The transformation from the parent function $y = |x|$ to the function $y = |x+4| - 2$ involves three distinct steps:

1. Horizontal Shift:

   • The expression $|x + 4|$ inside the absolute value indicates a horizontal shift.
   • Since it's $x + 4$, this represents a leftward shift of 4 units. The general rule is that $y = |x - h|$ shifts the graph $h$ units horizontally. Here, $h = -4$, so the graph shifts 4 units to the left.

2. Vertical Shift:

   • The "-2" outside the absolute value affects the vertical position of the graph.
   • This represents a downward shift of 2 units. The general rule is that $y = |x| + k$ shifts the graph $k$ units vertically. In this case, $k = -2$, so the graph shifts down 2 units.

3. No Stretch or Reflection:

   • There are no multiplicative factors affecting the steepness or orientation of the graph, so there is no vertical stretch or reflection.

Summary of Transformations:

• Shift the graph of $y = |x|$ 4 units to the left.
• Then, shift the resulting graph 2 units down.

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Do you have any questions or want further details?

Related Questions:

1. What would happen if the function was $y = |x-4| + 2$?
2. How do vertical stretches or compressions affect the graph of $y = |x|$?
3. What if there were a negative sign outside the absolute value, like $y = -|x+4| - 2$?
4. Can you explain how to find the vertex of an absolute value function from its equation?
5. How do you solve for $x$ when given an equation in the form $y = |x+4| - 2$?

Tip:

For absolute value transformations, focus first on what's inside (horizontal shifts) and then what's outside (vertical shifts and stretches).