The graph of 𝑦=𝑓(𝑥)
 is given below. Which of the following corresponds to the graph of 𝑔(𝑥)=2𝑓(3𝑥+9)+2
?

To determine which graph corresponds to \( g(x) = 2f(3x + 9) + 2 \), let’s analyze the transformations applied to the graph of \( f(x) \).

1. **Horizontal transformations**:
   - \( 3x + 9 \) implies a **horizontal compression** by a factor of \( \frac{1}{3} \) (due to the factor 3) and a **shift to the left** by 3 units (due to the \( +9 \)).
   
   This is because:
   - The function \( f(3x) \) compresses the graph horizontally by a factor of 3.
   - The \( +9 \) inside \( f(3x + 9) \) shifts the graph left by \( \frac{9}{3} = 3 \) units.

2. **Vertical transformations**:
   - The coefficient 2 in \( 2f(3x + 9) \) results in a **vertical stretch** by a factor of 2.
   - The \( +2 \) outside the function results in a **vertical shift** upwards by 2 units.

### Summary of transformations for \( g(x) = 2f(3x + 9) + 2 \):
- Compress the graph of \( f(x) \) horizontally by a factor of 3.
- Shift the graph 3 units to the left.
- Stretch the graph vertically by a factor of 2.
- Shift the graph upwards by 2 units.

To match the graph of \( g(x) \) with one of the options, look for a graph that appears:
- Compressed horizontally.
- Shifted left by 3 units.
- Stretched vertically.
- Shifted upwards.

Would you like to explore further on how these transformations apply to specific graph shapes?

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### Related questions:
1. What is the effect of multiplying the argument of a function by a constant?
2. How does adding a constant to the input of a function shift the graph?
3. Can you explain how a vertical stretch differs from a horizontal compression?
4. How do vertical shifts affect the graph of a function?
5. What is the general form for transformations of a function?

### Tip:
Always apply horizontal transformations (inside the function) before vertical ones when analyzing transformations systematically.

The graph of 𝑦=𝑓(𝑥) is given below. Which of the following corresponds to the graph of 𝑔(𝑥)=2𝑓(3𝑥+9)+2?

Analyze how the graph of g(x)=2f(3x+9)+2 is transformed from f(x). Learn about horizontal compression, vertical stretching, and shifts. Suitable for high school students in Grades 10-12.

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