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Chapter 4: Problem 56

Sketch the graph of the function. (Include two full periods.) $$y=2 \cos x-3$$

Short Answer

Expert verified
The graph of the function \(y=2 \cos x - 3\) is a cosine graph, vertically stretched by a factor of 2 and shifted down by 3 units. The function includes two complete periods from 0 to 4\pi.

Step by step solution

01

Identify amplitude, period, phase shift, and vertical shift

Start by identifying A, B, C and D from the given equation \(y=2 \cos x - 3.\) In this case, A=2 (amplitude), B=1 (so, the period is 2\pi), C=0 (no phase shift), and D=-3 (vertical shift).
02

Draw the basic cosine graph plot

Before adding in the changes to the function, draw the basic plot of the cosine function. This graph oscillates between -1 and 1, has a period of 2\pi, begins at the maximum point, and repeats after every 2\pi interval.
03

Apply amplitude and vertical shift

Now apply the amplitude and the vertical shift to the basic cosine graph. The amplitude, A=2, stretches the graph by a factor of 2 in the vertical direction. The vertical shift, D=-3, shifts the graph down by 3 units.
04

Plot the function on a graph

Finally, plot the function \(y=2 \cos x - 3.\) Include two full periods (from 0 to 4\pi).

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