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The graph shows the function \( f(x) = 0.5x - 2 \) and a transformed graph \( g(x) \). The equation given is \( g(x) = f(x) + k \), indicating a vertical shift by \( k \) units.

From the graph:

- The blue line represents \( f(x) \).
- The red line represents \( g(x) \).

To determine the value of \( k \), we compare the vertical shift between the two graphs. Both lines have the same slope (parallel), so the transformation is purely vertical. The red line is shifted upward from the blue line by comparing the y-intercepts. The y-intercept of \( f(x) \) is \( -2 \), while the y-intercept of \( g(x) \) is \( 0 \). Therefore, the vertical shift is 2 units up, meaning \( k = 2 \).

Thus, the correct choice is:

- **k is equal to 2. The graph of \( g(x) \) is translated 2 unit(s) up compared to the graph of \( f(x) \).**

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do vertical shifts affect the equation of a function?
2. How would the graph look if the shift was downward instead of upward?
3. How does the slope of a line affect its transformations?
4. What other types of transformations can be applied to a linear function?
5. How can we find the value of \( k \) if the transformation involved both a shift and a compression?

**Tip:** When transforming functions, vertical shifts affect the y-values but do not change the slope.

What value of k transforms the graph of f(x) = 0.5x - 2 into the graph g? Use the graph and the given equation to describe the transformation: g(x) = f(x) + k.

Learn how to find the value of k for the graph transformation of the function f(x) = 0.5x - 2 into g(x) = f(x) + k. Understand vertical shifts and how to compare y-intercepts in linear functions. Ideal for students in Grades 8-10.

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