Review of exponential functions

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What is an exponential function?

Back

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.

Front

What does the base of an exponential function represent?

Back

The base of an exponential function determines the rate of growth or decay. If the base is greater than 1, the function represents exponential growth; if the base is between 0 and 1, it represents exponential decay.

Front

How do you convert a percentage to a decimal?

Back

To convert a percentage to a decimal, divide the percentage by 100. For example, 5% becomes 0.05.

Front

What is the formula for continuous growth?

Back

The formula for continuous growth is A = Pe^(rt), where A is the amount after time t, P is the initial amount, r is the growth rate, and e is Euler's number (approximately 2.718).

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How do you model a situation with exponential growth?

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To model exponential growth, use the formula y = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate (as a decimal), and 't' is time.

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What is the formula for exponential decay?

Back

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

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If a population of bacteria doubles every 3 hours, how would you express this mathematically?

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You can express this as y = a(2^(t/3)), where 'a' is the initial population and 't' is time in hours.

Front

What is the growth factor in the equation y = 6(1.14)^x?

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The growth factor is 1.14, which indicates a 14% increase in the quantity for each unit increase in x.

Front

How do you find the amount after a certain time with a given growth rate?

Back

Use the formula 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 period.

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What does it mean if a population decreases by 30% each year?

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It means that each year, 30% of the population is lost, and 70% remains. This can be modeled as y = a(0.7)^t.