https://file.aiquickdraw.com/mathbot/file/1727104426752jhviajqm.jpg

Let's analyze the transformations and apply them to the given point \((-1, 6)\) on the graph of \(f(x)\).

### Given Information:
- The graph of \(f(x)\) contains the point \((-1, 6)\).
  
Now, we'll solve each transformation step by step:

### (a) The graph of \(f(x - 6)\):
This represents a **horizontal shift** to the **right** by 6 units. 
- When shifting horizontally, only the **x-coordinate** changes. 
- New point: \( (-1 + 6, 6) = (5, 6) \).

### (b) The graph of \(f(x) - 7\):
This represents a **vertical shift** **down** by 7 units.
- When shifting vertically, only the **y-coordinate** changes.
- New point: \((-1, 6 - 7) = (-1, -1)\).

### (c) The graph of \(f(x + 2) + 3\):
This represents two transformations:
1. A **horizontal shift** to the **left** by 2 units (due to \(x + 2\)).
2. A **vertical shift** **up** by 3 units (due to \(+3\)).

- Horizontal shift affects the **x-coordinate**: \(-1 - 2 = -3\).
- Vertical shift affects the **y-coordinate**: \(6 + 3 = 9\).
- New point: \((-3, 9)\).

### Final Answers:
- (a) \( (5, 6) \)
- (b) \( (-1, -1) \)
- (c) \( (-3, 9) \)

Let me know if you'd like further details or explanations. Here are some related questions for practice:

1. What happens to a point under a reflection across the x-axis?
2. How does a vertical stretch affect the coordinates of a point on a graph?
3. What point does the graph of \( f(x + 4) - 2 \) pass through if \( f(x) \) passes through \((2, 5)\)?
4. How do you describe the transformation \( f(x - 3) + 4 \)?
5. What happens to the point \((0, 0)\) when the graph undergoes a vertical shift up by 7?

**Tip**: Horizontal shifts affect the x-values, while vertical shifts affect the y-values.

The graph of f(x) contains the point (-1, 6). What point must be on each of the following transformed graphs?

Understand how to apply horizontal and vertical shifts to a graph. Given the point (-1, 6) on f(x), learn how to find corresponding points for f(x - 6), f(x) - 7, and f(x + 2) + 3. Suitable for students in Grades 9-12.

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