Find the asymptotes for the graph of y=2^(x+1) - 3

To find the asymptotes of the function \( y = 2^{(x+1)} - 3 \), we first identify the behavior of the function as \( x \) approaches certain values.

1. **Vertical Asymptotes**: 
   - Vertical asymptotes occur where the function approaches infinity or negative infinity. Since \( 2^{(x+1)} \) is defined for all \( x \), there are no vertical asymptotes for this function.

2. **Horizontal Asymptotes**:
   - Horizontal asymptotes are found by evaluating the limit of \( y \) as \( x \) approaches positive and negative infinity:
     - As \( x \to \infty \):
       \[
       y = 2^{(x+1)} - 3 \to \infty
       \]
     - As \( x \to -\infty \):
       \[
       y = 2^{(x+1)} - 3 \to 0 - 3 = -3
       \]
     - Therefore, there is a horizontal asymptote at \( y = -3 \).

### Summary of Asymptotes
- **Vertical Asymptotes**: None
- **Horizontal Asymptote**: \( y = -3 \)

If you would like further details or have any questions, feel free to ask! Here are some related questions for further exploration:

1. How does the function \( y = 2^{(x+1)} - 3 \) behave near the horizontal asymptote?
2. What transformations have been applied to the basic function \( y = 2^x \) to obtain \( y = 2^{(x+1)} - 3 \)?
3. How would the asymptotes change if the function were \( y = 2^{(x+1)} + 3 \)?
4. Can you graph the function to visually confirm the asymptotes?
5. What is the significance of the base of the exponential function in determining growth rates?

**Tip**: To analyze asymptotes effectively, consider both limits as \( x \) approaches infinity and negative infinity.

Discover how to find the asymptotes of the function y=2^(x+1) - 3. This guide includes the identification of vertical and horizontal asymptotes, demonstrating the absence of vertical asymptotes and the presence of a horizontal asymptote at y=-3. Ideal for high school students.

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