To analyze the transformation applied to the function \( f(x) = x \) to obtain \( g(x) = -f(x+2) \), we can break down the transformation step by step:
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Start with \( f(x) = x \): This is the original linear function, which is a straight line through the origin with a slope of 1.
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Horizontal Shift: The expression \( f(x + 2) \) represents a horizontal shift to the left by 2 units. This changes the function to \( f(x + 2) = x + 2 \).
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Reflection: The negative sign outside the function, \( -f(x + 2) \), reflects the function over the x-axis. This means that the output (y-value) is inverted. So, we have \( g(x) = -(x + 2) = -x - 2 \).
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Vertical Shift: The reflection does not introduce an upward translation; in fact, it moves the function downward because of the negative sign applied to the entire function \( f \).
Considering the transformations:
- The translation is to the left by 2 units due to \( (x + 2) \).
- The reflection is over the x-axis due to the negative sign in front of the function.
From the provided options, the correct description of the change in the graph of \( f(x) = x \) to \( g(x) = -f(x + 2) \) is:
A negative sign outside the function notation shows reflection over the x-axis. There is also a translation to the left on the x-axis by 2 units.
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