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.