The graph above shows both functions:

• The dashed blue line represents the parent function $f(x) = |x|$, which is a V-shaped graph with its vertex at $(0, 0)$.
• The red line represents the transformed function $y = |x - 3| - 3$, which has been shifted 3 units to the right and 3 units downward, with its vertex at $(3, -3)$.

These shifts confirm the transformations described earlier.

Do you have any questions or would you like more details?

Here are 5 related questions to deepen your understanding:

1. How would the graph change if the function was $y = |x + 3| - 3$?
2. What happens if we add a coefficient to the absolute value, like $y = 2|x - 3| - 3$?
3. Can you describe the effect of $y = |x - 3| + 3$ on the graph?
4. How would a reflection over the x-axis look in an absolute value function?
5. What is the effect of a vertical stretch or compression on $y = |x|$?

Tip: Horizontal shifts are counterintuitive—if you subtract a value from $x$, the graph shifts to the right.