Answer:
[tex]\textsf{D)}\quad y=2\cos \left(x-\dfrac{\pi}{2}\right)[/tex]
Step-by-step explanation:
The coefficient before a sine or cosine function represents its amplitude, which indicates the maximum distance of the function's values from its central axis. With an amplitude of 2, this means that the wave oscillates between -2 and 2.
When we add or subtract a value from the x-variable of a sine or cosine function, it translates the graph horizontally. Since all the given answer options involve adding or subtracting π/2 from the x-variable, this indicates that the graph has been translated horizontally from the position of its parent function y = sin(x) or y = cos(x).
The parent sine function intercepts the y-axis at (0, 0), which is the midpoint between its minimum and maximum values, while the parent cosine function intercepts the y-axis at its maximum value of (0, 1). Given that the graphed function intercepts the y-axis at (0, 0) but has been horizontally shifted, it must be a cosine function.
The peaks of the parent cosine function y = cos(x) occur at x = 2πn, where n is an integer. Since the peaks of the graphed function occur at x = (π/2)n + 2πn, the graph has been translated π/2 units to the right. To translate the graph k units to the right, we subtract k from the x-variable. Therefore, the equation that matches the graphed function is:
[tex]y=2\cos \left(x-\dfrac{\pi}{2}\right)[/tex]
