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Linear growth refers to the steady increase of a quantity over time at a constant rate. In the given scenario of Snapchat users, the number of users grew consistently over two specific intervals. This is a characteristic example of linear growth.

Linear growth can be easily modeled using straight lines when plotted on a graph. Each of these lines represents a fixed rate of increase known as the slope. The slope of a line reflects how steep the line is, thereby indicating how fast or slow a given quantity, such as the number of users, is growing.

In the case of the Snapchat data, we determined two separate growth rates - one from January 2014 to July 2015, and another from July 2015 to January 2017. The linear growth model for each of these periods was calculated using specific formulae to capture the consistent monthly increase in users.

Rates of change describe how one quantity changes concerning another. It's the "speed" of change, often expressed as the slope in linear functions. For Snapchat's user growth, the rates of change tell us how many million users were added each month.

To compute this, we calculated the difference in users between two points in time and divided it by the time elapsed. For instance, from January 2014 to July 2015, the rate of change was \(\frac{8}{3}\) million users per month. This was determined by taking the increment in users over the 18 months.

• Interval 1: A rate of \(\frac{8}{3}\) indicates slower growth, as it added fewer users per month.
• Interval 2: In contrast, \(\frac{11}{3}\) from July 2015 to January 2017 shows faster growth.

The changing rates across periods reveal the dynamic nature of user growth and show how much quicker users were added in the latter period.

A function is continuous if you can draw it without lifting your pen off the paper. Continuity in mathematical terms means that the function has no gaps, jumps, or breaks in its domain.

For the piecewise-linear function representing Snapchat's users, continuity at the critical point \(x = 18\) months was verified. This entails ensuring the values from each side of the piecewise definition meet at the same y-value.

We evaluated both formulas for \(x = 18\):

• First interval: \(S(18) = \frac{8}{3} \times 18 + 46 = 94\)
• Second interval: \(S(18) = \frac{11}{3} \times 18 + 26 = 94\)

The matching results confirm that there is no gap at this point, maintaining the function's continuity over the entire period from 0 to 36 months. This continuity reassures us that the model accurately reflects real-world user growth without missing connections between time periods.