Problem 29 - Precalculus Essentials

Question

Graph two periods of the given cosecant or secant function. $$y=3 \csc x$$

Step-by-Step Solution

Answer

Firstly identified the important properties of the function, such as the period and phase shift. Focused on determining the asymptotes of the function, which occur where the function is undefined. The asymptotes were found to be \(x=n\pi\), where \(n\) is an integer. Plotted key points between the asymptotes and lastly drew a graph for two periods approaching the asymptotes and going through the key points.

Step 1: Identifying function properties

The given function is \(y=3 \csc x\), which is a cosecant function. The multiplier 3 will affect the amplitude of the function, but since the amplitude is not a significant characteristic of the cosecant function, it mainly affects the maximum and minimum point's y-values. The function has no phase shift. The period of the function is \(2\pi\) since there's no coefficient attached to the 'x'.

Step 2: Determining the Asymptotes

The asymptotes of the function are at the values where sine function is 0 because cosecant is the reciprocal of sine. Asymptotes will occur where \(x=n\pi\), where \(n\) is an integer. The asymptotes for two periods can occur at \(x=0, \pi, 2\pi, 3\pi\). Plot these vertical lines to mark the asymptotes.

Step 3: Plotting the Key Points

Aside from the asymptotes, let's determine the key points between to draw a graph. For cosecant, the key points usually sit halfway between the asymptotes. So they would be at \(x=\pi/2, 3\pi/2, 5\pi/2, 7\pi/2\). The sine values at these points are 1, -1, 1, and -1 respectively. Since this is 3 times the cosecant function, the y-values at these points will be 3 and -3 alternatingly.

Step 4: Graphing the function

Draw concave open curves that approach the asymptotes and go through the key points. Again, remember the cosecant function never touches the x-axis, as the curve comes close, but never reaches it. This pattern will repeat for every period.