Modeling growth and decay with functions of the form Abx+c or Aebx+c, as well as the concept of half-life for exponential decay.
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An exponential model represents quantities that multiply repetetively by a constant factor b. The basic form of an exponential is bx, but any exponential can be written in the form Abx+k.
The graph of an exponential model is a curve that approaches a horizontal asymptote at y=k on one side, and has a y-intercept at (0,A+k). Because of the asymptote on an exponential graph, exponential models are good at describing behaviors that level off over time.
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Exponential growth describes quantities that increase by the same factor over a certain amount of time. Algebraically, exponential growth is modeled by functions of the form
where b>1. b is called the growth factor.
Note: Aekt is another model for exponential growth if the instantaneous growth rate, k, is positive.
Stewart EJ, Madden R, Paul G, Taddei F (2005), CC BY-SA 4.0
Exponential decay describes quantities that decrease by the same factor over a certain amount of time. Exponential decay is modeled by functions of the form
where 0<b<1. b is called the decay factor.
Note: Aekt is another model for exponential decay if the instantaneous growth rate, k, is negative.
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For any quantity that decays exponentially, the half-life is the amount of time it takes for the quantity to halve in value.
From any exponential decay model of the form f(t)=Abkt (0<b<1), the half-life, or time for the value of f to reach half of its current value, is given by t1/2=−klogb2.
Most commonly, given an equation of the form f(t)=Aekt, the half life is given by −kln2.