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Exponential Functions definitions

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  • Exponential Function
    A mathematical rule where a constant positive base is raised to a variable exponent, producing rapid growth or decay.
  • Base
    A fixed positive number, not equal to 1, that is repeatedly multiplied in an exponential expression.
  • Exponent
    A variable or expression in the power position, indicating how many times the base is used as a factor.
  • Continuous Graph
    A curve with no breaks or gaps, representing all real input values for the function.
  • Horizontal Asymptote
    A line that the graph approaches but never touches, often found at y = 0 for exponential decay.
  • Domain
    The complete set of possible input values, typically all real numbers for exponential functions.
  • Range
    The set of possible output values, usually all positive real numbers above the asymptote.
  • Growth
    A pattern where function values increase rapidly as the exponent increases, seen when the base is greater than 1.
  • Decay
    A pattern where function values decrease toward zero as the exponent increases, seen when the base is between 0 and 1.
  • e
    An irrational constant approximately 2.718, arising from continuous compounding and natural processes.
  • Compounding Interest
    A process where interest is calculated on both the initial amount and previously earned interest, leading to exponential growth.
  • Population Growth
    A real-world application where quantities increase exponentially, often modeled using a base of e.
  • Fractional Base
    A positive constant less than 1 used as the base, resulting in exponential decay.
  • Caret Key
    A calculator symbol (^) used to indicate exponentiation when evaluating exponential expressions.
  • Transformation
    A change applied to the basic graph, such as shifting or stretching, to create more complex exponential graphs.