Let’s break down the problem. The given graph is $y = f(x)$, and we need to determine how the graph changes with the specified transformations.

Part (a) $y = f(x+4) - 2$

1. Transformation Steps:
   • $f(x+4)$: Shift the graph 4 units to the left.
   • $-2$: Shift the graph 2 units down.

Match the Graph:

Looking at the choices (A, B, C, D), the correct graph will reflect these transformations:

• The vertex should shift 4 units left and 2 units down compared to the original graph.
• Based on the comparison, choice B is correct for $y = f(x+4) - 2$.

Would you like detailed analysis for parts (b) and (c), or help with additional questions? Let me know!

Related Questions:

1. What happens if we apply both horizontal and vertical stretches instead of shifts?
2. How do reflections (like $y = f(-x)$) affect the graph visually?
3. What is the vertex form of the transformed quadratic in this example?
4. Can you explain transformations using function composition?
5. How do these transformations affect the domain and range of the function?

Tip:

When analyzing transformations, always apply horizontal changes first, followed by vertical shifts or stretches. This order avoids confusion in multi-step transformations.