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The question asks you to describe the function \( g(x) \) in terms of \( f(x) \), given that the transformation involves a vertical stretch, a horizontal shift to the right, and a vertical shift upwards.

### Transformation details:
1. **Vertical stretch**: By a factor of 2, so the term outside of \( f(x) \) will be 2.
2. **Horizontal shift**: To the right by 8 units, so we replace \( x \) by \( (x - 8) \).
3. **Vertical shift**: Upward by 7 units, so we add 7 to the function.

The general form for a transformed function is:
\[
g(x) = A f(x + B) + C
\]
We will now plug in the values based on the transformations.

- \( A = 2 \) (vertical stretch by a factor of 2),
- \( B = -8 \) (since a shift to the right by 8 is represented as \( (x - 8) \)),
- \( C = 7 \) (shift upward by 7 units).

Thus, the function is:
\[
g(x) = 2 f(x - 8) + 7
\]

### Therefore, the values are:
- \( A = 2 \)
- \( B = -8 \)
- \( C = 7 \)

Let me know if you want more details or have any questions!

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### 5 Related Questions:
1. How would the equation change if the function was shifted left instead of right?
2. What if the vertical shift was downwards instead of upwards?
3. How does a horizontal compression affect the equation?
4. How would a reflection over the x-axis impact the function?
5. What if the vertical stretch factor was different, say 3 instead of 2?

### Tip:
When transforming functions, always remember that horizontal shifts affect the inside of the function (with opposite signs), while vertical shifts affect the outside directly.

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 2, then shifting the graph of f to the right 8 units and upward 7 units.

Learn how to describe the function g(x) in terms of f(x) through a vertical stretch by a factor of 2, a horizontal shift of 8 units to the right, and a vertical shift of 7 units upward. This problem covers algebraic transformations and function theory, suitable for students in Grades 9-12.

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