Problem 58
Question
How can you tell whether an exponential model describes exponential growth or exponential decay?
Step-by-Step Solution
Verified Answer
An exponential model describes exponential growth when its base is greater than 1, whereas it describes exponential decay when its base is between 0 and 1.
1Step 1: Identify the base of the exponential function
In an exponential function, the base is the number which is raised to the power of the variable. It is typically expressed in the general form of the exponential function as \(f(x) = ab^{x}\), where \(a\) is a constant that moves the graph up or down, \(b\) is the base, and \(x\) is the variable.
2Step 2: Determine if the base is greater than 1
If the base is greater than 1, this corresponds to a situation of exponential growth. In this scenario, the value of the function will increase as \(x\) increases.
3Step 3: Determine if the base is between 0 and 1
If the base is between 0 and 1, this corresponds to a situation of exponential decay. In this scenario, the value of the function decreases as \(x\) increases.
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