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).