https://file.aiquickdraw.com/mathbot/file/1725599754995w8sgnuaq.jpg

To solve the problem, we need to match each equation with its corresponding colored graph. The black graph represents \( y = 2^x \), and the other graphs are transformations of this basic exponential function.

### Analyzing each equation:
1. **\( y = 2^x + 1 \):**
   - This equation represents a vertical shift of \( y = 2^x \) by 1 unit upward.
   
2. **\( y = 2^x - 1 \):**
   - This represents a vertical shift of \( y = 2^x \) by 1 unit downward.

3. **\( y = 2^{x-1} \):**
   - This is a horizontal shift of \( y = 2^x \) by 1 unit to the right.

4. **\( y = 2^{x+1} \):**
   - This is a horizontal shift of \( y = 2^x \) by 1 unit to the left.

### Matching the graphs:

- **Black graph (K):** This is the standard \( y = 2^x \) without any transformations.
- **Orange graph (O):** This graph appears to be shifted vertically upward, which matches the equation \( y = 2^x + 1 \).
- **Red graph (R):** This graph is shifted to the left, matching \( y = 2^{x+1} \).
- **Green graph (G):** This graph is shifted to the right, corresponding to \( y = 2^{x-1} \).
- **Blue graph (B):** This graph is shifted downward, matching \( y = 2^x - 1 \).

### Final matching:
- \( y = 2^x + 1 \) → Orange (O)
- \( y = 2^{x+1} \) → Red (R)
- \( y = 2^{x-1} \) → Green (G)
- \( y = 2^x - 1 \) → Blue (B)

Would you like more details or have any questions? Here are some related questions to consider:

1. How do you identify vertical and horizontal shifts in exponential functions?
2. What happens if you shift an exponential function to the right or left by 2 units instead of 1?
3. How does changing the base of the exponential function affect the graph?
4. Can you describe the effect of multiplying the function by a constant?
5. What is the impact of reflecting an exponential function over the x-axis?

**Tip:** Vertical shifts occur by adding/subtracting constants outside the function, while horizontal shifts result from adding/subtracting constants inside the function.

Match each equation with its corresponding graph shown above. The graph of y = 2^x is shown in black (K).

Learn how to match exponential function equations with their corresponding graphs through transformations such as vertical and horizontal shifts. This guide covers concepts and formulas for exponential function shifts, suitable for students in Grades 9-11.

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