The problem involves transformations of the function $f(x)$, which is shown as a parabola with a vertex at $(1, 0)$ and passing through $(2, 1)$. The exercise asks to sketch the graphs of various transformations. Let's analyze each transformation and describe how to draw the new graph.

1. (a) $f(x) + 2$: Vertical Shift Upward

   • This transformation shifts the graph of $f(x)$ vertically by 2 units upward. The vertex at $(1, 0)$ will move to $(1, 2)$, and all other points will move 2 units up.

2. (b) $f(-x) - 1$: Reflection Across the y-axis and Vertical Shift Downward

   • The function $f(-x)$ reflects the graph of $f(x)$ across the y-axis.
   • Then, subtracting 1 shifts the entire graph 1 unit downward. The vertex will move to $(-1, -1)$ (reflection of $(1, 0)$ with the downward shift).

3. (c) $f(x - 2)$: Horizontal Shift Rightward

   • This transformation shifts the graph horizontally 2 units to the right. The vertex moves from $(1, 0)$ to $(3, 0)$.

4. (d) $-f(x) - 2$: Reflection Across the x-axis and Vertical Shift Downward

   • The negative sign reflects the graph across the x-axis.
   • Then, subtracting 2 shifts the reflected graph 2 units downward. The vertex will move from $(1, 0)$ to $(1, -2)$.

5. (e) $2f(x - 1) + 1$: Horizontal Shift, Vertical Stretch, and Vertical Shift

   • The $f(x - 1)$ shifts the graph 1 unit to the right, moving the vertex to $(2, 0)$.
   • The factor of 2 stretches the graph vertically, making it narrower.
   • Finally, adding 1 shifts the graph 1 unit upward, moving the vertex to $(2, 1)$.

6. (f) $f(2x)$: Horizontal Compression

   • The factor of 2 inside the function compresses the graph horizontally by a factor of 2. Points closer to the y-axis will be compressed, and the vertex will remain at $(1, 0)$.

These transformations change the graph based on the rules for shifting, reflecting, stretching, and compressing functions.

Would you like me to provide more details or clarify any specific transformation?

Here are 5 related questions to expand on this topic:

1. What happens to the graph of a function when it is reflected across the y-axis?
2. How does a vertical stretch affect the shape of a graph?
3. What is the difference between horizontal and vertical shifts in function transformations?
4. How do you determine the new vertex after a transformation involving both shifts and reflections?
5. How would the graph of $f(x+3) - 2$ differ from $f(x-3) + 2$?

Tip: When sketching transformations, focus first on how the vertex or key points move, then adjust the shape based on stretches or reflections.