To describe the transformations that produce the graph of \( g(x) = 5^{-x} + 2 \) from the parent function \( f(x) = 5^x \), let's analyze the transformations step by step:
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Reflection over the x-axis: The function \( g(x) \) can be rewritten as \( g(x) = \frac{1}{5^x} + 2 \). The term \( 5^{-x} \) indicates a reflection of \( f(x) \) over the y-axis, but we'll be careful here. A negative exponent reflects the graph horizontally; therefore, this transformation does not involve reflection over the x-axis.
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Reflection over the y-axis: The transformation from \( f(x) = 5^x \) to \( g(x) = 5^{-x} \) is indeed a reflection over the y-axis.
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Horizontal shift to the left 2 units: There is no horizontal shift in this transformation.
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Horizontal shift to the right 2 units: There is no horizontal shift in this transformation.
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Vertical shift up 2 units: The \( +2 \) in \( g(x) = 5^{-x} + 2 \) indicates a vertical shift upward by 2 units.
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Vertical shift down 2 units: There is no vertical shift downward in this transformation.
Therefore, the applicable transformations are:
- Reflection over the y-axis
- Vertical shift up 2 units
This means that the correct responses are:
- Reflection over the y-axis
- Vertical shift up 2 units