91Ó°ÊÓ

Since we’re working with a cotangent function, the period of the function is \( \pi \). Therefore, plotting two periods of the function will span the intervals \(0 \) to \( 2\pi \), including the usual points at which the cotangent function is undefined.

The function is shifted by \( \frac{\pi}{2} \) to the left, as evidenced by the term \( x+\frac{\pi}{2} \) in the cotangent function. This phase shift will affect the x-coordinates of the points in our graph.

The cotangent function is vertically stretched by a factor of 3, as indicated by the coefficient in front of the cotangent function. The vertical stretch affects the amplitude of the function but not its period or phase shift.

Consider the x-values across the span of two periods, taking into account the phase shift to the left. Mark the points on the x-axis where the original cotangent function is undefined (which are usually multiples of \( \frac{\pi}{2} \)), then shift these points to the left by \( \frac{\pi}{2} \). Plot the points of the cotangent function for these x-values, remembering to multiply all y-values by 3 due to the vertical stretch factor. Then, complete the graph by drawing in the 'U' shapes of the function between every two undefined points. Since the cotangent function approaches negative infinity as x approaches the points where it is undefined from the left, and it approaches positive infinity as x approaches these points from the right, ensure that the 'U' shapes reflect this behavior.