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 y = a(b^x), where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. It shows rapid growth or decay.

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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 (if greater than 1) or decay (if between 0 and 1) rate of the function.

Tags

CCSS.HSF-IF.C.8B

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How do you identify exponential growth?

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Exponential growth occurs when the base of the exponential function is greater than 1 (e.g., y = a(1 + r)^x, where r is the growth rate).

Tags

CCSS.HSF-IF.C.8B

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How do you identify exponential decay?

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Exponential decay occurs when the base of the exponential function is between 0 and 1 (e.g., y = a(1 - r)^x, where r is the decay rate).

Tags

CCSS.HSF-IF.C.8B

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What is the formula for calculating the amount after 't' years in exponential growth?

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The formula is A = P(1 + r)^t, where A is the amount after time t, P is the initial amount, r is the growth rate, and t is the time in years.

Front

What is the formula for calculating the amount after 't' years in exponential decay?

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The formula is A = P(1 - r)^t, where A is the amount after time t, P is the initial amount, r is the decay rate, and t is the time in years.

Front

If a quantity doubles every year, what is the growth factor?

Back

The growth factor is 2, meaning the function can be modeled as y = a(2^x).

Tags

CCSS.HSF-IF.C.8B

Front

What does the initial amount in an exponential function represent?

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The initial amount represents the value of the function when x = 0, indicating the starting point of the growth or decay.

Tags

CCSS.HSF.LE.B.5

Front

How can you determine the percentage increase from an exponential function?

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The percentage increase can be found by subtracting 1 from the base of the function and multiplying by 100. For example, if the function is y = a(1 + r)^x, then the percentage increase is r * 100.

Tags

CCSS.HSF-IF.C.8B

Front

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

Back

The exponent indicates the number of time periods that have passed, affecting the growth or decay of the function.

Tags

CCSS.HSF-IF.C.8B