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?

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?

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