Question

What is the value of k for the horizontal shift that occurred between the graph of the original function y=f(x)
and the graph of the translated function y=f(x−k)
Original Vertex point: (0,0) Other Points: (-1, 1), (1, 1), (-2, 4), (2, 4) (-3, 9), (3, 9)
Translated Vertex Point: (-1, 0), Other Points: (-2, 1), (0, 1), (-3, 4), (1, 4), (-4, 9), (2,9)

Answer 1

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.

1. **Original Vertex Point:** \( (0, 0) \)
2. **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 \( \boxed{1} \).