Problem 55 - Algebra and Trigonometry

Question

Use a vertical shift to graph one period of the function. $$y=\cos x-3$$

Step-by-Step Solution

Answer

The graph of the function \(y = \cos x - 3\) is a cosine wave that oscillates between -2 and -4 with a period of \(2π\), centered around \(y = -3\).

Step 1: Understanding the basic cosine function graph

Start with the fundamental graph of \(y = \cos x\). It's a wave that oscillates between -1 and 1, with a period of \(2π\) (meaning it repeats every \(2π\) units).

Step 2: Applying the vertical shift

The function \(y = \cos x - 3\) is a regular cosine function, but it's shifted down by 3 units. That implies every point on the \(y = \cos x\) graph will be moved three units downwards to get the graph of \(y = \cos x - 3\). This shifting doesn't affect the shape or the period of the cosine function; it only changes the location of the graph.

Step 3: Graphing the function

First, draw the basic cosine function. Then, move each point on the graph three units down. Another perspective is to think of the center line of the wave as being at \(y = -3\) instead of the x-axis. The wave still oscillates 1 unit above and below this center line, meaning it reaches a maximum of \( -3 + 1 = -2\) and a minimum of \(-3 - 1 = -4\) within each period.