To determine the type of function that would best model the relationship between time (years) and the number of referrals from social media websites, we can examine the behavior of the data provided.
The data shows the following pairs of time (in years) and cost:
- At year 5: $18,984
- At year 6: $28,477
- At year 7: $42,715
- At year 8: $64,072
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Identify the Pattern in the Data:
- We can calculate the changes in cost between each successive year to see if there's a pattern.
- From year 5 to year 6: \(28,477 - 18,984 = 9,493\)
- From year 6 to year 7: \(42,715 - 28,477 = 14,238\)
- From year 7 to year 8: \(64,072 - 42,715 = 21,357\)
It appears that the increase in cost is not constant but rather increasing at an increasing rate. This suggests that a linear model (constant rate of change) is not appropriate.
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Consider Exponential or Polynomial Growth:
- Since the differences in cost are increasing, this could suggest exponential growth, where the increase is proportional to the current value, or polynomial growth, particularly a quadratic function if the second differences (the differences of the differences) are constant.
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Analyze for Exponential Growth:
- To check for exponential growth, we note that exponential functions have the form \(y = a \cdot b^x\). In such functions, the rate of growth increases as \(x\) increases, which seems evident in the increases in cost.
Given these observations, the best fit would likely be an exponential function due to the nature of the increases in cost over time that suggest a proportional increase, indicating exponential growth patterns commonly seen in social media referals as a function of network effects or content virality.
In conclusion, the exponential function seems to be the best model for this situation, as it captures the rapidly increasing growth of referrals over time, which is characteristic of dynamics found in social media interactions.