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The problem asks to describe a function \( g(x) \) in terms of \( f(x) \), where:

- The graph of \( g(x) \) is obtained by vertically stretching \( f(x) \) by a factor of 3.
- Then the graph is shifted 8 units to the right and 7 units upward.

The general transformation form is:

\[
g(x) = A f(x + B) + C
\]

Where:
- \( A \) represents a vertical stretch/compression (positive for stretch).
- \( B \) represents a horizontal shift (positive for a left shift, negative for a right shift).
- \( C \) represents a vertical shift.

Now, applying the transformations step by step:

1. **Vertical stretch by a factor of 3**: This means \( A = 3 \), because the graph is stretched vertically by a factor of 3.
   
2. **Shift 8 units to the right**: This corresponds to shifting the function \( f(x) \) by \( -8 \) inside the argument, so \( B = -8 \).
   
3. **Shift 7 units upward**: This means we add 7 to the entire function, so \( C = 7 \).

Thus, the function \( g(x) \) is:

\[
g(x) = 3 f(x - 8) + 7
\]

To fill in the values:
- \( A = 3 \)
- \( B = -8 \)
- \( C = 7 \)

Would you like further details or have any questions about this transformation process? Here are some related questions:

1. How does a negative value of \( B \) affect the horizontal shift direction?
2. What happens to the graph of \( f(x) \) if the vertical stretch factor is less than 1?
3. Can you describe how the graph changes with a negative vertical stretch factor (e.g., \( A = -3 \))?
4. How do combined horizontal and vertical shifts affect the graph's overall position?
5. What is the effect of horizontal stretching or compressing on a function?

**Tip:** Always remember that horizontal shifts inside the function affect the \( x \)-axis in the opposite direction (positive values shift left, negative values shift right).

Describe a function g(x) in terms of f(x) if the graph of g is obtained by vertically stretching f by a factor of 3, then shifting the graph of f to the right 8 units and upward 7 units.

This problem explores how to transform a function by applying a vertical stretch, a horizontal shift to the right, and a vertical shift upward. Learn how to describe g(x) in terms of f(x) using the function transformation formula.

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