Problem 21 - Precalculus Essentials

Question

Graph two periods of the given cotangent function. $$y=-3 \cot \frac{\pi}{2} x$$

Step-by-Step Solution

Answer

The graph of \(y = -3 \cot(\frac{\pi}{2}x)\) consists of periods from 0 to 4 and then from 4 to 8 each consisting of an upward curve from -3 to 3 separated by vertical asymptotes.

Step 1: Identify the Period

The period of the cotangent function is affected by the multiplier of x. Usually, the period of y = cot(x) is π but here we have \(y = -3 \cot(\frac{\pi}{2}x)\), so the period is \(\frac{2\pi}{\frac{\pi}{2}}\), which simplifies to 4.

Step 2: Identify the Vertical Stretch

The key number that vertically stretches a cotangent function is the multiplier of the cotangent function. In this instance, the number is -3 for \(y = -3 \cot(\frac{\pi}{2}x)\). Because it's negative, the graph would be reflected about the x-axis, hence this value not only stretches the graph but also flips it.

Step 3: Plot the Key Points and Asymptotes for One Period

Draw a sketch labeling the x-axis from 0 to 4, this is one period for this given function. Make a note of where the asymptotes should be: at the start and end of the period, hence at x = 0 and x = 4. Then, plot the points at one quarter and three quarters through the period, which would be x = 1 and x = 3. Since it's a negative cotangent function, at x = 1 (which is one quarter into the period), the function equals to -3 (the minimum value for this function), and at x = 3 (which is three quarters into the period), the function equals to 3 (maximum value).

Step 4: Sketch the Graph

To sketch the graph for one period, start with an open circle at x = 0 (an asymptote, so the function is undefined here) at y = 3, sketch a curve going down to the point (1, -3), then going up to an open circle at (4, 3), where x = 4 is again an asymptote for the next period.

Step 5: Repeat for more periods

To sketch two periods of the function, repeat the same pattern from x = 4 to x = 8. Thus, the graph completes two periods between x = 0 and x = 8.