The problem asks to identify the type of function that describes the increase in the number of sheep copies over time as shown in the table.

Let's first interpret the data from the table:

• Years After Cloning: 0, 1, 2, 3, 4, 5, 6, 7
• Number of Sheep Copies: 100, 120, 145, 175, 210, 250, 295, 345

To determine the type of function that models this situation, we need to observe the pattern in the increase in the number of sheep copies.

Analyzing the Type of Function:

1. Linear Function: If the increase between consecutive years is constant, it would suggest a linear function. Let's check the differences:

   • From Year 0 to Year 1: $120 - 100 = 20$
   • From Year 1 to Year 2: $145 - 120 = 25$
   • From Year 2 to Year 3: $175 - 145 = 30$
   • From Year 3 to Year 4: $210 - 175 = 35$
   • From Year 4 to Year 5: $250 - 210 = 40$
   • From Year 5 to Year 6: $295 - 250 = 45$
   • From Year 6 to Year 7: $345 - 295 = 50$

   The increase is not constant but increases by 5 more sheep each time. This indicates that the function is not linear.

2. Quadratic Function: If the first differences are not constant but the second differences are, the function might be quadratic.

   • First differences: $20, 25, 30, 35, 40, 45, 50$
   • Second differences: $5, 5, 5, 5, 5, 5$ (all are constant)

   Since the second differences are constant, this suggests a quadratic function.

3. Exponential Function: For an exponential function, the ratio between consecutive values would be constant, but that is not the case here.

4. Absolute Value Function: This type of function is characterized by a "V" shape graph and would not fit the given data.

Conclusion:

The data fits a quadratic function because the second differences are constant.

The correct answer is:

• C. a quadratic function with a positive second difference.

Would you like more details or have any other questions?

Related Questions to Explore:

1. How would the data look if the function was linear?
2. What are the characteristics of an exponential function?
3. Can a quadratic function model both increasing and decreasing trends?
4. How do you determine the equation of a quadratic function given a table of values?
5. What are real-world examples of phenomena that follow quadratic patterns?

Tip: When analyzing tables of values to determine the type of function, check both the first and second differences to identify linear, quadratic, or other types of functions.