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.