To determine how far the original function \( f(x) = |x + 1| \) has been shifted horizontally to create the translated function, we need to compare the key points of both functions.
The key points of the original function are:
- Vertex at \((-1, 0)\)
- Other points: \((0, 1)\), \((-2, 1)\), \((1, 2)\), \((-3, 2)\)
The translated function has the following key points:
- Vertex at \((1, 0)\)
- Other points: \((0, 1)\), \((2, 1)\), \((-1, 2)\)
Now, let's analyze the horizontal shift:
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Vertex Shift: The vertex of the original function moved from \((-1, 0)\) to \((1, 0)\). The horizontal shift can be calculated as: \[ \text{Shift} = 1 - (-1) = 1 + 1 = 2 \] This indicates a shift of \(2\) units to the right.
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Validation with Other Points:
- The point \((0, 1)\) remains unchanged in both functions.
- The point \((-2, 1)\) from the original function would correspond to \((0, 1)\) in the translated function.
- The point \((1, 2)\) from the original corresponds to \(((3, 2)\) in the translated function.
Since we verified that the vertex shifts from \((-1, 0)\) to \((1, 0)\) shows a consistent translation, we can conclude that the original function has shifted 2 units to the right.
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