Question

Original function f(x) = |x + 1| shape is a v with points (-1,0), (0,1),(-2,1),(1,2),(-3,2)
Translated function
V shape points (1,0), (0,1), (2,1), (-1,2)how far has the original function shifted horizontally

Answer 1

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:

1. **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.

2. **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**.