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 $y = 19t + 5$:

For the exponential function $y = 3^t$:

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.