Exponential Characteristics

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What is exponential growth?

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Exponential growth occurs when a quantity increases by a consistent 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 consistent percentage over a period of time, leading to 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 of the exponential (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 rate of growth or decay. If the base is greater than 1, the function represents growth; if the base is between 0 and 1, it represents decay.

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

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As x approaches negative infinity (x → -∞), the value of y approaches the horizontal asymptote, which is typically a constant value.

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

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As x approaches negative infinity (x → -∞), the value of y approaches infinity (y → ∞).

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What is an asymptote in the context of exponential functions?

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An asymptote is a line that a graph approaches but never touches. For exponential functions, the horizontal asymptote is often a constant value that the function approaches as x approaches infinity or negative infinity.

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How do you identify the horizontal asymptote of an exponential function?

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The horizontal asymptote of an exponential function can be identified by examining the constant added or subtracted from the function. For example, in y = a(b^x) + k, the horizontal asymptote is y = k.

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

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The y-intercept of an exponential function is the value of y when x = 0. It represents the initial value of the function and can be found by evaluating the function at x = 0.

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How does changing the base of an exponential function affect its graph?

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Changing the base of an exponential function affects the rate of growth or decay. A larger base results in a steeper growth curve, while a smaller base (greater than 1 but less than 2) results in a more gradual increase.