Week 16 KA: Identifying and Evaluating Exponential Function

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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. The function grows or decays at a constant percentage rate.

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How can you identify an exponential function from a table of values?

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An exponential function can be identified if, as 'x' increases by 1, 'y' is multiplied by a constant factor. This means the ratio of consecutive 'y' values remains constant.

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What is a constant factor in the context of exponential functions?

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A constant factor is the fixed number by which 'y' is multiplied as 'x' increases by 1 in an exponential function. For example, if y is multiplied by 3 as x increases by 1, the constant factor is 3.

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What is a linear function?

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A linear function is a function that creates a straight line when graphed. It can be expressed in the form f(x) = mx + b, where 'm' is the slope and 'b' is the y-intercept.

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How can you differentiate between exponential and linear functions?

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Exponential functions grow or decay at a constant percentage rate, while linear functions grow at a constant rate. In a table, exponential functions show a constant factor, while linear functions show a constant difference.

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What does it mean for a function to have a constant rate of change?

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A constant rate of change means that the difference between consecutive 'y' values remains the same as 'x' increases. This is characteristic of linear functions.

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

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The base in an exponential function determines the rate of growth or decay. If the base is greater than 1, the function grows; if the base is between 0 and 1, the function decays.

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Can an exponential function ever be negative?

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No, an exponential function is always positive if 'a' (the initial value) is positive and 'b' (the base) is positive. The output of an exponential function cannot be negative.

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What happens to the graph of an exponential function as 'x' approaches infinity?

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As 'x' approaches infinity, the graph of an exponential function with a base greater than 1 will rise steeply towards infinity, while a base between 0 and 1 will approach zero.

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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 value of 'y' when 'x' is 0. It is equal to 'a' in the function f(x) = a * b^x.