https://file.aiquickdraw.com/mathbot/file/17286628233277ogelqs9.jpg

Let's solve each transformation applied to the point \( A(-1, 5) \) on the curve \( y = f(x) \).

1. **\( f(x - 6) \)**:  
   This transformation shifts the function 6 units to the right. The new x-coordinate becomes:
   \[
   x' = -1 + 6 = 5
   \]
   The y-coordinate remains the same, so the image of \( A \) is \( (5, 5) \).

2. **\( -2f(x) \)**:  
   This transformation reflects the function across the x-axis and then scales it by a factor of 2. The new y-coordinate becomes:
   \[
   y' = -2 \cdot 5 = -10
   \]
   The x-coordinate remains the same, so the image of \( A \) is \( (-1, -10) \).

3. **\( f(x) + 4 \)**:  
   This transformation shifts the function up by 4 units. The new y-coordinate becomes:
   \[
   y' = 5 + 4 = 9
   \]
   The x-coordinate remains the same, so the image of \( A \) is \( (-1, 9) \).

4. **\( f(-x) \)**:  
   This transformation reflects the function across the y-axis. The new x-coordinate becomes:
   \[
   x' = -(-1) = 1
   \]
   The y-coordinate remains the same, so the image of \( A \) is \( (1, 5) \).

### Summary of Transformed Coordinates:
1. \( (5, 5) \)
2. \( (-1, -10) \)
3. \( (-1, 9) \)
4. \( (1, 5) \)

Would you like any further details or explanations?

### Related Questions:
1. What happens when you apply a vertical shift \( f(x) - c \) instead of \( f(x) + c \)?
2. How does a horizontal shift \( f(x + a) \) differ from \( f(x - a) \)?
3. What effect does scaling the function by a factor \( c \cdot f(x) \) have on the graph?
4. What is the result when you reflect the function over both axes, as in \( -f(-x) \)?
5. Can you determine the image of a point under a transformation like \( f(2x) \) or \( f\left(\frac{x}{2}\right) \)?

### Tip:
Remember, horizontal transformations affect the x-values and are usually counterintuitive: \( f(x - a) \) moves right, while \( f(x + a) \) moves left.

The point A(-1, 5) lies on the curve with equation y = f(x). State the coordinates of the images of point A on the curve after the following transformations: 1. f(x - 6) 2. -2f(x) 3. f(x) + 4 4. f(-x)

Learn how to find the new coordinates of point A(-1, 5) on the curve y = f(x) after applying transformations like horizontal and vertical shifts, reflections, and function scaling. Suitable for students in Grades 9-11.

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