4.1 INTRODUCTION TO THE FAMILY OF EXPONENTIAL FUNCTIONS 1

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Growing at a Constant Percent Rate Example 2 During the 2000s, the population of Mexico increased at a constant annual percent rate of 1.2%. Since the population grew by the same percent each year, it can be modeled by an exponential function. Let’s calculate the population of Mexico for the years after In 2000, the population was 100 million. The population grew by 1.2%, so Pop. in 2001 = Pop. in % of Pop. in 2000 = (100) = = million. 3

Growing at a Constant Percent Rate Example 2 continued Population of Mexico The population of Mexico increases by slightly more each year than it did the year before, because each year the increase is 1.2% of a larger number. Year ΔP, % increase in population P, population (millions) 2000— The projected population of Mexico, assuming 1.2% annual growth year P, population (millions) 4

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Growth Factors and Percent Growth Rates The Growth Factor of an Increasing Exponential Function In Example 2, the population grew by 1.2%, so New Population = Old Population + 1.2% of Old Population = ( ) ˑ Old Population = ˑ Old Population We call the growth factor. 1.2% is called the growth rate. 6 The Growth Factor of a Decreasing Exponential Function In Example 3, the carbon-14 changes by −11.4% every 1000 yrs. New Amount = Old Amount −11.4% of Old Amount = (1 −.114) ˑ Old Amount = ˑ Old Amount Although represents a decay factor, we use the term “growth factor” to describe both increasing and decreasing quantities.

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A General Formula for the Family of Exponential Functions An exponential function Q = f(t) has the formula f(t) = a b t, a ≠ 0, b > 0, where a is the initial value of Q (at t = 0) and b, the base, is the growth factor. The growth factor is given by b = 1 + r or r = b- 1 where r is the decimal representation of the percent rate of change. If there is exponential growth, then r > 0 and b > 1. If there is exponential decay, then r < 0 and 0 < b < 1. An exponential function Q = f(t) has the formula f(t) = a b t, a ≠ 0, b > 0, where a is the initial value of Q (at t = 0) and b, the base, is the growth factor. The growth factor is given by b = 1 + r or r = b- 1 where r is the decimal representation of the percent rate of change. If there is exponential growth, then r > 0 and b > 1. If there is exponential decay, then r < 0 and 0 < b < 1. 8

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Applying the General Formula for the Family of Exponential Functions Example 6 Using Example 2, find a formula for P, the population of Mexico (in millions), in year t where t = 0 represents the year Solution In 2000, the population of Mexico was 100 million, and it was growing at a constant 1.2% annual rate. The growth factor is b = = 1.012, and a = 100, so P = 100(1.012) t. Because the growth factor may change eventually, this formula may not give accurate results for large values of t.

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