Sketch one complete cycle of the following function by first graphing the appropriate sine or cosine curve and then using the reciprocal relationships.
A)
B)
C)
D)
E)

Question

Sketch one complete cycle of the following function by first graphing the appropriate sine or cosine curve and then using the reciprocal relationships.  ​
A)​ 
B)​ 
C)​ ​ 
D)​ ​ 
E)​ ​

Quizplus · Accepted Answer

For each of the functions given, we can first determine if it is a sine or cosine function based on the "hump" in the middle of the interval. We can then determine the amplitude and period of the function to graph it appropriately.

For function (C), we have a cosine function since the maximum occurs at x=1. The amplitude is 3 and the period is 2π/5. We can then sketch the cosine curve and label the intercepts, maximum, and minimum points.

Next, we can use the reciprocal relationships to graph one complete cycle. The reciprocal of cosine is secant, so we can graph the secant function by reflecting the cosine curve over the y-axis and drawing vertical asymptotes at the x-intercepts. We can then label the intercepts and vertical asymptotes.

Finally, we can use the other reciprocal relationships to complete the cycle. The reciprocal of secant is cosine, so we can flip the secant curve over the x-axis to get the cosine curve for the second part of the cycle. We can then use the other reciprocal relationships to complete the rest of the cycle.

Option C is the best choice because it allows for the clearest and most straightforward use of the reciprocal relationships to graph one complete cycle.

Sketch One Complete Cycle of the Following Function by First