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Chapter 4: Problem 48

Graph two periods of each function. $$y=\csc \left(2 x-\frac{\pi}{2}\right)+1$$

Short Answer

Expert verified
With a period of \(\pi\), phase shift of \(\frac{\pi}{4}\) to the right, and vertical shift upward by 1 unit, the two periods of the function \(y = \csc(2x - \frac{\pi}{2})+1\) can be accurately graphed.

Step by step solution

01

Finding the period

The standard formula for the period of a transformed trigonometric function is given by \( P = \frac{2\pi}{|b|} \), where \(b\) is a coefficient bound with the argument of the considered function. In this case, \(b = 2\), so the period of the function \( y = \csc(2x - \frac{\pi}{2})+1 \) will be \( P = \frac{2\pi}{2} = \pi \) - the function repeats every \( \pi \) units.
02

Identifying the phase shift

The phase shift of the function can be found by setting the argument of the function equal to zero. In this case, setting \(2x - \frac{\pi}{2}\) equal to zero gives \(x = \frac{\pi}{4}\). Therefore the function is shifted \(\frac{\pi}{4}\) units to the right.
03

Identifying the vertical shift

The function is shifted upwards by 1 unit, as indicated by the '+1' at the end of the function.
04

Sketching the function

Utilizing the information, it is known that every \(\pi\) units the function will repeat itself. Moreover, we know that it has been horizontally shifted by \(\frac{\pi}{4}\) to the right and vertically upward by 1 unit. Now, draw a clear and comprehensible graph accordingly. Remember that cosecant function will have asymptotes where corresponding sine function crosses x-axis. Use correct labeling.

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Most popular questions from this chapter

Use a graphing utility to graph two periods of the function. $$y=0.2 \sin \left(\frac{\pi}{10} x+\pi\right)$$

The number of hours of daylight in Boston is given by $$ y=3 \sin \frac{2 \pi}{365}(x-79)+12 $$ where \(x\) is the number of days after January 1 a. What is the amplitude of this function? b. What is the period of this function? c. How many hours of daylight are there on the longest day of the year? d. How many hours of daylight are there on the shortest day of the year? e. Graph the function for one period, starting on January 1

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