Skills Check 5.1b: Using Exponential Functions

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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.

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What is the horizontal asymptote of an exponential function?

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The horizontal asymptote of an exponential function is a horizontal line that the graph approaches as x approaches positive or negative infinity. For functions of the form f(x) = a * b^x + k, the horizontal asymptote is y = k.

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What does it mean for a function to exhibit exponential growth?

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Exponential growth occurs when a quantity increases at a rate proportional to its current value, resulting in a rapid increase over time. The graph of an exponential growth function rises steeply.

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What does it mean for a function to exhibit exponential decay?

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Exponential decay occurs when a quantity decreases at a rate proportional to its current value, leading to a rapid decrease over time. The graph of an exponential decay function falls steeply.

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What is the y-intercept of an exponential function?

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The y-intercept of an exponential function is the value of the function when x = 0. For f(x) = a * b^x, the y-intercept is f(0) = a.

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What is the range of an exponential growth function?

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The range of an exponential growth function is (k, ∞) where k is the horizontal asymptote.

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What is the range of an exponential decay function?

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The range of an exponential decay function is (0, k) where k is the horizontal asymptote.

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How do you determine the horizontal asymptote of the function f(x) = 2 * 3^x + 2?

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The horizontal asymptote is y = 2, which is the value added to the exponential term.

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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 f(x) = a * b^x, where 'a' is a constant, 'b' is the base, and 'x' is the exponent.

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What is the effect of changing the base 'b' in an exponential function?

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Changing the base 'b' affects the rate of growth or decay. If b > 1, the function exhibits growth; if 0 < b < 1, it exhibits decay.