To find the value of \( k \) that characterizes the horizontal shift in the graph of \( y = f(x) \) to the graph of \( y = f(x - k) \), we need to analyze the given points on both graphs.
The original graph points (on the left) are:
- \( (15, 45) \)
- \( (20, 40) \)
- \( (25, 35) \)
The shifted graph points (on the right) are:
- \( (5, 45) \)
- \( (10, 40) \)
- \( (15, 35) \)
The transformation from \( y = f(x) \) to \( y = f(x - k) \) involves a horizontal shift by \( k \) units to the right. Thus, a point \( (a, b) \) from \( y = f(x) \) will move to \( (a + k, b) \) in \( y = f(x - k) \).
Now, let's compare points:
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The point \( (15, 45) \) shifts to \( (5, 45) \): \[ 15 - k = 5 \] \[ k = 15 - 5 = 10 \]
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The point \( (20, 40) \) shifts to \( (10, 40) \): \[ 20 - k = 10 \] \[ k = 20 - 10 = 10 \]
-
The point \( (25, 35) \) shifts to \( (15, 35) \): \[ 25 - k = 15 \] \[ k = 25 - 15 = 10 \]
In all cases, we find that \( k = 10 \).
Thus, the value of \( k \) that correctly characterizes the horizontal shift is \( \boxed{10} \).