To determine the transformations needed to go from $f(x) = |x|$ to $g(x) = -|x + 2| + 1$, let's analyze the components of the equation.

1. $|x + 2|$: This represents a horizontal shift. The graph is shifted to the left by 2 units because adding 2 inside the absolute value moves the graph left.

2. $-|x + 2|$: The negative sign outside the absolute value represents a reflection about the x-axis. This flips the graph upside down.

3. $+1$: This represents a vertical shift. The graph is shifted 1 unit upwards because of the "+1" outside the absolute value.

Thus, the transformations needed are:

• Horizontal shift (left by 2 units)
• Reflection about the x-axis
• Vertical shift (up by 1 unit)

Correct answers:

• A. Horizontal shift
• D. Reflection about the x-axis
• E. Vertical shift

Let me know if you'd like more details!

Here are five related questions to consider:

1. How would the graph change if the "+1" were "-1" instead?
2. What would happen if the negative sign were removed in front of $|x+2|$?
3. How would the transformations change if $f(x)$ were $|x-3|$ instead of $|x|$?
4. Can you describe how the graph shifts when the constant inside the absolute value changes?
5. How would the graph look if there were a coefficient in front of the absolute value (e.g., $2|x+2|$)?

Tip: When analyzing transformations, always break them down by focusing on changes to the inside and outside of the absolute value function separately!