Characteristics of 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 is the y-intercept of an exponential function?

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The y-intercept of an exponential function is the point where the graph crosses the y-axis, which occurs at (0, a) for the function f(x) = a * b^x.

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What is the range of an exponential function?

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The range of an exponential function f(x) = a * b^x is all real numbers greater than 0 if a > 0, or all real numbers less than 0 if a < 0.

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What does it mean if an exponential function is increasing?

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An exponential function is increasing (exponential growth) if the base 'b' is greater than 1, meaning the function's value increases as x increases.

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What does it mean if an exponential function is decreasing?

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An exponential function is decreasing (exponential decay) if the base 'b' is between 0 and 1, meaning the function's value decreases as x increases.

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What is the general shape of the graph of an exponential growth function?

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The graph of an exponential growth function rises steeply to the right and approaches the horizontal asymptote to the left.

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What is the general shape of the graph of an exponential decay function?

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The graph of an exponential decay function falls steeply to the right and approaches the horizontal asymptote to the left.

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How do you determine if a function is exponential growth or decay from its equation?

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If the base 'b' in the function f(x) = a * b^x is greater than 1, it is exponential growth. If 'b' is between 0 and 1, it is exponential decay.

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

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The base 'b' determines the rate of growth or decay of the function. A larger base results in faster growth, while a base less than 1 results in decay.