We are given two functions to graph and compare:

1. $f(x) = |x|$, which is the absolute value function.
2. $g(x) = |x| - 5$, which is a transformation of the absolute value function.

Step 1: Graph of $f(x) = |x|$

The graph of $f(x) = |x|$ is a V-shaped graph with its vertex at the origin (0,0). It has two linear pieces:

• For $x \geq 0$, $f(x) = x$, which is a line with a slope of 1.
• For $x < 0$, $f(x) = -x$, which is a line with a slope of -1.

Step 2: Graph of $g(x) = |x| - 5$

The function $g(x) = |x| - 5$ is obtained by shifting the graph of $f(x)$ downward by 5 units. This is because the "-5" outside the absolute value shifts the entire graph vertically.

Step 3: Relationship between the Graphs

• The graph of $g(x)$ is the same V-shape as $f(x)$, but it is shifted down by 5 units.
• The vertex of $g(x)$ is at $(0, -5)$, whereas the vertex of $f(x)$ is at $(0, 0)$.
• The slopes of the two arms of the V-shape for both functions remain the same as in $f(x)$.

I can also generate the graph if needed. Would you like a visual representation of these functions?

Relative Questions:

1. What other transformations can we apply to absolute value functions?
2. How does shifting a graph horizontally differ from shifting it vertically?
3. How would the graph of $g(x) = |x| + 5$ compare to $f(x) = |x|$?
4. How does the slope of the arms of the V-shaped graph affect its appearance?
5. What happens to the graph of $f(x) = |x|$ if we reflect it across the x-axis?

Tip: Vertical shifts in a function do not affect the slope or the horizontal position of the graph, only the vertical positioning.