Exponential Functions Word Problems

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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 represents growth or decay processes.

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What does the term 'growth factor' mean in exponential functions?

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The growth factor is the base 'b' in the exponential function y = a(b)^x. It indicates how much the quantity increases for each unit increase in 'x'.

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What is the formula to calculate the future value of an investment with exponential growth?

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The future value can be calculated using the formula: FV = P(1 + r)^t, where P is the principal amount, r is the growth rate, and t is the time in years.

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How do you determine the initial value in an exponential growth problem?

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The initial value is represented by 'a' in the equation y = a(b)^x. It is the value of the quantity at time x = 0.

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What is the difference between growth rate and growth factor?

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The growth rate (r) is the percentage increase per time period, while the growth factor (b) is the multiplier used in the exponential function, calculated as b = 1 + r.

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If a quantity decreases by a certain percentage, how is the growth factor calculated?

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The growth factor for a decrease is calculated as b = 1 - (decrease percentage). For example, a 25% decrease gives a growth factor of b = 1 - 0.25 = 0.75.

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What is the equation for exponential decay?

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The equation for exponential decay is y = a(b)^x, where 'b' is less than 1. It represents a decrease in quantity over time.

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How do you find the value of an investment after a certain number of years with exponential growth?

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To find the value, use the formula: V = P(1 + r)^t, where V is the future value, P is the initial value, r is the growth rate, and t is the number of years.

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

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The exponent 'x' in an exponential function indicates the number of time periods that have passed, affecting the growth or decay of the quantity.

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How can you express a percentage increase as a growth factor?

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To express a percentage increase as a growth factor, convert the percentage to a decimal and add 1. For example, a 15% increase becomes 1 + 0.15 = 1.15.