6.10 Functions and Their Graphs 395 Exponential Functions and Their Graphs Many real-life problems, including population growth, growth of bacteria, and decay of radioactive substances, increase or decrease at a very rapid rate. For example, the graph in Fig. 6.48 shows the growth of the world population. Notice how the graph is increasing rapidly. This is an example in which the graph is increasing exponentially . The equation of a graph that increases or decreases exponentially is called an exponential equation or an exponential function . An exponential equation or exponential function is of the form = y ax or = f x a ( ) ,x where in both cases > ≠ a a 0, 1. When graphing exponential functions we often use the form = y a ,x and when working with applications we often use the form = f x a ( ) .x However, since = y f x( ), we can use either form to represent exponential functions. World Population Year Population (billions) 1650 1700 1750 1800 1850 1900 1950 2000 2050 7 6 5 3 4 1 2 8 Source: Census.gov Figure 6.48 MATHEMATICS TODAY Population Growth Why This Is Important Knowing the population growth rate helps government agencies determine the number of people for whom they will need to provide future services. Population growth during certain time periods can be described by an exponential function. Whether it is a population of bacteria, fish, flowers, or people, the same general trend emerges: a period of rapid (exponential) growth, which is then followed by a leveling-off period. In an exponential function, letters other than x and y may be used for the variables. The following are examples of exponential functions: ( ) = = f x A r () 2, () x r 1 2 and = P t( ) 2.3 .t Note in exponential functions that the variable is the exponent of some positive constant that is not equal to 1. In many real-life applications, the variable t will be used to represent time. Problems involving exponential functions can be evaluated much more easily if you use a calculator containing a y , x x , y or ∧ key. The following function, referred to as the exponential growth or decay formula , is used to solve many real-life problems. Exponential Growth or Decay Formula = > ≠ P t P a a a ( ) , 0, 1 kt 0 In the exponential growth or decay formula, P0 represents the original amount present, P t ( ) represents the amount present after t years, and a and k are constants. When > > k a P t 0, and 1, ( ) increases as t increases and we have exponential growth. When < k P t 0, ( ) decreases as t increases and we have exponential decay.
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