Exponential Functions and Geometric Sequences Review

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What is an exponential function?

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An exponential function is a mathematical function of the form f(x) = a(b)^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. It shows rapid growth or decay.

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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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What is a geometric sequence?

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A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

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How do you find the common ratio in a geometric sequence?

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To find the common ratio, divide any term in the sequence by the previous term. For example, in the sequence 3, 15, 75, the common ratio is 15/3 = 5.

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What is the formula for the nth term of a geometric sequence?

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The nth term of a geometric sequence can be found using the formula: a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.

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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, represented by an exponential function with a base greater than 1.

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What is exponential decay?

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Exponential decay occurs when a quantity decreases by a consistent percentage over time, represented by an exponential function with a base between 0 and 1.

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How can you identify exponential growth or decay from a function?

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You can identify exponential growth if the base of the function is greater than 1 (e.g., f(x) = 2(1.25)^x). If the base is between 0 and 1, it indicates exponential decay (e.g., f(x) = 2(0.5)^x).

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What is the common ratio of the function f(x) = 2(4)^x?

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The common ratio of the function f(x) = 2(4)^x is 4.

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What are the next three terms in the geometric sequence: 512, 256, 128?

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The next three terms are 64, 32, and 16.