Exponential Characteristics

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What is exponential growth?

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Exponential growth occurs when a quantity increases by a fixed percentage over a period of time, resulting in a rapid increase that can be represented by the function y = a(1 + r)^t, where a is the initial amount, r is the growth rate, and t is time.

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What is exponential decay?

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Exponential decay occurs when a quantity decreases by a fixed percentage over a period of time, resulting in a rapid decrease that can be represented by the function y = a(1 - r)^t, where a is the initial amount, r is the decay rate, and t is time.

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What is the general form of an exponential function?

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The general form of an exponential function is y = ab^x, where a is a constant, b is the base (b > 0), and x is the exponent.

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What does the base of an exponential function indicate?

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The base of an exponential function indicates the growth or decay factor. If the base is greater than 1, the function represents exponential growth; if the base is between 0 and 1, it represents exponential decay.

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How do you determine the y-intercept of an exponential function?

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The y-intercept of an exponential function can be found by evaluating the function at x = 0. For y = ab^x, the y-intercept is (0, a).

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What is the left end behavior of an exponential function with a positive base?

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For an exponential function with a positive base, as x approaches negative infinity (x → -∞), y approaches 0 (y → 0).

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What is the right end behavior of an exponential growth function?

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For an exponential growth function, as x approaches positive infinity (x → ∞), y approaches positive infinity (y → ∞).

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What is the right end behavior of an exponential decay function?

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For an exponential decay function, as x approaches positive infinity (x → ∞), y approaches 0 (y → 0).

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How can you express repeated multiplication using exponents?

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Repeated multiplication can be expressed using exponents by writing the base multiplied by itself a certain number of times as base^n, where n is the number of times the base is multiplied.

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What is the significance of the y-intercept in an exponential function?

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The y-intercept represents the initial value of the function when x = 0, providing insight into the starting point of the growth or decay process.