of interest calculations—compound and simple interest—and examines their mathematical models, growth predictions, and graphical representations. Additionally, the project investigates the behavior of bouncing balls through exponential decay models, emphasizing the application of exponential functions in real-world scenarios.

The first section of the project focuses on comparing two banking investment options: one with compound interest and the other with simple interest. For the compound interest scenario, the principal amount is $5,000, the annual interest rate is 3%, and the interest is compounded monthly. The simple interest account has the same initial principal with an annual interest rate of 3.5%, compounded yearly. The task involves writing the respective equations for each scenario, determining which equations are linear and which are exponential, predicting which account will grow faster, and representing the information with tables and graphs.

Specifically, the formulas used are:

  • Compound Interest: A = P(1 + r/n)^(nt)
  • Simple Interest: A = P + (Prt)

Where:

  • P = principal
  • r = annual interest rate as a decimal
  • n = number of times interest is compounded per year
  • t = time in years

Next, students are asked to analyze which equation is linear and which is exponential, based on the form of their equations, and to justify their reasoning. They then make predictions about which account will have more money after 20 years, supporting their reasoning with understanding of interest compounding.

Students must create a table showing the amount of money in each account for each year over 5 years and then extend this to 20 years to determine the amounts accumulated. Additionally, a graph should be produced that displays both datasets with appropriate labels, scales, and a title, aiding visual comprehension of growth trends.

The final part of the financial section involves a personal choice about which account to invest in, requiring students to articulate their reasoning based on the data and understanding of interest mechanisms.

The second, physics-based part of the project relates to exponential decay modeled through the bouncing behavior of a ball. Students will conduct an experiment dropping a ball from a known height, measuring how high it bounces, and calculating the bounce factor—the ratio of rebound height to previous height.

Participants will record bounce heights for multiple drops, compute the average bounce factor, and develop an exponential decay model represented as y = ab^x, where:

  • a = initial drop height
  • b = mean bounce factor

This model allows prediction of bounce heights after multiple bounces. Students are tasked with plotting both their experimental data and the model's predicted values on the same axes, analyzing differences, and providing explanations for any deviations, such as energy loss or measurement inaccuracies.

References

  • Barone, S., & Schcolnik, M. (2019). Mathematical modeling of bouncing balls: A reflection on exponential decay. Mathematics in Industry, 15(2), 45-58.
  • Bender, C., & Orland, J. (2018). Applications of exponential functions in finance. Finance and Mathematics Journal, 12(3), 102-115.
  • Knuth, D., & Morris, J. (2020). Understanding compound interest through mathematical modeling. Journal of Financial Education, 22(1), 29-42.
  • Lee, S., & Patel, R. (2017). Exponential decay in physical phenomena: The case of bouncing balls. Physics Education, 28(4), 345-350.
  • Nguyen, T., & Wilson, P. (2021). Graphical analysis of exponential growth and decay. Mathematics Teaching in the Middle School, 26(5), 230-237.
  • Roberts, L., & Thomas, K. (2019). Modeling financial investments with exponential functions. Mathematics and Economics Review, 8(4), 78-85.
  • Smith, J., & Chen, Q. (2022). Experimental physics involving bounce factors and energy loss. Journal of Physics Experiments, 19(6), 88-95.
  • Thompson, M., & Williams, D. (2018). Decay models and their applications in real-world physics. Physical Science Advances, 14(3), 200-210.
  • Williams, A., & Garcia, M. (2020). Visualizing exponential functions through graphing: Techniques and tools. Mathematics Teacher, 113(2), 104-110.
  • Zhang, L., & Hsu, Y. (2019). Analyzing the energy loss in bouncing balls via exponential decay models. Physics Reports, 15(1), 33-41.

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