Exponential growth and decay

Today’s Agenda First, we will review exponential growth. Second, we will review exponential decay. Third, we will learn to recognize if a function models exponential growth or decay. Finally, we will learn to solve problems that represent exponential growth or decay.

Exponential growth and decay You can use exponential functions to model the way certain populations increase or decrease over time. Increasing populations can be modeled by exponential growth function.

Exponential growth and decay Same way depreciation is the decline in an item’s value resulting from age and/or wear. When an item loses about the same percent of its value each year, you can use an exponential function to model the depreciation or decay.

Exponential growth and decay What are exponential growth and decay? Exponential growth and decay are rates; that is, they represent the change in some quantity through time.

Exponential growth and decay Exponential growth/decay is growth in which the rate of growth is proportional to the current size.

Exponential growth Can you tell by looking at an equation whether the function models exponential growth or decay? When b > 1, the function models exponential growth, and b is the growth factor.

Exponential decay When 0 < b < 1, the function models exponential decay, and b is the decay factor. It also called negative growth.

Exponential growth and decay Now tell me just by looking at the equation if it represent exponential growth or decay. y = 15(1.45)x y = 7.3(0.8)x

Exponential growth and decay What percent is grow or decay? y = 15(1.45)x y = 7.3(0.8)x

Exponential growth and decay Let say a stereo costs $900. Its value is estimated to decrease by 12% each year. Which exponential function models the value of the stereo? y = abx, a = $900, b = 12% y = 900(0.88)x ………… 1.00 - .12 = 0.88

Exponential growth and decay Suppose you buy a computer that costs $2000 and expect to depreciate by 30% each year. What will be the computer’s resale value be in 4 years? y = abx a = initial cost = $20000; b = (0.7); x = 4 y = 20000(0.7)4 = 480.20

Exponential growth and decay An insect population of two insects increases at a rate of 450% each week. What will the insect population be in 4 weeks? y = abx a = initial population = 2; b = (5.5); x = 4 y = 2(5.5)4 = 1830

Activity: Group Work The population of the United States in 1994 was about 260 million, with an average annual rate of increase of about 0.7%. 1. What is the growth factor for the United States population?

Group Work After one year the population would be 260 + 260(0.7%) million. = 260 + 260(0.007)……… 0.7% = 0.007 = 260( 1 + 0.007)…………Distributive property = 260(1.007) Therefore, growth factor is 1.007, where in abx, a is 260 and constant, and b is 1.007, which is a growth factor.

Group Work 2. Suppose this rate of growth continues. Write an equation that model the future growth of the United States population.

Group Work To write the equation, find a and b. a = 260, and b = 1.007 Let say, x = number of years after 1994 and y = the population (in million) The population increases exponentially. Standard form of exponential function is y = abx y = 260(1.007)x Therefore, the equation y = 260(1.007)x models the future growth of the United States population.

Group Work 3. Use the equation, y = 260(1.007)x , to predict the United States population in 2001.

Group Work y = 260(1.007)x, = 261.82, [ 261.82 – 260 = 1.82], every year population grows by 1.82 million. Total year [ 2001 – 1994 ] = 7 years, therefore, 1.82*7= 12.74 million. 260 + 12.74 = 272.74 ~ 273 million.

Group Work 4. Using the model, every year population grows by 1.82 million, when will the United States population be closest to 289 million.

Group Work 1994 population = 260, every year increases by 1.82 million. Expected population 289, increase, 289 – 260 = 29 million. Total growth 29 million, every year grows 1.82 million, therefore, 29/1.82 ~ 15 years. 1994 + 15 = 2009.

Summary Exponential growth occurs when some quantity regularly increases by a fixed percentage. Exponential decay occurs whenever the size of a quantity is decreasing by the same percentage each unit of a time.

Summary If the factor b is greater than 1, then we call the relationship Exponential Growth. If the factor b is greater less than 1, then we call the relationship Exponential Decay.