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

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How do you identify the base in an exponential function?

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The base 'b' in an exponential function y = ab^x is the number that is raised to the power of 'x'. It determines the growth or decay rate of the function.

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What does it mean for a function to triple every day?

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If a function triples every day, it means that the value of the function is multiplied by 3 for each unit increase in the independent variable (usually time). This can be represented as y = a(3^x), where 'a' is the initial amount.

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How do you write an exponential function from a word problem?

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To write an exponential function from a word problem, identify the initial value and the growth/decay factor, then use the form y = a(b^x), where 'a' is the initial value and 'b' is the growth/decay factor.

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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 if a function is exponential from a table of values?

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A function is exponential if the ratio of consecutive outputs (y-values) is constant when the inputs (x-values) increase by the same amount.

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

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The initial value 'a' represents the starting amount or value of the function when x = 0.

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How do you graph an exponential function?

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To graph an exponential function, plot the initial value at x = 0, then use the base to determine the growth or decay for other values of x, creating a smooth curve.

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What is the effect of changing the base 'b' in an exponential function?

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Changing the base 'b' affects the rate of growth or decay; a base greater than 1 indicates growth, while a base between 0 and 1 indicates decay.