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 does the base 'b' in an exponential function represent?

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The base 'b' represents the growth factor in an exponential function. If b > 1, the function models exponential growth; if 0 < b < 1, it models exponential decay.

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How do you identify the initial value in an exponential function?

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The initial value is represented by 'a' in the function f(x) = a * b^x, which is the value of the function when x = 0.

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What is the general form of an exponential growth equation?

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The general form of an exponential growth equation is y = a * b^x, where 'a' is the initial amount and 'b' is the growth factor.

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How can you find the value of 'a' in an exponential function from a table?

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You can find the value of 'a' by looking at the value of y when x = 0 in the table.

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

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The x-intercept is the point where the graph crosses the x-axis, which occurs when y = 0. In exponential functions, this point is not typically reached.

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How do you determine the growth factor from a table of values?

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The growth factor can be determined by finding the ratio of consecutive y-values in the table.

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What does it mean if an exponential function has a base of 1.67?

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A base of 1.67 indicates that the quantity is increasing by a factor of 1.67 for each unit increase in x.

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How can you use an exponential function to model real-world scenarios?

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Exponential functions can model scenarios such as population growth, radioactive decay, and compound interest, where quantities grow or shrink at a constant percentage rate.

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What is the formula to calculate the time it takes for an exponential function to reach a certain value?

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To find the time, you can use the equation y = a * b^x and solve for x when y is the desired value.