Exponential functions are crucial for understanding growth and decay in various contexts. They are defined by the formula y = ab^x, where 'a' is the initial value, 'b' is the base, and 'x' is the exponent. These functions model phenomena such as population growth, investment returns, radioactive decay, and asset depreciation. Adjustments to the model allow for different compounding frequencies, making them versatile for financial calculations and biological studies.
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Exponential functions are mathematical expressions that describe the growth or decay of quantities at a rate proportional to their current value
Base
The base of an exponential function is a constant number raised to a variable exponent
Initial Amount
The initial amount in an exponential function is represented by the coefficient 'a'
Exponent
The exponent in an exponential function indicates the number of time intervals
The base of an exponential function must be a positive real number other than 1 to ensure the function is truly exponential
Exponential growth and decay are two fundamental behaviors modeled by exponential functions
Exponential growth occurs when the base is greater than 1, indicating a fixed percentage increase over equal time intervals
Exponential decay occurs when the base is between 0 and 1, signifying a consistent percentage decrease over time
Exponential functions are used in finance to model compound interest and depreciation of assets
Exponential functions have tangible applications in fields such as biology and environmental science
When dealing with exponential growth or decay that does not occur annually, the function must be adjusted to reflect the correct frequency
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Exponential Function Base 'b'
Base 'b' is a positive real number, not equal to 1, representing growth (>1) or decay (0<b<1).
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Initial Amount 'a' in Exponential Functions
'a' is the starting value of the quantity before exponential change occurs.
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Exponent 'x' in Exponential Functions
'x' is the variable indicating the number of time intervals for growth or decay.
