To determine which absolute value equation creates the lower function in the graph, we need to analyze the vertical shifts and the coefficients in front of the absolute value expressions.
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y = |x| - 4: This is a standard absolute value function shifted down by 4 units. The vertex is at (0, -4).
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y = 2|x| - 2: This function is also an absolute value function, but it stretches the graph vertically by a factor of 2 and shifts it down by 2 units. The vertex is at (0, -2).
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y = |x - 4| - 1: This function is an absolute value function shifted to the right by 4 units and down by 1 unit. The vertex is at (4, -1).
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y = 12|x| - 4: This function is stretched vertically by a factor of 12 and shifted down by 4 units. The vertex is at (0, -4).
From these evaluations, we can compare the minimum points (vertices) of the equations:
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y = |x| - 4 has its vertex at (0, -4).
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y = 2|x| - 2 has its vertex at (0, -2).
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y = |x - 4| - 1 has its vertex at (4, -1).
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y = 12|x| - 4 has its vertex at (0, -4).
The lowest point among these vertices is (0, -4), which corresponds to both y = |x| - 4 and y = 12|x| - 4.
Since we're asked to identify which equation can create the lower function, either of these two could fit that description. However, both yield the same vertical position for their vertex at (0, -4) when graphed.
If you have to choose one from the given options, it may depend on other context, such as the scales involved in the graph. But technically, both y = |x| - 4 and y = 12|x| - 4 are valid answers since they both yield the same minimum value for \( y \).
Therefore, the correct response could be:
y = |x| - 4 or y = 12|x| - 4 as both create the lower function, with the vertex at (0, -4). If you have to select only one, typically y = |x| - 4 is the simpler standard form usually expected.
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