To determine the line of reflection between the function \( f(x) = x - 2 \) and its reflection, we can analyze how the original function and its reflection relate to each other.
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The original function is: \[ f(x) = x - 2 \] It is a linear function with a slope of 1 and a y-intercept of -2.
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The derivative (slope of the tangent) is given by: \[ f'(x) = -2 - x \] This is also a linear function, meaning the slope of the tangent line at any point depends on x.
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To determine the line of reflection, let's find the image of a point on the function when reflected across the line. The reflection would generally maintain a perpendicular relationship to the line of reflection.
We can test the options given:
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Option #1: the x-axis
- Reflecting over the x-axis would change the sign of \( y \). Therefore, reflecting \( f(x) \) would change it to \( f'(x) = -(x - 2) = -x + 2 \), which does not match \( f'(x) = -2 - x \).
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Option #2: the y-axis
- Reflecting over the y-axis would change the sign of \( x \). The function would become \( f(-x)=-x - 2 \), which does not match the derivative.
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Option #3: y = x
- Reflecting over the line \( y = x \) exchanges \( x \) and \( y \) coordinates. This means \( f(x) = y \) would become \( f^{-1}(y) = x + 2 \). However, this does not yield the original function.
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Option #4: y = -x
- Reflecting over \( y = -x \) would transform \( (x,y) \) to \( (-y,-x) \), effectively changing the function to \( f(-y) = -x + 2 \). This could match a relationship based on symmetry.
After considering the properties of the function and how derivatives relate in reflection, it appears that the line \( y = x \) serves as the transformation between the function and its reflection correctly.
Thus, the line of reflection is Option #3: y = x.
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