Thursday, February 7, 2013

The properties of exponential functions

Robert was having trouble visualizing the graphs of exponential functions and the ways an exponential function can be generalized (the characteristics that every function of the form f(x) = a * b^x will share). I started by drawing a few diagrams for him:

Robert was able to understand that in a function f(x) = b^x, regardless of the base b, when x is 0, the function has a value of 1. This is because any base raised to an exponent of 0 is equal to 1.

I asked Robert to consider how the y-intercept would change if the function b^x times a constant a. First I gave him an example of f(x) = 2 * b^x. He told me that the y-intercept is (0, 2). I then asked him to extend this idea to any constant a. He was able to see that the y-intercept would be (0, a).

I then asked Robert to consider the range of a function f(x) = b^x. He was able to see that the function will never reach 0, or be less than 0. I then asked him to consider what would happen if he added five on the end, as in the function g(x) = b^x + 5. After the previous question, Robert quickly concluded that the range of the function is all numbers greater than 5.

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