The problem presents two functions, $f(x) = |x|$ and $g(x) = |x| - 5$, and asks to graph them together and describe how the graph of $g(x)$ is related to that of $f(x)$.

1. Function $f(x) = |x|$ is the basic absolute value function. Its graph is a V-shaped curve with the vertex at $(0, 0)$ and slopes of 1 and -1 for positive and negative $x$-values, respectively.

2. Function $g(x) = |x| - 5$ is a vertical translation of the graph of $f(x)$. The effect of subtracting 5 from $f(x)$ is that the entire graph of $f(x)$ is shifted downward by 5 units.

Thus, the graph of $g(x)$ will have the same V-shape but will have its vertex at $(0, -5)$ instead of $(0, 0)$.

Graphing and Relation:

• The graph of $g(x)$ is the same as $f(x)$, but every point on the graph of $f(x)$ is shifted 5 units down to get the graph of $g(x)$.

Would you like to explore the graphs more or have any questions? Here are some related questions for deeper understanding:

1. How would the graph change if the transformation were $g(x) = |x| + 5$?
2. What is the geometric significance of adding/subtracting constants outside the absolute value function?
3. What would happen if we transformed $f(x) = |x|$ horizontally, such as $f(x) = |x - 3|$?
4. How do reflections affect the graph of $f(x) = |x|$, such as $f(x) = -|x|$?
5. Can we combine horizontal and vertical transformations in one function? How would it look?

Tip: Vertical translations modify the $y$-coordinate of all points on the graph by the same amount.