The amount of sulfur dioxide pollutant from heating fuels...

The amount of sulfur dioxide pollutant from heating fuels released in the atmosphere in a city varies seasonally. Suppose the number of tons of pollutant released into the atmosphere during the $n$ th week after January 1 for a particular city is given by $$ A(n)=1.5+\cos \frac{n \pi}{26} \quad 0 \leq n \leq 104 $$ Graph the function over the indicated interval and describe what the graph shows.

The amount of sulfur dioxide pollutant from heating fuels released in the atmosphere in a city varies seasonally. Suppose the number of tons of pollutant released into the atmosphere during the $n$ th week after January 1 for a particular city is given by
$$
A(n)=1.5+\cos \frac{n \pi}{26} \quad 0 \leq n \leq 104
$$
Graph the function over the indicated interval and describe what the graph shows.

Key Concepts

Periodic Functions

Periodic functions are those that repeat their values in regular intervals or cycles. They are essential for modeling phenomena that exhibit regular, repeating patterns over time, such as seasonal variations, sound waves, and light waves. In this context, the function shows how the sulfur dioxide pollutant levels repeat over a yearly period.

Cosine Function

The cosine function is a fundamental trigonometric function that models oscillatory behavior. In its basic form, cosine produces a wave-like pattern that is symmetric and periodic. It is particularly useful in illustrating natural cyclic phenomena due to its smooth, continuous oscillations.

Amplitude and Vertical Shift

Amplitude refers to the height of the wave from the midline, indicating the extent of the oscillation's deviation from its average value. A vertical shift moves the entire function up or down, altering the midline around which the oscillations occur. These transformations help tailor the cosine function to match observed data by adjusting its maximum and minimum values.

Frequency and Period

Frequency refers to the number of cycles the function completes in a unit interval, while the period is the length of one complete cycle. They are inversely related. In the function provided, the argument of the cosine determines the period, which is crucial for modeling the timing of the seasonal changes in pollutant levels.

Graphical Representation of Functions

Graphing functions involves plotting the output of the function against its input to visualize trends, patterns, and behaviors. For a periodic function, the graph highlights the repeated cycles, amplitude, period, and any transformations, enabling a clear understanding of how the modeled phenomenon varies over time.

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