The problem provided asks to map the point $(4, 1)$ on the graph of $f(x)$ to its corresponding point on the graph of $p(x)$, where $p(x) = -2f(x - 3) + 4$.

Step-by-Step Solution:

1. Transformation Formula: The transformation involves:

   • A horizontal shift of $3$ units to the right because of $(x - 3)$,
   • A vertical stretch by a factor of $-2$ (reflection and scaling),
   • A vertical shift up by $4$ units because of $+4$.

2. Original Point: Start with $(4, 1)$ from the graph of $f(x)$.

3. Horizontal Shift: The transformation $f(x - 3)$ means shifting the $x$-coordinate 3 units to the right: $x = 4 \implies x - 3 = 1$ So, the new point becomes $(1, 1)$.

4. Vertical Stretch and Reflection: Apply the vertical transformation $-2$. This means multiplying the $y$-coordinate by $-2$: $y = 1 \implies y \times (-2) = -2$ Now, the point is $(1, -2)$.

5. Vertical Shift: Finally, add 4 to the $y$-coordinate because of the $+4$: $y = -2 \implies y + 4 = 2$ So, the final transformed point is $(1, 2)$.

Final Answer:

The point $(4, 1)$ on the graph of $f(x)$ maps to $(1, 2)$ on the graph of $p(x)$.

Would you like further explanation or any details on the steps?

Follow-up Questions:

1. How does a vertical stretch affect the graph of a function?
2. What would happen if we had a transformation like $f(x + 3)$ instead of $f(x - 3)$?
3. How does the reflection caused by a negative coefficient affect the shape of the graph?
4. Can you describe what a horizontal shift does to the domain of a function?
5. What happens to a point if only a vertical shift is applied without any stretch or reflection?

Tip:

When working with transformations, always apply horizontal shifts before vertical transformations (stretches and shifts) for clarity!