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In the realm of trigonometry, understanding the relationship between functions is crucial for grasping concepts like the secant graph. The secant function, denoted as \(sec(x)\), is one of the reciprocal trigonometric functions, meaning it is the inverse of another fundamental trigonometric function. Specifically, \(sec(x)\) is the reciprocal of the cosine function, expressed as \(sec(x) = 1/\cos(x)\).

As with any reciprocal, when \(\cos(x)\) is zero, its reciprocal—\(sec(x)\)---is undefined because division by zero is not possible. This occurs at specific intervals where \(\cos(x) = 0\), leading to vertical asymptotes on the graph of secant at the corresponding \(x\) values. Recognizing these reciprocal relationships and their implications is essential when graphing functions like \(f(x)=\sec 2x\).

The periodicity of trigonometric functions refers to the property that allows the functions to repeat their values at regular intervals. For the secant function, this is particularly important when understanding its graph. Normally, \(\sec(x)\) and \(\cos(x)\) share the same period of \(2\pi\). This means they complete a full cycle and begin to repeat every \(2\pi\) radians.

However, when the function is altered, as we see with \(f(x)=\sec 2x\), the period changes. In this case, due to the inner multiplication by '2', the period is halved, resulting in a new period of \(\pi\). This means that \(f(x)=\sec 2x\) will repeat its pattern every \(\pi\) radians, leading to more frequent cycles on the graph. Understanding the concept of periodicity is necessary for marking the correct intervals and shapes when plotting the graph of a secant function.

Graphing transformations involve shifts, stretches, compressions, and reflections of the basic function graph. They are essential for visualizing how alterations in the function formula reflect on the graph. The secant function graph undergoes transformations when multiplied by a factor or when the argument is changed, such as with \(f(x)=\sec 2x\).

The inner multiplication by '2' within \(\sec(2x)\) causes the function to have a horizontal compression. This compression results in the aforementioned halving of the period, leading to a more frequent repetition of its 'U' shaped curves. Additionally, it also means that the vertical asymptotes and extrema occur at different \(x\)-values compared to the basic \(\sec(x)\) function. In effect, the graph will be a series of alternating 'U' and upside-down 'U' shapes, compressed horizontally to fit the new period. When plotting the graph, we account for these transformations to accurately portray the function's behavior over its domain.