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The transformation \( f(x) = 2^{x+1} \) compared to \( f(x) = 2^x \) involves a horizontal shift and a vertical stretch.

To analyze the effect:

1. **Starting with \( f(x) = 2^x \)**: This is a basic exponential function with base 2.
  
2. **Transformation \( f(x) = 2^{x+1} \)**: This can be rewritten as:

   \[
   f(x) = 2^{x+1} = 2^x \cdot 2^1 = 2 \cdot 2^x
   \]

   This shows that the transformation is equivalent to multiplying the original function by 2.

3. **Effect on the graph**:
   - The function \( f(x) = 2^{x+1} \) is vertically stretched by a factor of 2 compared to \( f(x) = 2^x \). This means that for every point on the graph of \( 2^x \), the corresponding point on the graph of \( 2^{x+1} \) is twice as high.
   
   - Alternatively, it can be interpreted as a shift to the left by 1 unit in the input domain, which is another perspective since adding 1 inside the exponent effectively shifts the function to the left.

**Summary**: The main effect of the transformation \( 2^{x+1} \) is a vertical stretch by a factor of 2 compared to \( 2^x \), which can also be viewed as a left shift by 1 unit.

Would you like more details or have any questions?

Here are some related questions:

1. What happens when we add or subtract a constant outside the exponent in an exponential function?
2. How does changing the base of an exponential function affect its graph?
3. What is the effect of adding a constant to the exponent of an exponential function?
4. How do you graph exponential functions with transformations?
5. How would \( 2^{-x} \) compare to \( 2^x \) in terms of transformation?

**Tip**: Always consider rewriting transformed functions to compare them directly with the original function, making it easier to identify shifts, stretches, or compressions.

What is the effect of the transformation f(x) = 2^(x+1) compared to f(x) = 2^x?

Analyze the effect of the transformation f(x) = 2^(x+1) on the exponential function f(x) = 2^x. Learn how the function shifts and stretches, including a detailed explanation of vertical stretch by a factor of 2 or a left shift by 1 unit.

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