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The cosine function, often written as \( \cos x \), is a fundamental trigonometric function that describes the relationship between the angle in a right triangle and the length of the adjacent side over the hypotenuse. Imagine plotting \( \cos x \) on a graph; the shape you will see is a periodic wave that repeats every \( 2\pi \) radians.
The cosine function starts at 1 when \( x = 0 \), dips down to -1, and then returns to 1, creating a smooth curve. Importantly, it oscillates between -1 and 1. This makes it particularly useful for modeling cyclical phenomena such as waves.

• Range: -1 to 1
• Periodicity: Repeats every \( 2\pi \)
• Symmetry: Even function

The squared cosine function \( \cos^2 x \) is derived from \( \cos x \) and takes each value of \( \cos x \), squares it to produce a range from 0 to 1. Even though it looks similar, \( \cos^2 x \) doesn’t go below zero because squaring a negative number results in a positive value. Therefore, the feature of squaring gives us a slightly different wave pattern, which stays positive and reaches a maximum of 1.

Vertical transformations occur when alterations are applied to modify the output of a function, in this case, a trigonometric function. Generally, these changes involve shifting the function vertically on the graph, either upwards or downwards.
For the function \( y = 6 \cos^2 x - 3 \):

• The term \( 6 \cos^2 x \) effectively stretches the cosine squared function vertically. The amplitude of the cosine function, which originally ranges between 0 and 1, is multiplied by 6, expanding its range to 0 to 6.
• The "-3" indicates a vertical downward shift of the entire graph by 3 units. This offset means every point on the original \( 6 \cos^2 x \) graph is translated 3 units down. Thus, the final range of \( y \) is from -3 to 3.

This vertical transformation modifies both the amplitude and the midline of the function, altering the appearance and placement of the usual cosine wave pattern on the graph.

Plotting trigonometric functions, especially when transformations are applied, involves carefully noting the changes in position and shape of the graph. Here’s a step-by-step guide to plotting a function like \( y = 6 \cos^2 x - 3 \):

• **Identify Key Points:** First, understand and calculate specific values for certain angles like \( 0, \pi/2, \pi, 3\pi/2, \text{and} \, 2\pi \). These values often correspond to pivotal points where cosine functions reach local maxima or minima.
• **Plot Calculated Points:** Using the computes from the exercise, mark each point on graph paper. For instance, at \( x=0 \), \( y=3 \); at \( x=\pi/2 \), \( y=-3 \); continue with other angles.
• **Draw the Curve:** Connect the points smoothly with a wave-like pattern, ensuring it reflects the nature of a horizontally squished and vertically stretched cosine function.
• **Review Amplitude Changes:** Check how the vertical transformation adjusted the graph. Here, the amplitude is multiplied by 6, creating taller peaks before it is moved down by 3.

With these steps, you can sketch the curve accurately, which visually communicates how transformations impact trigonometric functions.