What transformation is used to graph y=2^x+b from y=2^x

To graph the function \( y = 2^x + b \) starting from the basic function \( y = 2^x \), you apply a **vertical translation**.

### Transformation:
- **Vertical Translation:** The function \( y = 2^x + b \) is obtained by shifting the graph of \( y = 2^x \) vertically by \( b \) units.
  - If \( b > 0 \), the graph shifts **up** by \( b \) units.
  - If \( b < 0 \), the graph shifts **down** by \( |b| \) units.

This transformation does not affect the shape of the graph, only its vertical position. The horizontal asymptote of the graph also shifts from \( y = 0 \) to \( y = b \).

Would you like more details or have any questions?

Here are 5 related questions:
1. How would you graph \( y = 2^{x-1} + b \) starting from \( y = 2^x \)?
2. What effect does multiplying the function by a constant have on the graph?
3. How do you determine the new asymptote after a vertical translation?
4. What are the differences between vertical and horizontal translations?
5. How does the graph change if you reflect \( y = 2^x + b \) over the x-axis?

**Tip:** When graphing transformations, always consider the order of operations—horizontal shifts and reflections are applied before vertical shifts.

Explore how to graph the function y = 2^x + b starting from y = 2^x by applying vertical translations. Learn about shifting the graph vertically and its effects on the graph's position without altering its shape. Suitable for Grades 10-12 students studying exponential functions.

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