Mth141s52 - Sec. 5.2 Exponential Functions and Graphs Up to...

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Sec. 5.2 – Exponential Functions and Graphs Up to this point, we have mainly concentrated on polynomial functions. Examples of polynomial functions are : constant functions (f(x) = 4), linear functions (f(x) = 3x -2), quadratic functions (f(x) = 2 3 2 6 x x - + ), cubic functions (f(x) = 3 2 2 7 2 1 x x x + - + ), and higher degree functions. In these types of functions, the terms contain a base value of the variable x , and an exponent that is a whole number. Now we are going to look at some functions where the base is a constant , and the exponent is a variable . For example, 2 ( ) 3 4 x f x + = - . Notice that the variable is in the exponent!! These will be called exponential functions . Now for the definition : The function ( ) x f x a = , where x is a real number, a > 0 and 1 a , is called the exponential function , base a. Note that the base must be greater than 0, but cannot equal 1. Why can’t the base = 1??? Well, if the base is 1, then you have f(x) = 1 x , which is 1 for all real numbers of x. This is our constant function f(x) = 1 !! So we exclude a being 1. Let’s look at the graphs of some exponential functions: The graph of ( ) 2 x f x = looks like the following : 2 x Things to notice about the graph: 1) As x gets larger, the graph goes up steeper. We say it increases exponentially. 2) As x gets smaller in the negatives, the graph approaches the negative x-axis as a horizontal asymptote. It will never touch the negative x-axis! 3) The graph crosses the y-axis at (0, 1). Now let’s look at some other bases. Say 3 , 4, and 10 .
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10 x 4 x 3 x 2 x Things to notice about these exponential functions: 1) As x gets larger, all the graphs increase rapidly. They all have the same basic shape, starting close to negative x-axis, going thru (0, 1), and increasing rapidly on the right. 2) The bigger the base, the quicker the graph gets steeper. 3) The bigger the base, the quicker the graph approaches the negative x-axis. 4) All the graph go thru the same y-int value of (0, 1). What happens as the base gets closer to 1? Let’s see. Let’s graph bases of 2 , 1.5 , and 1.1 . 2 x 1.5 x 1.1 x The graph shows : ( ) 2 x f x = (in red), ( ) 1.5 x f x = (in blue), ( ) 1.1 x f x = (in black). So as the base approaches 1, the right-hand part of the graph increases less
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This note was uploaded on 06/06/2011 for the course MTH mth141 taught by Professor Stean during the Spring '10 term at Moraine Valley Community College.

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Mth141s52 - Sec. 5.2 Exponential Functions and Graphs Up to...

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