Exponential functions - General Mathematics
Download as PPTX, PDF8 likes5,588 views
This document discusses exponential functions and their real-world applications. It defines exponential functions as functions of the form f(x) = bx, where b is the base. Exponential functions can model situations involving population growth, radioactive decay, compound interest, and other phenomena where change occurs at a constant rate. The document provides examples of using exponential functions to model population growth over time, radioactive decay based on half-life, and compound interest on an investment over multiple years.
Exponential functions - General Mathematics
- 1. Exponential Functions! Cris A Capilayan
- 2. Representing Real-life Situations Using Exponential Functions • Exponential functions occur in various real world situations • Exponential functions are used to model real-life situations such as population growth, radioactive decay, carbon dating, growth of an epidemic , loan interest rates, and investments.
- 3. Definition of an exponential function • An exponential function with base b is a function of the form f(x) = bx or y = bx, where b>0 , b 1
- 4. • Complete a table of values for x = -3, -2, -1, 0, 1, 2, and 3 for the exponential function y = (1/3)x , y = 10x and y = (0.8)x.
- 5. X -3 -2 -1 0 1 2 3 y = (1/3)x 27 9 3 1 1/3 1/9 1/27 y = 10x 1/1000 1/100 1/10 1 10 100 1000 y = (0.8)x 1.9531 25 1.5625 1.25 1 .8 .64 .512
- 6. • If f(x) = 3x, evaluate f(2), f(-2), f(1/2), • f(2) = 9 • f(-2) = 1/9 • F(1/2) = √3
- 7. • Let b be a positive number not equal to 1. • The transformation of an exponential function with base b is a function of the form g(x) = a ∙ b x-c + d
- 8. • Many applications involve transformations of exponential functions. Some of the most common applications in real life of exponential functions and their transformations are: • POPULATION GROWTH • EXPONENTIAL DECAY • COMPOUND INTEREST
- 9. Example • Scientists will start with a certain number of bacteria or animals and watch how the population grows. • Ex. If the population doubles every 3 days, this can be represented as an exponential function. • Let t = time in days, at t = 0, there were 20 bacteria. Suppose the bacteria doubles every 100 hours. Give an exponential model for the bacteria as a function.
- 10. • t = 0 no. of bacteria is 20 • T= 100 no. of bacteria is 20 (2) • t = 200 no. of bacteria is 20 (2)2 • T = 300 no. of bacteria is 20(2)3 • T = 400 no. of bacteria is 20(2)4 • Exponential model : y = 20(2) t/100
- 11. Exponential decay • The half life of a radioactive substance is the time it takes for half the substance to decay. • Suppose that the half-life of a certain radioactive substance is 10 days and there are 10 g initially, determine the amount of substance remaining after 10 days.
- 12. Compound interest • Principal= starting amount of money • Principal can be invested at a certain rate that is earned at the end of a given period of time (such as 1 year). If the interest is compounded, the interest earned at the end of the period is added to the principal, and this new amount will earn interest in the next period.
- 13. Ex. • Mrs. Dela Cruz invested P 100, 000 in a company that offers 6% interest compounded annually. How much will this investment be worth at the end of the year for the next 5 years?