Average Rate of Change Exponential Functions

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What is the average rate of change of a function over an interval?

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The average rate of change of a function over an interval [a, b] is calculated as (f(b) - f(a)) / (b - a). It represents the change in the function's value divided by the change in the input value.

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CCSS.HSF.IF.B.6

CCSS.8.F.B.4

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How do you calculate the average rate of change for an exponential function?

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To calculate the average rate of change for an exponential function f(x) = a * b^x over an interval [c, d], use the formula: (f(d) - f(c)) / (d - c). Substitute the values of f(d) and f(c) using the exponential function.

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CCSS.HSF.IF.B.6

CCSS.8.F.B.4

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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 a positive real number, and x is the exponent. The base b determines the growth or decay rate.

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What does it mean for a function to have a negative average rate of change?

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A negative average rate of change indicates that the function's value decreases over the specified interval. This means that as the input increases, the output decreases.

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CCSS.HSF.IF.B.6

CCSS.8.F.B.4

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

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The base in an exponential function determines the growth or decay rate. 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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CCSS.HSF-IF.C.8B

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How can you interpret the average rate of change in a real-world context?

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In a real-world context, the average rate of change can represent how a quantity changes over time, such as the depreciation of a car's value or the increase in population.

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CCSS.HSF.IF.B.6

CCSS.8.F.B.4

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What is the average rate of change of the function f(x) = 3(2)^x from x = 1 to x = 5?

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To find the average rate of change, calculate f(5) and f(1): f(5) = 3(2)^5 = 96, f(1) = 3(2)^1 = 6. The average rate of change is (96 - 6) / (5 - 1) = 22.5.

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CCSS.8.F.B.4

CCSS.HSF.IF.B.6

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How do you find the average rate of change for a function defined by a table of values?

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To find the average rate of change from a table of values, select two points (x1, y1) and (x2, y2) and use the formula: (y2 - y1) / (x2 - x1).

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CCSS.8.F.B.4

CCSS.HSF.IF.B.6

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What is the average rate of change of a car's value modeled by f(x) = 22,000(0.9)^x over 4 years?

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Calculate f(0) = 22,000 and f(4) = 22,000(0.9)^4. The average rate of change is (f(4) - f(0)) / (4 - 0) = -1,891.45/year.

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CCSS.8.F.B.4

CCSS.HSF.IF.B.6

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What is the average rate of change from day 15 to day 20 if the function is decreasing?

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If the function is decreasing, the average rate of change will be negative, indicating a loss in value or quantity over that interval.

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CCSS.8.F.B.4

CCSS.HSF.IF.B.6