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Chapter 4: Problem 67

Use a graphing utility to graph the function. (Include two full periods.) Be sure to choose an appropriate viewing window. $$y=-2 \sin (4 x+\pi)$$

Short Answer

Expert verified
The graph of the function \(y=-2 \sin (4 x+\pi)\) with the appropriate viewing window displays two full periods where the y-values oscillate between -2 and 2 from x= 0 to x= 蟺. The function is a sinusoidal wave that peaks at y = -2, rises to y = 2, then decreases back to y = -2, repeating this pattern twice between x=0 and x=蟺.

Step by step solution

01

Identify the amplitude, frequency, period, and phase shift of the function

In our function, \(y=-2 \sin (4 x+\pi)\), the amplitude, A, is -2, suggesting the function will have a peak value of -2 and a minimum value of 2. The coefficient of x inside the sine function is B=4, so the frequency of the function is 4, meaning it will repeat itself four times in the interval \([0,2蟺]\). The period of the function, which is the length over which the function repeats, can be calculated as \(T = \(\frac{2蟺}{abs(B)}\) = \(\frac{2蟺}{4}\)= 蟺/2 \). The term +蟺 inside the sine function represents a phase shift, indicating that the function will be shifted to the left by 蟺 units. Note that if we had a negative sign instead of plus it would be to the right.
02

Choose an appropriate viewing window

Since our function has a period of 蟺/2 and repeats four times in a full cycle of \([0,2蟺]\), we will need to select a viewing window that shows at least two full periods of the function. This suggests we need to see the function from at least 0 to 蟺 as that gives us two full periods (two repetitions of the function). The y-values of the function will oscillate between -2 and 2 due to the amplitude of the function. So the viewing window on graphing utility should be set accordingly.
03

Graph the function

Once we have chosen our viewing window, we can graph the function \(y=-2 \sin (4 x+\pi)\) on the graphing utility. The graph should display two full periods between 0 and 蟺 as our x-values and should oscillate between -2 and 2 for the y-values. The graph will be a wave that peaks at y = -2, increases to y = 2, then decreases back to y = -2 before repeating this pattern.

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