4. Exponential Growth and Decay
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There are some exponential functions that increase by a constant factor or percentage over equal intervals of time, while others decrease by a constant factor or percentage instead.
See solution.
From what we can see in Exploration 1, we know that there are some specific groups of exponential functions that increase by a constant factor or percentage over equal intervals of x. For these cases, we say that the functions exhibit exponential growth.
y = ab^x If we consider a>0, the exponential function will be increasing over time only if b>1. Therefore, we can rewrite the exponential function as shown below. y = ab^x ⇔ y = a(1+r)^x Here r represents how fast the function is growing as a percentage, written in decimal form. This is why r is known as the growth rate in these cases. On the other hand, if the function is decreasing by a constant factor or percentage over equal intervals of x, we say that it exhibits exponential decay.
For this to happen, we need that 0