What value of k would create the function y = f(kx) on the graph if it is a transformed version of the function f(x)= x+2

To find the value of \( k \) for the transformed function \( y = f(kx) \) given the function \( f(x) = x + 2 \), we need to analyze the transformation due to \( k \).

1. Identify the original function: The original function \( f(x) = x + 2 \) is a straight line with a slope of 1 and a y-intercept of 2.

2. Determine the effect of the transformation: The function \( y = f(kx) = kx + 2 \) modifies the slope of the function. The transformed function's slope will be \( k \).

3. Given points for new function: We have some points (-2, -4), (-1, -1), (0, 2), (1, 5), and (2, 8) that lie on the graph of \( f(kx) \).

   Let's check if these points are indeed on the line corresponding to a specific slope and intercept.

4. Calculate the slope between two points: We can take the points (0, 2) and (1, 5): \[ \text{slope} = \frac{5 - 2}{1 - 0} = \frac{3}{1} = 3 \]

   By checking the slope between other pairs of points such as (-1, -1) and (0, 2): \[ \text{slope} = \frac{2 - (-1)}{0 - (-1)} = \frac{3}{1} = 3 \]

   It confirms that this is a consistent slope across the points.

5. Finding \( k \): Since the slope of the transformed function \( f(kx) \) is \( k \), we have: \[ k = 3 \]

6. Conclusion: The value of \( k \) that creates the transformed function \( y = f(kx) \) corresponding to the points given is \( k = 3 \).

Thus, the answer is: \[ \boxed{3} \]