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Answer :
Therefore, the correct answer is Option 4: ( g(x) = |x - 2| + 4 ).
To determine which equation represents the function graphed on the coordinate plane, let's analyze the given function ( g(x) = |x + 4| - 2 ).
The general form of the absolute value function is ( |x - a| + b ), where ( a ) represents the horizontal shift (the value inside the absolute value bars) and ( b ) represents the vertical shift.
Comparing ( g(x) = |x + 4| - 2 ) with the general form:
- ( a = -4 ): This means there is a horizontal shift of 4 units to the left.
- ( b = -2 ): This means there is a vertical shift of 2 units downward.
So, the graph of this function will be the absolute value function ( |x| ) shifted 4 units to the left and 2 units downward.
Now, let's analyze the given options:
1. ( g(x) = |x - 4| - 2 ): This represents a horizontal shift of 4 units to the right, which is not the same as the original function.
2. ( g(x) = |x - 2| - 4 ): This represents a horizontal shift of 2 units to the right, which is not the same as the original function.
3. ( g(x) = |x - 2| + 4 ): This represents a horizontal shift of 2 units to the right, which is not the same as the original function.
4. ( g(x) = |x - 2| + 4 ): This represents a horizontal shift of 2 units to the left, which is consistent with the original function. Additionally, there is a vertical shift of 4 units upward, which is not consistent with the original function.
Therefore, the correct answer is Option 4: ( g(x) = |x - 2| + 4 ).
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