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To find the value of \( k \) that transforms \( f \) into \( g \) by shifting the line vertically, let’s examine the positions of \( f \) and \( g \) on the coordinate plane.

The equation of the transformation is \( g(x) = f(x) + k \), where \( k \) represents the vertical shift.

1. **Determine the slope and y-intercept of line \( f \):**  
   Observing line \( f \), it passes through points like \((-4, -4)\) and \((0, -2)\), which gives a slope of:
   \[
   \text{slope} = \frac{-2 - (-4)}{0 - (-4)} = \frac{2}{4} = \frac{1}{2}
   \]
   This makes the equation of line \( f \):
   \[
   f(x) = \frac{1}{2}x - 2
   \]

2. **Equation of line \( g \):**  
   Line \( g \) is parallel to \( f \), which means it has the same slope of \( \frac{1}{2} \). The line \( g \) intersects the y-axis at \( y = 2 \). Therefore, the equation of \( g \) is:
   \[
   g(x) = \frac{1}{2}x + 2
   \]

3. **Determine the value of \( k \):**  
   Since \( g(x) = f(x) + k \), we substitute \( f(x) = \frac{1}{2}x - 2 \) and \( g(x) = \frac{1}{2}x + 2 \):
   \[
   \frac{1}{2}x + 2 = \frac{1}{2}x - 2 + k
   \]
   Solving for \( k \):
   \[
   2 = -2 + k \Rightarrow k = 4
   \]

Thus, the value of \( k \) that transforms \( f \) into \( g \) is \( \boxed{4} \).

Would you like more details or have any questions?

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### Related Questions
1. What would happen if \( k \) were negative? How would that affect the transformation?
2. How would you find \( k \) if the two lines weren’t parallel?
3. Can transformations be both horizontal and vertical? If so, how would they affect the equations?
4. What if the transformation included a scaling factor in addition to \( k \)? How would you find \( k \) then?
5. How would you describe a transformation that reflects \( f(x) \) over the x-axis?

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**Tip:** For linear transformations, parallel lines indicate only a vertical or horizontal shift, while changes in slope indicate additional transformations like rotations or scalings.

Identify a value of k that transforms f into g, where g(x) = f(x) + k.

Learn how to find the value of k that shifts a linear function f(x) to g(x) using vertical transformations. This problem involves understanding the properties of parallel lines and applying the equation of a line. Ideal for students in Grades 8-10.

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