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The task involves comparing two functions representing the growth rates of the values of two paintings. The first painting's value is modeled by the linear function \( y = 19t + 5 \), while the second painting's value is modeled by the exponential function \( y = 3^t \). Both are expressed in terms of the variable \( t \), representing the number of years.

### Let's fill in the tables based on the given functions.

#### For the linear function \( y = 19t + 5 \):

| \( t \) | \( y = 19t + 5 \) |
|--------|-------------------|
| 0      | \( 19(0) + 5 = 5 \) |
| 1      | \( 19(1) + 5 = 24 \) |
| 2      | \( 19(2) + 5 = 43 \) |
| 3      | \( 19(3) + 5 = 62 \) |
| 4      | \( 19(4) + 5 = 81 \) |

#### For the exponential function \( y = 3^t \):

| \( t \) | \( y = 3^t \) |
|--------|----------------|
| 0      | \( 3^0 = 1 \)  |
| 1      | \( 3^1 = 3 \)  |
| 2      | \( 3^2 = 9 \)  |
| 3      | \( 3^3 = 27 \) |
| 4      | \( 3^4 = 81 \) |

### Comparison:

- **Constant growth rate (Linear function):** The first painting's value grows by a constant amount of 19 dollars per year. This is evident from the linear nature of the equation, where the slope is 19.
- **Increasing growth rate (Exponential function):** The second painting's value follows an exponential pattern, meaning its growth rate accelerates as \( t \) increases. Initially, the value grows slowly but starts increasing much faster as \( t \) grows larger.

### Conclusion:
- The painting represented by \( y = 19t + 5 \) has a **constant growth rate**.
- The painting represented by \( y = 3^t \) has an **increasing growth rate** because it's an exponential function.

Let me know if you'd like more details or have any questions!

Here are 5 follow-up questions you might find helpful:
1. What happens to the exponential function's growth if the base is changed from 3 to another number?
2. How do linear and exponential growth rates differ over long periods?
3. Can you describe a real-world situation where an exponential function is more appropriate than a linear function?
4. How would the linear function change if the slope were negative?
5. What would happen if you added a constant to the exponential function?

**Tip:** In exponential functions, the rate of growth depends on both the base and the exponent, making them grow much faster than linear functions over time.

How can you compare the growth rates of linear and exponential functions? The task involves comparing two functions: y = 19t + 5 and y = 3^t, representing the growth of two paintings' values over time.

Learn how to compare the growth rates of linear and exponential functions using the equations y = 19t + 5 and y = 3^t. Discover how the value of two paintings changes over time, one with a constant growth rate and the other with an increasing growth rate. Suitable for Grades 8-10.

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