The equation you've provided is a transformation of the function \( f(x) \) involving both a horizontal and a vertical shift. 

Here's a breakdown:

1. **Horizontal Shift:** The term \( x - 2 \) inside the function \( f(x) \) indicates a horizontal shift. Specifically, it shifts the graph of \( f(x) \) **2 units to the right**.

2. **Vertical Shift:** The term \( -1 \) outside of \( f(x - 2) \) indicates a vertical shift. Specifically, it shifts the graph of \( f(x - 2) \) **1 unit downward**.

So, the function \( y = -1 + f(x - 2) \) represents the graph of the function \( f(x) \) shifted 2 units to the right and 1 unit downward.

Would you like more details or have any questions about this transformation? Here are some related questions you might consider:

1. What would the graph look like if the function was \( y = 1 + f(x + 3) \)?
2. How would the equation change if the function was shifted 3 units to the left instead of 2 units to the right?
3. What happens to the function if you add a vertical stretch to it, for example, \( y = 2f(x - 2) - 1 \)?
4. Can you describe the transformation for \( y = f(2x - 4) - 1 \)?
5. How does the graph change if the function \( f(x) \) is reflected over the x-axis, e.g., \( y = -f(x - 2) - 1 \)?

**Tip:** When analyzing transformations, remember that changes inside the function argument (like \( x - 2 \)) affect horizontal shifts, while changes outside the function (like \( -1 \)) affect vertical shifts.

Explore how function transformations work with the equation 𝑦 = −1 + 𝑓(𝑥 − 2). Learn about horizontal and vertical shifts, key concepts in function transformations, suitable for high school students.

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