Transformations of Exponential Functions

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What is a vertical translation in exponential functions?

Back

A vertical translation shifts the graph of the function up or down without changing its shape. For example, f(x) = a^x + k translates the graph vertically by k units.

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What is a horizontal translation in exponential functions?

Back

A horizontal translation shifts the graph of the function left or right. For example, f(x) = a^(x - h) translates the graph horizontally by h units.

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What is exponential growth?

Back

Exponential growth occurs when a quantity increases by a constant percentage over time, represented by the function f(x) = a * b^x, where b > 1.

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What is exponential decay?

Back

Exponential decay occurs when a quantity decreases by a constant percentage over time, represented by the function f(x) = a * b^x, where 0 < b < 1.

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What is the range of an exponential function?

Back

The range of an exponential function f(x) = a^x is (0, ∞) if a > 0, meaning y is always greater than 0.

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How does a vertical translation affect the range of an exponential function?

Back

A vertical translation shifts the range of the function. For example, f(x) = a^x + k has a range of (k, ∞) if k > 0.

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What does it mean for a function to have a horizontal asymptote?

Back

A horizontal asymptote is a line that the graph of the function approaches as x approaches infinity or negative infinity. For exponential functions, the horizontal asymptote is typically y = 0.

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What is the effect of a negative exponent in an exponential function?

Back

A negative exponent indicates a reciprocal. For example, a^(-x) = 1/(a^x), which reflects the graph across the y-axis.

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What is the base of an exponential function?

Back

The base of an exponential function is the constant factor that is raised to a variable exponent. It determines the rate of growth or decay.

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How do you identify exponential growth from a table of values?

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Exponential growth can be identified if the ratio of consecutive outputs (y-values) is constant and greater than 1.