The graph of each function in Exercises oscillates between...

The graph of each function in Exercises oscillates between two parallel lines, as in Example 5. Find the equations of the two lines and graph the lines and the function in the same viewing window.
$$y=1+x+\cos 3 x$$

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Key Concepts

Amplitude

The amplitude is the maximum displacement of the oscillatory (cosine) part of the function from its midline. In a function of the form f(x) = linear part + A cos(Bx), the cosine term oscillates between -A and A, so the amplitude is A.

Midline/Linear Component

The midline or baseline of the oscillation is given by the non-oscillatory part of the function. When a cosine function is added to a linear function (or any other function), the oscillatory behavior occurs around the graph of that linear component. This midline is defined by the linear expression present in the function.

Envelope Functions

Envelope functions are the two boundary lines that the oscillatory graph does not cross. For a function in the form f(x) = g(x) + A cos(Bx), the upper and lower envelope lines are represented by y = g(x) + A and y = g(x) - A, respectively. These lines represent the maximum and minimum bounds of the function's graph.

Graphing Oscillatory Functions

Graphing functions that combine both a linear (or other slowly varying) component and a periodic oscillatory component requires plotting the base function along with its envelope functions. This visualization helps in understanding the overall behavior of the function, where the oscillations occur about a steadily increasing or decreasing base, bounded by two parallel lines.

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