Write and Evaluate 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. It represents growth or decay at a constant percentage rate.

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How do you evaluate an exponential function?

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To evaluate an exponential function, substitute the given value of 'x' into the function and calculate the result. For example, for f(x) = 5^x, to find f(4), calculate 5^4 = 625.

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What does the base of an exponential function represent?

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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 the formula for exponential growth?

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The formula for exponential growth is f(t) = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate (as a decimal), and 't' is the time period.

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What is the formula for exponential decay?

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The formula for exponential decay is f(t) = a(1 - r)^t, where 'a' is the initial amount, 'r' is the decay rate (as a decimal), and 't' is the time period.

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How do you identify an exponential function from a table of values?

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An exponential function can be identified from a table of values if the ratio of consecutive outputs (y-values) is constant when the inputs (x-values) are increased by the same amount.

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What is the difference between linear and exponential growth?

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Linear growth increases by a constant amount over equal intervals, while exponential growth increases by a constant percentage over equal intervals, leading to faster growth as time progresses.

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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 the function when x = 0, which represents the initial amount before any growth or decay occurs.

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How do you graph an exponential function?

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To graph an exponential function, plot points for various values of x, noting that the graph will rise rapidly for positive x (if the base > 1) or approach zero for negative x.

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What is the exponential growth model used for?

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The exponential growth model is used to describe situations where a quantity increases at a rate proportional to its current value, such as population growth, compound interest, and certain biological processes.