To determine the value of \( k \) for the horizontal shift that occurs between the graph of the original function \( y = f(x) \) and the translated function \( y = f(x - k) \), we can analyze the vertex points and other points given.
- Original Vertex Point: \( (0, 0) \)
- Translated Vertex Point: \( (-1, 0) \)
The vertex of the original function \( y = f(x) \) is at \( (0, 0) \), and the vertex of the translated function \( y = f(x - k) \) is at \( (-1, 0) \).
This indicates that the entire graph of the original function has been shifted horizontally. To find \( k \), we observe how far the vertex has moved from the original position to the new position:
- The original vertex \( x \)-coordinate was \( 0 \).
- The new vertex \( x \)-coordinate is \( -1 \).
To shift from \( 0 \) to \( -1 \), we need to find \( k \) such that: \[ x - k = -1 \] substituting \( x = 0 \) (the location of the original vertex), we have: \[ 0 - k = -1 \] Solving for \( k \): \[ -k = -1 \ k = 1 \]
Thus, the value of \( k \) for the horizontal shift is \( k = 1 \).
Verification with Other Points:
To confirm, let's verify with the other points:
- The original point \( (-1, 1) \) translates to \( (-2, 1) \) (since \( -1 - k = -1 - 1 = -2 \)).
- The original point \( (1, 1) \) translates to \( (0, 1) \) (since \( 1 - k = 1 - 1 = 0 \)).
- The original point \( (-2, 4) \) translates to \( (-3, 4) \) (since \( -2 - k = -2 - 1 = -3 \)).
- The original point \( (2, 4) \) translates to \( (1, 4) \) (since \( 2 - k = 2 - 1 = 1 \)).
- The original point \( (-3, 9) \) translates to \( (-4, 9) \) (since \( -3 - k = -3 - 1 = -4 \)).
- The original point \( (3, 9) \) translates to \( (2, 9) \) (since \( 3 - k = 3 - 1 = 2 \)).
All points align perfectly with the translated function.
Therefore, the value of \( k \) is indeed \( \boxed{1} \).
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