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Let's analyze the four given exponential functions and their graphs. Each exponential function has a specific behavior based on its growth rate and vertical shift.

### The Functions:

1. \( p(x) = 4(1.5)^x + 3 \)
   - Exponential growth due to \(1.5^x\).
   - Shifted up by 3 units.

2. \( m(x) = 4(1.5)^{-x} - 6 \)
   - Exponential decay (since exponent is negative).
   - Shifted down by 6 units.

3. \( t(x) = 1.5^x + 3 \)
   - Exponential growth (no multiplication factor for \( 1.5^x \)).
   - Shifted up by 3 units.

4. \( w(x) = 1.5^{-x} - 6 \)
   - Exponential decay (due to \(1.5^{-x}\)).
   - Shifted down by 6 units.

### Graph Behavior:
- **Exponential Growth**: Curves moving upward and becoming steeper as \( x \) increases.
- **Exponential Decay**: Curves approaching 0 as \( x \) increases, but steeper when \( x \) is negative.

### Matching the Graphs:

1. **Top-left graph**: This graph shows an exponential growth that is shifted up. So, it should be either \( p(x) \) or \( t(x) \). Since this graph appears steeper, it likely corresponds to \( p(x) = 4(1.5)^x + 3 \).

2. **Top-right graph**: This graph shows exponential growth with a similar shape but a smaller vertical shift. Hence, this matches \( t(x) = 1.5^x + 3 \).

3. **Bottom-left graph**: This graph displays exponential decay with a downward shift, matching \( w(x) = 1.5^{-x} - 6 \).

4. **Bottom-right graph**: This graph shows a sharper exponential decay and downward shift, which matches \( m(x) = 4(1.5)^{-x} - 6 \).

### Final Matching:

- Top-left: \( p(x) = 4(1.5)^x + 3 \)
- Top-right: \( t(x) = 1.5^x + 3 \)
- Bottom-left: \( w(x) = 1.5^{-x} - 6 \)
- Bottom-right: \( m(x) = 4(1.5)^{-x} - 6 \)

Would you like more details on how to identify these transformations or any other questions?

Here are 5 related questions to explore:
1. What are the general properties of exponential growth and decay functions?
2. How do vertical shifts affect the graphs of exponential functions?
3. How can you distinguish between exponential growth and decay based on the function?
4. What is the role of the base of the exponential (like 1.5 in this case)?
5. How do negative exponents influence the shape of exponential graphs?

**Tip**: When matching exponential functions to their graphs, always observe whether the function is growing or decaying and note any shifts up or down on the y-axis.

Match each exponential function to its graph.

Learn how to match exponential functions like p(x) = 4(1.5)^x + 3 to their respective graphs by analyzing exponential growth, decay, and graph transformations. Suitable for students in Grades 9-11, this task involves exponential function shifts and graph behavior.

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