To analyze the function \( f(x) = 8|8x - 3| - 4 \), we'll break down the transformations applied to the base function \( |x| \).
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Vertical scaling: The coefficient of \( 8 \) in front of the absolute value indicates a vertical stretch of factor 8.
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Horizontal translation: The term \( 8x - 3 \) can be rewritten to identify the horizontal transformation. This can be expressed as: \[ 8(x - \frac{3}{8}) \] This shows that the function is translated \( \frac{3}{8} \) units to the right.
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Vertical translation: The \( -4 \) at the end indicates a downward translation of 4 units.
Now looking at the options provided:
- a) A reflection on the x-axis. (Incorrect) - There is no negative sign in front of the function.
- b) A horizontal stretch by the factor 6. (Incorrect) - This is not correct.
- c) A horizontal translation 2 units to the right. (Incorrect) - The translation is \( \frac{3}{8} \), not 2 units.
- d) A vertical translation 2 units down. (Incorrect) - It translates down 4 units.
None of the options match accurately what we found. Based on the breakdown:
- Correct analysis: Vertical stretch of 8, horizontal translation \( \frac{3}{8} \) units to the right, and a vertical translation down 4 units.
If I had to pick the closest, option d) conveys a vertical translation, but the distance mentioned is not correct. Therefore, none match correctly.