https://file.aiquickdraw.com/mathbot/file/1732056820037h41t6stz.jpg

Here’s the solution for each part of the assessment:

1. **Graph with the curved line**:  
   - **Type**: Exponential  
   - **Reason**: Exponential functions show a constant ratio between consecutive values, represented by the steep upward curve.

2. **Table labeled \( g(x) \)**:  
   - **Type**: Linear  
   - **Reason**: The differences between \( g(x) \) values are constant (\(-8, -5, -2, 1, 4, 7\)), showing a constant difference.

3. **Table labeled \( f(x) \)**:  
   - **Type**: Exponential  
   - **Reason**: The \( f(x) \) values double each time (e.g., \( 6 \to 12 \to 24 \to 48 \)), showing a constant ratio.

4. **Equation \( y = -2x + 8 \)**:  
   - **Type**: Linear  
   - **Reason**: Linear equations have the form \( y = mx + b \) with a constant rate of change (\(-2\) here).

5. **Recurrence relation \( f(n) = f(n-1) + 4 \)**:  
   - **Type**: Linear  
   - **Reason**: Adding a constant value (\(+4\)) for every step results in a constant difference.

6. **Equation \( f(x) = 3(2^x) \)**:  
   - **Type**: Exponential  
   - **Reason**: The function has a base of \( 2 \) raised to a power (\( x \)), showing exponential growth with a constant ratio.

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Would you like further details or help with any specific part of this solution?  

Here are 5 related questions to expand your understanding:  
1. What is the general difference between linear and exponential functions?  
2. How can you verify whether a table shows a linear or exponential function?  
3. What happens when the base of an exponential function is less than 1?  
4. How would you describe the graph of a negative linear slope like \( y = -2x + 8 \)?  
5. Can a recurrence relation represent an exponential function? If so, how?

**Tip**: To quickly identify an exponential pattern, check for a consistent multiplication factor (constant ratio) between outputs!

Label each function as LINEAR or EXPONENTIAL and state if the function will have a constant difference or a constant ratio.

Identify linear and exponential functions with examples. Learn the differences, including constant difference for linear functions and constant ratio for exponential functions, with equations and tables.

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