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Chapter 4: Problem 104

Use a graphing utility to graph two periods of the function. $$y=3 \sin (2 x-\pi)+5$$

Short Answer

Expert verified
The function \(y=3 \sin (2x-\pi)+5\) has an amplitude of 3, a period of \(\pi\), phase shift of \(\pi\) units to the right, and a vertical shift of 5 units upwards. The graph oscillates between 2 and 8, repeating every \(\pi\) units for two periods.

Step by step solution

01

Identify Values

In the given function \(y=3 \sin (2x-\pi)+5\), we have \(a = 3\), \(b = 2\), \(c = \pi\), and \(d = 5\). This indicates that the amplitude of the function is 3, there is a horizontal compression by a factor of 1/2, there is a phase shift of \(\pi\) to the right, and there is a vertical shift of 5 units upwards.
02

Calculate the Period

For the function \(y = \sin(bx)\), the period is \(2\pi / |b|\). So, for the function \(y=3 \sin (2x-\pi)+5\), the period would be \(2\pi / 2 = \pi\). So, the function repeats every \(\pi\) units.
03

Determine Key Points

For a single period of the shifted and scaled sine function, the key points are usually taken at the end points and the maximum and minimum values. The end points for two periods of the given function will be at \(x = 0, \pi, 2\pi\). The maximum and minimum occur midway between these, at \(x = \pi/2, 3\pi/2\). The function values at these points are: \[y(0) = 3 \sin(-\pi) + 5 = 5 - 3 = 2\] \[y(\pi/2) = 3 \sin(\pi) + 5 = 5 + 3 = 8\] \[y(\pi) = 3 \sin(2\pi - \pi) + 5 = 5 - 3 = 2\] \[y(3\pi/2) = 3 \sin(3\pi - \pi) + 5 = 5 + 3 = 8\] \[y(2\pi) = 3 \sin(4\pi - \pi) + 5 = 5 - 3 = 2\]
04

Plot the Graph

Using a graphing utility and the key points determined in Step 3, plot the function for two periods. The horizontal compression will make the sine curve 'move' faster between the maximum and minimum. The phase shift moves the entire function one unit to the right and the vertical shift moves all points up by 5 units. Finally, the amplitude stretches the function by a factor of 3. The function will oscillate between 2 and 8, repeating every \(\pi\) units.

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