The function \( g(x) = -f(x) - 5 \) represents a transformation of the original function \( f(x) \). Let's break down the transformations step-by-step:
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Reflection: The term \( -f(x) \) indicates a reflection of the function \( f(x) \) across the x-axis. This means that all the y-values of \( f(x) \) are multiplied by -1. For example, if \( f(x) \) has a point (x, y), after this transformation it will have the point (x, -y).
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Vertical Shift: The term \( -5 \) indicates a downward shift of the graph by 5 units. This means that after reflecting \( f(x) \) across the x-axis, every point on this new graph is then moved down 5 units. So if you had a point (x, -y) from the first transformation, it will now be at (x, -y - 5).
In summary, the transformation of the function \( f(x) \) to \( g(x) = -f(x) - 5 \) involves:
- Reflecting \( f(x) \) over the x-axis.
- Shifting the resulting graph downward by 5 units.