To determine which absolute value equation creates the lower function in the graph, we need to analyze each option and consider their implications on the y-values of the function based on the x-values.
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y = 2|x| - 2
- This function has a vertex at (0, -2). The slope increases as we move away from the vertex, resulting in a V-shape.
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y = (1/2)|x| - 4
- This function has a vertex at (0, -4). It opens wider (less steep) than the first function, which means that for large |x| values, the y-values will be higher than other options but stays lower than some functions for smaller values of x.
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y = |x - 4| - 1
- This function has a vertex at (4, -1). It opens up and reaches (-1) at x=4. Thus for smaller x this can still be higher than y = |x| - 4.
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y = |x| - 4
- This function has a vertex at (0, -4). It reaches its minimum value there, making it the lowest function when x is near 0.
Comparative Analysis:
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At \(x=0\):
- \(y = 2|0| - 2 = -2\)
- \(y = (1/2)|0| - 4 = -4\)
- \(y = |0 - 4| - 1 = 3\)
- \(y = |0| - 4 = -4\)
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Therefore, for x = 0, \(y = (1/2)|0| - 4\) and \(y = |0| - 4\) both give us -4, which is lower than the other functions which give -2 and 3 at x = 0.
As we further evaluate, we see \(y = |x| - 4\) has a steep rise compared to y = 2|x| - 2 which increases more quickly.
Conclusion: The equation that creates the lowest function in the graph is y = |x| - 4.
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