Haberman MTH Introduction to Eponential Functions (plus Eponential Models) Eponential functions are functions in which the variable appears in the eponent. For eample, f( ) 80 (0.35) is an eponential function since the independent variable,, appears in the eponent. One way to characterize eponential functions is to say that they represent quantities that change at a constant percentage rate. EXAMPLE: When Rodney first got his job in 003 he earned $,000 per year. After every year, Rodney receives a 0% raise. After one year, Rodney gets a 0% raise. His salary becomes: After his second year, Rodney gets another raise of 0%. His salary becomes: After his third year, Rodney gets another raise of 0%. His salary becomes We can now write a formula for Rodney s salary st ()(in dollars) after he has worked at the job t years: This is obviously an eponential function (since the variable is in the eponent). Thus, we can see why eponential functions represent quantities that change at a constant percent rate. Note that this function works when t 0 because
In the function st ( ) 000(.0) t, where did.0 come from? The number.0 is called the since... Below is another eample that shows us that eponential functions represent quantities that change at a constant percentage rate. EXAMPLE: Suppose that the population of the Epo Nation this year is 50,000. If the population decreases at a rate of 8% each year, find a function p that represents the population of the Epo Nation t years from now. population this year: 50000 population after year: 50000 50000(0.08) 50000 0.08 population after years: 50000(0.9) 50000(0.9) 50000(0.9)(0.08) 50000(0.9) 0.08 population after 3 years: 50000(0.9)(0.9) 50000(0.9) 50000(0.9) 50000(0.9) (0.08) 50000(0.9) 0.08 50000(0.9) (0.9) 50000(0.9) Observing the pattern above, we can deduce that the population of the Epo Nation after t years is given by the function: 3
3 What is the annual growth factor and the rate-of-change represented by the function p? Discuss the structure of both of the eponential functions ( ) 000(.0) t st and pt ( ) 50000(0.9) t. DEFINITION: An eponential function has the form f ( ) C a where C is the initial value (i.e., C f(0) ) and a is the growth factor, and a r where r is the decimal representation of the percent rate of change per unit of. NOTE: If r 0, then a, and the resulting function ehibits eponential growth. If r 0, then a, and the resulting function ehibits eponential decay. ALSO: a is always positive. ( a r, and we know that r since the rate of change cannot be less than 00%, i.e., we cannot lose more than 00% per unit of time.)
4 Graphs of Eponential Functions We already know what happens to the graphs of functions when we multiply their rules by positive and negative constants. Thus, all we need to determine is the shape of simple eponential function and we will then be able to determine the shape of more complicated eponential function. There are basically two classes of eponential functions:. f ( ) C a with a. f ( ) C a with 0 a Let's investigate the shape of the graphs of these two classes of eponential functions: EXAMPLE: Sketch a graph of h ( ). Note that this is an eponential function of the form h( ) C a where C and a. h ( ) A graph of h ( ). EXAMPLE: Sketch a graph of g ( ) form g( ) C a where C and g ( ). Note that this is an eponential function of the a. A graph of g ( ).
5 DEFINITION: A horizontal asymptote is a horizontal line that the graph of a function gets arbitrarily close to as the input values get very large (or very small). The functions ( ) h and ( ) g have a horizontal asymptote at y 0 (the -ais). Based on the two eamples above, we can conclude that the graph of an eponential function of the form f ( ) C a is increasing if a and decreasing if 0 a. (Technically we need to also state that C is positive for this increasing/decreasing behavior; the net eample might clarify why this is so.) EXAMPLE: Use your understanding of graph transformations to predict how the graph of m ( ) 4 compares with the graph of h ( ). On a graphing calculator or other graphing utility, sketch graphs of h and m to confirm your predictions. Finally, draw a conclusion about the role of C on the graph of an eponential function of the form f ( ) C a.
Eponential Models 6 EXAMPLE: Suppose that the function mt ( ) 500(0.945) t models the population of a certain species of monkeys in South America t years after January, 08. Describe this monkey population. EXAMPLE: Suppose that the population of a certain species of snakes in South America was 36 on January, 08. If this snake population is increasing at the rate of.7% per year, find a function that models this snake population. EXAMPLE: Find the rule for an eponential function passing through the two points (, 6) and (3, 4).
Practice for Eponential Functions and Models 7. If an investment worth $,000 January, 08 declines in value at the rate of % per year, create a function that models the value of the investment and use that function to determine the value of the investment on January, 05.. The population of a certain species of rodents was 300 on January, 04 and 04,800 on January, 08. If the rodent population is increasing eponentially, find a function, R, that models the population t years after January, 04. 3. Find the algebraic rule for an eponential function, f, passing through the two points (, 8) and (3, 79). 4. Find an algebraic rule for an eponential function f if f ( 3) 5 and f () 0. 8