Linear Vs Exponential Review

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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 (a positive real number), and 'x' is the exponent. It represents a situation where a quantity grows or decays at a rate proportional to its current value.

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What is a linear function?

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A linear function is a polynomial function of degree one, which can be expressed in the form f(x) = mx + b, where 'm' is the slope and 'b' is the y-intercept. It represents a constant rate of change.

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How do you identify an exponential growth pattern in a table of values?

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In a table of values, an exponential growth pattern is identified by a consistent multiplicative increase between consecutive values, rather than a consistent additive increase.

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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 equal intervals, while exponential growth increases by a constant percentage over equal intervals, leading to much faster growth as time progresses.

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What is the general form of an exponential growth function?

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The general form of an exponential growth function is y = a * b^x, where 'a' is the initial value, 'b' is the growth factor (b > 1), and 'x' is the time or number of intervals.

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What does the base of an exponential function represent?

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

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How can you determine if a function is linear or exponential from a word problem?

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To determine if a function is linear or exponential from a word problem, look for keywords indicating constant addition (linear) or constant percentage increase (exponential). Also, analyze the context for growth patterns.

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What is the formula for exponential decay?

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The formula for exponential decay is y = a * e^(-kt), where 'a' is the initial amount, 'e' is the base of the natural logarithm, 'k' is the decay constant, and 't' is time.

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What is a common real-world example of exponential growth?

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A common real-world example of exponential growth is population growth, where the number of individuals increases at a rate proportional to the current population.

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What is a common real-world example of linear growth?

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A common real-world example of linear growth is saving money at a fixed interest rate, where the amount saved increases by a constant amount over time.