Unit 6 Review: 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. It describes growth or decay processes.

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What does it mean for a population to increase at a constant rate?

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A constant rate of increase means that the population grows by the same percentage each year. For example, if a population increases by 5% annually, it will grow by 5% of its current size each year.

Front

How do you calculate the future population using exponential growth?

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To calculate future population, use the formula: P = P0 * (1 + r)^t, where P0 is the initial population, r is the growth rate (as a decimal), and t is the number of years.

Front

If a population is 725,000 and grows at 5.2% per year, what is the growth rate in decimal form?

Back

The growth rate in decimal form is 0.052.

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What is the formula to find the population after a certain number of years?

Back

The formula is P = P0 * (1 + r)^t, where P0 is the initial population, r is the growth rate, and t is the number of years.

Front

Calculate the population after 5 years if the initial population is 725,000 and the growth rate is 5.2%.

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P = 725000 * (1 + 0.052)^5 = 934150.

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

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The base 'b' determines the rate of growth or decay. If b > 1, the function represents exponential growth; if 0 < b < 1, it represents exponential decay.

Front

What is the difference between linear and exponential growth?

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Linear growth increases by a constant amount over time, while exponential growth increases by a constant percentage, leading to much larger values over time.

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What is a real-world example of exponential growth?

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A real-world example of exponential growth is the spread of a virus, where the number of infected individuals can double over a fixed period.

Front

What does 't' represent in the exponential growth formula?

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In the formula P = P0 * (1 + r)^t, 't' represents the number of time periods (usually years) over which the growth occurs.