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 f(x) = a(b^x), where 'a' is a constant, 'b' is the base of the exponential (b > 0), and 'x' is the exponent.

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What does it mean for a quantity to grow exponentially?

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Exponential growth occurs when a quantity increases by a fixed percentage over a consistent time period, resulting in a rapid increase.

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What does it mean for a quantity to decay exponentially?

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Exponential decay occurs when a quantity decreases by a fixed percentage over a consistent time period, leading to a rapid decrease.

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How do you write the equation for exponential decay?

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The equation for exponential decay can be written as y = a(1 - r)^x, where 'a' is the initial amount, 'r' is the decay rate, and 'x' is time.

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How do you write the equation for exponential growth?

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The equation for exponential growth can be written as y = a(1 + r)^x, where 'a' is the initial amount, 'r' is the growth rate, and 'x' is time.

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What is the y-intercept of an exponential function?

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The y-intercept of an exponential function is the initial value, which is the value of the function when x = 0.

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How do you calculate the future value of an investment with exponential growth?

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To calculate the future value, use the formula FV = P(1 + r)^t, where FV is the future value, P is the principal amount, r is the growth rate, and t is the time in years.

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What is the formula for calculating the population after a certain time with exponential growth?

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The formula is P(t) = P0 * e^(rt), where P(t) is the population at time t, P0 is the initial population, r is the growth rate, and e is Euler's number.

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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 the same period.

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

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The base determines the rate of growth or decay; a base greater than 1 indicates growth, while a base between 0 and 1 indicates decay.