To analyze the change from the original function \( f(x) = |x| + 1 \) to the final function \( f(x) = \frac{1}{2}|x| + \frac{1}{2} \), we can compare the coefficients of the absolute value terms in both functions.
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Original Function: \( f(x) = |x| + 1 \)
- The coefficient of \( |x| \) is \( 1 \).
- The function has a vertical shift of \( +1 \).
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Final Function: \( f(x) = \frac{1}{2}|x| + \frac{1}{2} \)
- The coefficient of \( |x| \) is \( \frac{1}{2} \).
- The function has a vertical shift of \( +\frac{1}{2} \).
The change in the coefficient of \( |x| \) from \( 1 \) to \( \frac{1}{2} \) indicates a vertical compression. Specifically, the graph of the final function will be "squished" vertically towards the x-axis compared to the original function.
Additionally, the vertical shift changes from \( +1 \) to \( +\frac{1}{2} \), which means the entire function moves downwards.
Overall, the most significant change affecting the shape of the function, aside from the vertical shift, is the vertical compression.
Thus, the best option to describe the change from the original to the final function is: A vertical compression.