Let’s analyze the given problem step-by-step:

The original parent function is $f(x)$, and its transformed function is: $g(x) = 2f(x - 1) + 2$

Breaking down the transformations:

1. Horizontal Shift:

   • The term $f(x - 1)$ indicates a shift to the right by 1 unit.

2. Vertical Stretch:

   • The multiplication by $2$ (the coefficient of $f(x - 1)$) indicates a vertical stretch by a factor of 2.

3. Vertical Shift:

   • The addition of $+2$ outside the function indicates a shift upward by 2 units.

Combined Transformation:

The graph of $g(x)$ is transformed from $f(x)$ by:

• A stretch along the y-axis (vertical stretch),
• A shift along the x-axis (right by 1 unit), and
• A shift along the y-axis (up by 2 units).

Correct Answer:

The second option:
Stretch, shift along the y-axis, and a shift along the x-axis.

Would you like me to explain any step in detail or provide examples?

Related Questions:

1. How does the term $f(x+c)$ differ from $f(x-c)$ in horizontal shifts?
2. What does a vertical compression (e.g., $\frac{1}{2}f(x)$) do to a graph?
3. How would the transformation change if $g(x) = -2f(x-1) + 2$?
4. What are the effects of adding a constant $+k$ inside versus outside $f(x)$?
5. How would $g(x) = 2f(x+1) - 2$ transform the graph of $f(x)$?

Tip:

Remember:

• Inside the parentheses affects the x-axis (horizontal changes).
• Outside the parentheses affects the y-axis (vertical changes).