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Chapter 4: Problem 69

Use a graphing utility to graph the function. (Include two full periods.) Be sure to choose an appropriate viewing window. $$y=\cos \left(2 \pi x-\frac{\pi}{2}\right)+1$$

Short Answer

Expert verified
The graph of the function is a cosine wave with amplitude of 1, period of 1, shifted \(\frac{1}{4}\) units right, and shifted 1 unit upwards. The appropriate view window for the graph is from 0 to 2 for both x and y. This would depict two full periods of the function.

Step by step solution

01

Identify and understand the different parts of the function

The given function is \(y=\cos(2 \pi x - \frac{\pi}{2}) + 1\). Let us identify: A = 1, which is the amplitude. B = \(2\pi\), which determines the period. C = \( \frac{\pi}{2} \), which is the phase shift. D = 1, which is the vertical shift.
02

Compute the period of the function

The period of a cosine or sine function is given by \( \frac{2\pi}{|B|} \). Therefore, the period of the function is \( \frac{2 \pi}{|2 \pi|} = 1 \)
03

Determine the phase shift of the function

The phase shift of the function is obtained by solving the equation \(Bx - C = 0\), which gives \(x = \frac{C}{B} = \frac{\pi/2}{2 \pi} = \frac{1}{4}\). This means the graph of the function will be shifted to the right by \(\frac{1}{4}\).
04

Identify the vertical shift of the function

The vertical shift D is 1. This means that the function will be shifted upward by 1.
05

Choose view window and graph the function

Given the period is 1, and we need to include two full periods in our graph, we choose a view window from 0 to 2 for x-values. Since the amplitude is 1 and there's a vertical shift of 1, the y-value ranges from 0 to 2. The graph will be a wave that repeats every 1 unit (period), starting from \(\frac{1}{4}\) units right (phase shift), with the whole wave shifted 1 unit upward (vertical shift).

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