Problem 104 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The function \(y=3 \sin (2 x-\pi)+5\) is graphed by understanding the amplitude, period, phase shift and vertical shift of the function and applying these modifications to the basic sinusoidal graph. The final graph depicts a sine function with amplitude of 3, a period of \(\pi\), a phase shift of \(\pi\), and an upward shift of 5 units.

Step 1: Identify & Understand the Function Components

The function given is \(y=3 \sin (2 x-\pi)+5\). Here, '3' is the amplitude, '2' is the frequency which determines the period of the function, \(-\pi\) is the phase shift and '5' is the vertical shift. The effect of these components is as follows: Amplitude ('3') will scale the function vertically. Frequency ('2') will adjust the period of the function. It is inversely proportional to the period or the distance of one full cycle of the wave. Phase shift (\(-\pi\)) will move the function horizontally on the x-axis. And finally, the vertical shift ('5') will move the function up or down along the y-axis.

Step 2: Graphing the Basic Function

Before modifications, graph a basic sine function which has an amplitude of 1, no phase shift, no vertical shift and a period of \(2\pi\). It starts from the origin (0,0), peaks at \(\pi/2\), goes to zero at \(\pi\), reaches its minimum at \(3\pi/2\), and goes back to zero at \(2\pi\), completing one period. Now, realize that the modifications will alter this standard pattern of the sine wave.

Step 3: Apply the Amplitude

This function's amplitude is 3. So, the maximum and minimum values of the function, which were 1 and -1, respectively, will now be 3 and -3. Change the range on your graph to reflect this.

Step 4: Apply the Period

The period of the function can be calculated as \(2\pi / |2| = \pi\). Thus, each cycle of our function will now be completed over a range of \(\pi\) instead of \(2\pi\). Adjust your graph to depict this.

Step 5: Apply the Phase Shift

The phase shift of the function is -π, which will move our function π units to the right. Incorporate this shift into your graph.

Step 6: Apply the Vertical Shift

Lastly, add the vertical shift of 5 units upwards. All points from the previously drawn graph should be shifted up by 5 units.

Step 7: Plot the Graph

Now plot the final graph with the graphing utility using the parameters and adjustments defined in the previous steps. Remember to plot two periods of the function.