Which of the following could be the equation of one cycle of...

Which of the following could be the equation of one cycle of the graph in the figure above?
I. $y=\sin 4 x$
II. $y=\cos \left(4 x-\frac{\pi}{2}\right)$
III. $y=-\sin (4 x+\pi)$
(A) only I
(B) only I and II
(C) only II and III
(D) only II
(E) I, II, and III

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Key Concepts

Trigonometric Functions

These are periodic functions like sine and cosine that exhibit regular repeating patterns, which makes them ideal for modeling cycles. Understanding their standard behavior is key to analyzing how modifications such as stretching, compressing, or shifting affect their graphs.

Period

The period of a trigonometric function is the horizontal length required for the function to complete one full cycle. For functions of the form y = sin(bx) or y = cos(bx), the period is calculated as 2?/|b|. This concept is essential for matching a given cycle of a graph with its algebraic representation.

Phase Shift

Phase shift refers to the horizontal translation of the graph of a trigonometric function. It comes into play when the function’s argument is modified by an addition or subtraction (e.g., sin(bx - c) or cos(bx + c)). Recognizing and calculating phase shifts helps in aligning the starting points of cycles correctly.

Amplitude and Reflection

Amplitude indicates the vertical stretch or compression of a trigonometric graph, and a negative sign in front of the function indicates a reflection across the horizontal axis. Both amplitude changes and reflections can alter the appearance of the cycle, so understanding these concepts is important when matching equations to their graphs.

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