To discuss the properties and determine the equation for the horizontal asymptote of an exponential decay function, let's go through the characteristics of such a function.
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General Form: The general form of an exponential decay function can be expressed as: \[ y = a \cdot b^{x} \] where:
- \(a\) is the initial value (when \(x=0\)),
- \(b\) (the base) is a positive number between 0 and 1 (for decay).
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Characteristics:
- The function decreases rapidly at first and then levels off.
- As \(x\) increases, \(y\) approaches 0 but never actually reaches it.
- The graph is continuous and smooth.
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Horizontal Asymptote: The horizontal asymptote of an exponential decay function represents a value that the function approaches as \(x\) approaches positive infinity. For all exponential decay functions, the horizontal asymptote is at: \[ y = 0 \] This means that the function will never actually touch or cross the line \(y = 0\).
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Example with Specific Point (0, -2): If you have the point (0, -2), this suggests the function has been translated or altered. The value at \(x = 0\) is -2 instead of the expected initial value from the standard form. This implies that the function can be adjusted to: \[ y = a \cdot b^{x} - 2 \] where \(a\) is the initial value that gives -2 when \(x = 0\) (i.e., setting \(a \cdot b^0 - 2 = -2\), which means \(a - 2 = -2\) therefore \(a = 0\)). So the function could look something like: \[ y = 0 \cdot b^{x} - 2 \]
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Conclusion: Regardless of the transformations applied, the horizontal asymptote remains at: \[ y = -2 \] This means that as \(x\) increases, the function will approach the line \(y = -2\).
In summary, the properties of the exponential decay function suggest it decreases but approaches a horizontal asymptote of \(y = 0\) unless shifted; in this case, based on the point (0, -2), we can deduce that the horizontal asymptote is at \(y = -2\).