06Iincreasingtrends

= Trig Functions with an Increasing Trend =

A particular example of adding two functions to model a situation is where a linear function and a trig function is added to get a cyclic function with an increasing trend. This might occur, for example, in economic growth cycles, or with a seasonal population that is steadily increasing over time.

In general, such a situation can be modelled using the equation:

y = ax + b + m**sin**(nx)

With this equation, ax + b represents the general increasing trend (the median line) and m**sin**(nx) represents the regular variation above and below that median line.


 * Example 1 **

Consider an island where the population of rabbits varies according to the seasons but is showing an increasing trend over time. The population (P) at a time, t months after 1st January, 2010, is modelled by the function: math P \big( t \big) = 2t+100-60\sin \left( \dfrac{\pi t}{6} \right) math
 * a) ** With the aid of a CAS calculator, sketch the function for 5 years from 1st January, 2010. Also sketch the underlying trend line for the same period.

{ 5 years but t = months, so t Î [0, 60] }

{median line = 2t + 100}


 * b) ** According to the model, what is the population at the end of May in 2012.

{end of May, 2012, is t = 29. Use Analysis- Trace}

P(29) = 128


 * c) ** According to the model, what is the year and month when the population first reaches 210?

{Solve P(t) = 210 or Graph y = 210 and find first point of intersection}

Solution: t = 31.7 months

{This is 2 years (24 months) after the starting point, (so 2012) and during the 8th month of that year (August)}

Solution: August, 2012. Go to top of page flat .