SALES The projected monthly sales S (in thousands of units)...

SALES The projected monthly sales $S$ (in thousands of units) of lawn mowers (a seasonal product) are modeled by $S =\ 74\ +\ 3t\ -\ 40\ cos(\pi t/6)$, where $t$ is the time (in months), with $t=1$ corresponding to January. Graph the sales function over 1 year.

Key Concepts

Graphing Composite Functions

Graphing a composite function involves plotting a function that is made up of multiple parts, such as the combination of a linear trend and a periodic function. This requires understanding how each component influences the overall shape of the graph, particularly when interpreting the seasonal variations superimposed on a continuous growth or decline.

Linear Trends

A linear trend represents a constant rate of change. In the sales function, the linear component (74 + 3t) captures the general upward (or downward) tendency in sales over time, independent of the seasonal fluctuations introduced by the cosine term.

Cosine Function Parameters

The cosine function in the model contributes key features such as amplitude and period. The amplitude determines the extent of the seasonal variation, while the argument of the cosine function (?t/6) sets the period of the cycle, ensuring that the repeated pattern aligns with the seasonal behavior typical for a year.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals over time. In the context of the sales model, the cosine term introduces a seasonal fluctuation, meaning that sales follow a repeating pattern within each year.

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