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04.04.2019: One Lesson of Math - Logarithms and Exponents, 9/10: Graphs of Exponential Functions


Today's soundtrack is The Johns: EP, a 27-minute indie rock album that I bought many years back, and I have no recollection of the circumstances regarding that purchase. I'm glad I bought it, though; it's a nice album, a bit reminiscent of Ben Folds.


This morning, I'm learning how to make graphs of exponential functions.


The most basic exponential function is y = 2ˣ, which starts near its asymptote, y=0, then gets higher and higher as it goes further right, rising exponentially. When calculating the y-values of corresponding -x values, we just invert the exponent to get a rational answer.


Since 1/2 is the inverse of 2¹, and 1/2 = 2⁻¹, a graph of the function y = 2⁻ˣ (also can be written as y=[1/2]ˣ) will fall in from the left at a steep angle, and will level out as it gets closer to its asymptote, y=0. Functions with whole number coefficients are representative of exponential growth; functions with rational coefficients or negative exponents represent exponential decay.


For functions of exponential growth, the higher the value of the coefficient, the steeper will be the rise of the graph. For example, the function y=15ˣ will have a much steeper rise than the function y=2ˣ.

The inverse is true for functions of exponential growth. The lower the value of the coefficient, the steeper its fall will be. For example, the function y=(1/10)ˣ is much more steep than the function y=(6/7)ˣ.


Shortcut: If we are told to graph a function with modifications on it, first graph the function without modifications, then make the modifications. It's much easier to do it that way.

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