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The cosine function, represented as \(y = \cos x\), is a fundamental trigonometric function. It’s one of the building blocks for understanding periodic functions. The graph of the basic cosine function resembles a wave, and it repeats its pattern every \(2\pi\) radians, constituting one full period. Typically, the cosine wave starts at its maximum point, then decreases to zero, continues to the minimum, and rises back to the maximum point within this interval. This pattern is common and can be identified by these key points along the x-axis:

• Starts at (0,1)
• Reaches zero at (\(\frac{\pi}{2}\),0)
• Lowest at (\(\pi\),-1)
• Returns to zero at (\(\frac{3\pi}{2}\),0)
• Ends at (\(2\pi\),1)

Understanding and plotting the cosine function helps with visualizing its transformations, like shifting or stretching. This particular exercise deals with moving the graph up and down, known as the vertical shift.

A vertical shift in a graph occurs when the entire graph of a function moves either up or down along the y-axis. This shift changes the y-values of the graph but maintains the shape of the function. For the function \(y = \cos x - 3\), each point on the graph of the basic cosine function \(y = \cos x\) is moved down by 3 units. This means:

• The original starting point (0,1) will move to (0,-2)
• The zero crossing (\(\frac{\pi}{2}\),0) will move to (\(\frac{\pi}{2}\),-3)
• The lowest point (\(\pi\),-1) will shift to (\(\pi\),-4)
• The next zero crossing (\(\frac{3\pi}{2}\),0) will change to (\(\frac{3\pi}{2}\),-3)
• The end point (\(2\pi\),1) will be at (\(2\pi\),-2)

Vertical shifts do not alter the period or horizontal placement of the cosine wave, but they do change its range. In this case, the wave maintains its periodic behavior while all points are lowered by 3 units.

The Cartesian plane is a two-dimensional coordinate system that organizes and displays the location of points using an x-axis (horizontal) and y-axis (vertical). Each point is represented by an ordered pair (x, y). When graphing functions, such as \(y = \cos x - 3\), the Cartesian plane allows us to visually interpret the relationship between the x-values (inputs) and their corresponding y-values (outputs). When plotting the vertically shifted cosine function, we start by marking points calculated for the transformation, like those outlined in the previous sections. For instance, we first locate (0,-2), followed by (\(\frac{\pi}{2}\),-3), and so on, before smoothly connecting them. This resulting graph allows students to grasp how the function remains periodic but sits at a lower vertical position due to the vertical shift applied.