Problem 23 - Precalculus Essentials

Question

Graph two periods of the given cotangent function. $$y=3 \cot \left(x+\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

To graph two periods of the function \( y=3 \cot \left(x+\frac{\pi}{2}\right) \), recognize that it has a period of \( \pi \), a phase shift of \( \frac{\pi}{2} \) to the left, and a vertical stretch factor of 3. Plot the x-values corresponding to two periods of the function, consider the phase shift, and adjust the y-values by the stretch factor. The function's graph characteristics include undefined points at shifted multiples of \( \frac{\pi}{2} \), and between every two undefined points, there is a 'U' shape in the function's curve.

Step 1: Identify the period of the function

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.

Step 2: Identify the phase shift of the function

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.

Step 3: Identify the vertical stretch factor

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.

Step 4: Plot the graph

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.