Exponential functions, defined as f(x) = a * b^x, are key in modeling growth and decay in real life. Understanding their characteristics, like the base, growth factors, and asymptotes, is essential for mastering concepts in AP Precalculus.
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Definition of an exponential function: f(x) = a * b^x
- An exponential function is defined by the equation f(x) = a * b^x, where 'a' is a constant and 'b' is the base.
- The variable 'x' is the exponent, which indicates the power to which the base 'b' is raised.
- Exponential functions model growth or decay processes in various real-world scenarios.
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Base (b) and its properties (b > 0, b ≠ 1)
- The base 'b' must be greater than 0 to ensure the function is defined for all real numbers.
- The base cannot equal 1, as this would result in a constant function rather than an exponential function.
- Different values of 'b' affect the steepness and direction of the graph (growth vs. decay).
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Growth factor and decay factor
- The growth factor is represented by 'b' when b > 1, indicating that the function increases as 'x' increases.
- The decay factor is represented by 'b' when 0 < b < 1, indicating that the function decreases as 'x' increases.
- The growth or decay factor determines how rapidly the function changes.
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Vertical asymptote (y-axis)
- The vertical asymptote of an exponential function is the line y = 0 (the x-axis).
- As 'x' approaches negative infinity, the function approaches the asymptote but never touches it.
- This characteristic indicates that the function's values will never be negative.
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y-intercept (always positive)
- The y-intercept occurs at f(0) = a * b^0 = a, which is always positive if 'a' is positive.
- This point represents the initial value of the function when x = 0.
- The y-intercept is crucial for understanding the starting point of the exponential function.
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Domain and range
- The domain of an exponential function is all real numbers, (-∞, ∞).
- The range is (0, ∞) if 'a' is positive, indicating that the function only takes positive values.
- This reflects the nature of exponential growth or decay, which cannot produce negative outputs.
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End behavior
- As 'x' approaches positive infinity, f(x) approaches infinity for growth functions (b > 1).
- As 'x' approaches negative infinity, f(x) approaches 0 for both growth and decay functions.
- Understanding end behavior helps predict the long-term trends of the function.
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Exponential growth vs. decay
- Exponential growth occurs when the base 'b' is greater than 1, leading to rapid increases in function values.
- Exponential decay occurs when the base 'b' is between 0 and 1, resulting in a decrease in function values over time.
- Identifying whether a function represents growth or decay is essential for modeling real-world scenarios.
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Compound interest as an application
- Exponential functions are used to model compound interest, where the amount of money grows over time based on a fixed interest rate.
- The formula A = P(1 + r/n)^(nt) represents the future value of an investment, where 'P' is the principal, 'r' is the interest rate, 'n' is the number of times interest is compounded per year, and 't' is the time in years.
- Understanding this application helps students relate exponential functions to financial contexts.
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Natural exponential function (e^x)
- The natural exponential function is defined as f(x) = e^x, where 'e' is approximately 2.71828.
- It is a special case of exponential functions with unique properties, such as its derivative being equal to the function itself.
- The natural exponential function is widely used in calculus, finance, and natural sciences due to its continuous growth rate.