To determine the best type of function to model the relationship between time and the number of referrals (y), we should examine the rate of change in the costs over time (x).
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Growth Pattern: The costs provided increase significantly as time progresses. Let's look at the total increments in cost:
- From 5 to 6 years: \(28,477 - 18,984 = 9,493\)
- From 6 to 7 years: \(42,715 - 28,477 = 14,238\)
- From 7 to 8 years: \(64,072 - 42,715 = 21,357\)
The increments themselves are increasing, suggesting that as time progresses, the cost increases at an accelerating rate.
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Exponential Growth: The pattern of increasing costs suggests that the number of referrals might exhibit exponential growth, where the rate of growth is proportional to the current value. This is typical in scenarios where social media presence compounds over time—meaning that as more refers are gained, they further lead to more referrals through a network effect.
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Comparison with Polynomial: If we were to model this with a polynomial function (for example, a quadratic or cubic function), we would expect the differences in increments to become constant or to follow a non-linear pattern (but not necessarily in the same accelerating way observed here). The rapid increase in costs could be more aligned with an exponential function rather than a polynomial.
In summary, the function that best models the situation of referrals from a social media website over time, given the data on costs increasing significantly and non-linearly, would likely be an exponential function. This decision stems from observing how the increments in the data indicate accelerating growth rather than constant or decreasing rates of change typically associated with polynomial functions.