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STEM13-math 108 - Math Learning Center Boise State ©2010...

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Unformatted text preview: Math Learning Center Boise State ©2010 Exponential functions: week 13 STEM As we have seen, exponential functions describe events that grow (or decline) at a constant percent rate, such as placing finances in a savings account. Exponential functions will have a role to play in almost every subject. In science, exponential functions describe population growth rates, radioactive decay, Richter scale for earthquakes, Fiber optics, space exploration and more. As this is the first experience for many into exponential functions we will spend today focusing on developing a deeper understanding of exponential functions, increasing the learning opportunities when exponential functions reappear. The simplest form of an exponential function is gG¡¢ £ ¤ ¥ where u ¦ ¤ ¦ U or ¤ § U . Note: the letter ¤ is used in an exponential function as it represents the “base” of the exponential function. Let’s start with ¤ § U , let ¤ £ ¨©ª «¬­ ®¯ Graph the three functions: Describe the characteristics that all three graphs have in common. Did you catch the relationship of the value of g G¡¢ °±²¬ ¡ £ u³ One relationship that is generally not seen by most students when they begin to study exponential functions is that as ¡ ´ µ¶ the answer, gG¡¢ gets closer and closer to the x-axis but never reaches it. This type of behavior is called asymptotic behavior. And it is said that these three functions are asymptotic to the line · £ u ¸¹ ¡ ´ µ¶ . gG¡¢ £ ¨ ¥ ¡ gG¡¢-2 -1 0 1 2 gG¡¢ £ ª ¥ ¡ gG¡¢-2 -1 0 1 2 gG¡¢ £ ® ¥ ¡ gG¡¢-2 -1 0 1 2 The notation ¡ ´ µ¶ is read “as x goes to negative “infinity”. The arrow means “goes to”. The sentence “as x goes to negative infinity implies that we are considering what happens to the graph as x moves further and further to the left on the x-axis. Math Learning Center Boise State ©2010 So we have an idea as to what happens if g G u . Let’s now consider when U ¡ g ¡ u . Graph the following functions. Describe the characteristics of these three functions; include a discussion on when ¢ £ U and on asymptotic behavior. A critical question that is very important for next week: Are exponential functions one-to-one? (One-to one means that for each input value there is exactly one output value and for each output value there is exactly one input value.) The last two graphing activities were designed to look at values of g where U ¡ g ¡ u or g G u...
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