Question

Identify the value of k that would correctly characterize the horizontal shift in the graph if the line on the left represents y = f(x) and the line to the right represents y = f(x -k)
Left has values of (15,45), (20,40), (25,35)
Right values (5,45),(10,40), (15,35)

Answer 1

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:

1. The point \( (15, 45) \) shifts to \( (5, 45) \):
\[ 15 - k = 5 \]
\[ k = 15 - 5 = 10 \]

2. The point \( (20, 40) \) shifts to \( (10, 40) \):
\[ 20 - k = 10 \]
\[ k = 20 - 10 = 10 \]

3. 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 \( \boxed{10} \).