Writing 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 y = a(b^x), where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent.

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What does the base of an exponential function represent?

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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.

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How do you write an exponential function from a graph?

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To write an exponential function from a graph, identify key points on the graph, determine the base and the initial value, and use the form y = a(b^x) to formulate the equation.

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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(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

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What is the general form of an exponential decay function?

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The general form of an exponential decay function is y = a(1 - r)^t, where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

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How can you determine the equation of an exponential function from a table of values?

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To determine the equation from a table, look for a constant ratio between consecutive y-values as x increases, which indicates the base of the exponential function.

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What is the significance of the initial value in an exponential function?

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The initial value in an exponential function represents the starting amount before any growth or decay occurs.

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How do you model real-world situations using exponential functions?

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Real-world situations can be modeled using exponential functions by identifying the initial value and the growth or decay rate, then applying the appropriate exponential function form.

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What is the formula for compound interest, and how does it relate to exponential functions?

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The formula for compound interest is A = P(1 + r/n)^(nt), where A is the amount, P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. This formula is an example of an exponential function.

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What is the difference between exponential growth and linear growth?

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Exponential growth increases at a rate proportional to its current value, leading to faster increases over time, while linear growth increases by a constant amount.