Problem 57 - Precalculus: A Concise Course

Question

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

Step-by-Step Solution

Answer

The graph of the function \(y=3\cos(x+\pi)-3\) is a cosine function with amplitude 3, shifted \(\pi\) units to the left and 3 units down. The complete sketch would show these characteristics over two full periods of the function.

Step 1: Identify function attributes

The given function is \(y=3 \cos (x+\pi)-3 \). The number 3, multiplicative to the cosine function, is the amplitude of the function. The \(\pi\) added to \(x\) inside the cosine function is a horizontal shift (a phase shift) of \(\pi\) units to the left. The minus 3 after the cosine function is a vertical shift of 3 units downwards.

Step 2: Sketch the unshifted function

First, sketch the graph of \(y=3\cos(x)\) without taking the shifts into account. Since the period of the cosine function is \(2\pi\), the graph will have a maximum value of 3 at \(x=0\) and \(x=2\pi\) (and all multiples of \(2\pi\)), will cross the x-axis at \(x=\pi/2, 3\pi/2\) (and these values plus any multiples of \(2\pi\)), and will have a minimum value of -3 at \(x=\pi\) (and \(x=\pi\) plus multiples of \(2\pi\)).

Step 3: Apply the shifts

Next, apply the mentioned shifts to this graph. The phase shift should be applied first by shifting each x-coordinate \(\pi\)-units to the left. Then, apply the vertical shift by subtracting 3 from each y-coordinate.

Step 4: Sketch two full periods

The last step is to sketch two complete periods of the function. Remember, the period of the cosine function is \(2\pi\), so two full periods will extend from \(0\) to \(4\pi\). Because there is a phase shift, this will effectively extend from \(-\pi\) to \(3\pi\). Remember that the graph should still maintain the same shapes and feature points as the unshifted function.