To describe the transformations that result in the function \( h(x) = f(-x) + 5 \), we need to break down the transformations applied to the original function \( f(x) \).
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Reflection across the y-axis: The term \( f(-x) \) indicates that the function \( f(x) \) is reflected across the y-axis. This means that for every point \( (a, f(a)) \) on the original graph of \( f(x) \), the new point \( (-a, f(a)) \) will be on the graph of \( f(-x) \).
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Vertical Shift: The term \( +5 \) indicates that the entire graph of \( f(-x) \) is then shifted vertically upwards by 5 units. This means that every point on the graph of \( f(-x) \) will have its y-coordinate increased by 5.
In summary, the transformations that result in \( h(x) = f(-x) + 5 \) are:
- Reflection of \( f(x) \) across the y-axis (producing \( f(-x) \)).
- Vertical shift upwards by 5 units (producing \( h(x) = f(-x) + 5 \)).