To determine the best type of function to model the situation of website referrals received from social media over time (with time periods of 5, 6, 7, and 8 years resulting in costs of $18, $28, $42, and $64 respectively), we should examine the pattern of the data.
Given Data:
- \( x = 5 \), Cost = $18
- \( x = 6 \), Cost = $28
- \( x = 7 \), Cost = $42
- \( x = 8 \), Cost = $64
Step 1: Analyze the differences
First, calculate the first differences (change in cost per year) and then the second differences:
1st Differences:
- From 5 to 6: \( 28 - 18 = 10 \)
- From 6 to 7: \( 42 - 28 = 14 \)
- From 7 to 8: \( 64 - 42 = 22 \)
So, the first differences are: 10, 14, 22.
Next, calculate the second differences:
- From 10 to 14: \( 14 - 10 = 4 \)
- From 14 to 22: \( 22 - 14 = 8 \)
The second differences are not constant; however, there is an increasing pattern.
Step 2: Determine the Function Type
The first differences suggest that the function might be quadratic, as the second differences are increasing but not constant. Quadratic functions often have the general form \( y = ax^2 + bx + c \).
Step 3: Fit a Quadratic Model
To confirm this, you could use polynomial regression to fit a quadratic model through the points. However, since we are performing a manual analysis here, we might suggest that a quadratic function would be a suitable approximation because of the changing increments in your data.
Conclusion
In conclusion, the best type of function to model the number of referrals received from social media websites over time would likely be a quadratic function. This conclusion arises from the analysis of the differences, which indicate that the relationship is not linear, leading us to suggest that the relationship may be best captured by a quadratic equation. If further data points were available, a more definitive model could be established through statistical analysis.