This is an example of an Exponential Function. An exponential function has a constant growth factor, f(x+1)/f(x). The variable (x) and the output (y) gets multiplied by a fixed amount. The equation for exponential function is f(x)=ba^x where a>0 and a does not equal 1. The y-intercept or starting value is b and the growth factor is a, which is also called the base of the function. A growth factor greater than 1 gives exponential growth and a growth factor between 0 and 1 gives exponential decay. The domain is all real numbers and the range is f(x)>0. F(x)=a^x is increasing for all x when the growth factor, a, is greater than 1. F(x)=a^x is decreasing for all x when the growth factor, a, is between 0 and 1. It is always concave up when f(x)=a^x. Vertical translation of an exponential function is when f(x)=b^x +c, which shifts the graph up however many units C is. The horizontal translation of an exponential function is when f(x)=b ^x+c, which shifts the graph to the left however many units C is. There is a reflection over the x axis when f(x)=-b^x and there is a reflection over the y axis when f(x)=b^x. Another equation used for exponential functions is A(t)=a(1+/- r)^t. This equation can be used for population and money problems.
Jacob Bernoulli was the first to understand the way that the log function is the inverse of the exponential function. On the other hand the first person to make the connection between logarithms and exponents was James Gregory. Johann Bernoulli began the study of the calculus of the exponential function in 1697 when he published Principia calculi exponentialium seu percurrentium. The work involves the calculation of various exponential series and many results are achieved with term by term integration. In 1884 Boorman calculated e to 346 places and found that his calculation agreed with that of Shanks as far as place 187 but then became different. In 1887 Adams calculated the logarithm of e to the base 10 to 272 places.
Jacob Bernoulli was the first to understand the way that the log function is the inverse of the exponential function. On the other hand the first person to make the connection between logarithms and exponents was James Gregory. Johann Bernoulli began the study of the calculus of the exponential function in 1697 when he published Principia calculi exponentialium seu percurrentium. The work involves the calculation of various exponential series and many results are achieved with term by term integration. In 1884 Boorman calculated e to 346 places and found that his calculation agreed with that of Shanks as far as place 187 but then became different. In 1887 Adams calculated the logarithm of e to the base 10 to 272 places.
Exponential Function Examples.docx | |
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Real Life Exponential Functions.docx | |
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