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Understanding the period of a trigonometric function is crucial to analyzing its behavior over an interval. The period refers to how long it takes for the function to repeat its pattern.
For a function like \( f(x) = \cos 2x \), the period is determined by the formula: \[ T = \frac {2\pi} { |k|}, \]where \( k \)is the coefficient of \( x \)in \( \cos kx \). In our example, \( k = 2 \).Thus, the period is \( \pi \), meaning \( \cos 2x \)repeats its pattern every \( \pi \)units along the x-axis.
What does this mean?

• The graph will have one complete cycle within the range of \((0, \pi)\).
• If you extend the graph, it will show an identical cycle from \((\pi, 2\pi)\), and so on.

This periodic nature is fundamental in many applications, including engineering and signal processing.

Amplitude is the measure of how "tall" or "short" the peaks and valleys of the wave are. It largely affects how much the function oscillates above and below the central axis.
In our function \( f(x) = \cos 2x \), amplitude is determined by the coefficient \( A \) in the general form \( A\cos(kx) \). Hence, for \( f(x) = \cos 2x \), \( A \) is 1.
This means:

• The highest point on the graph is 1.
• The lowest point on the graph is -1.

Amplitude tells us not just how high or low the function can go, but gives us insights about the energy or intensity of the wave. Even when the function stretches or compresses based on different amplitudes, it maintains the essential pattern and repeats itself periodically.

The concept of phase shift helps us understand how a trigonometric function may be shifted horizontally, impacting where the cycle starts. Typically, the horizontal shift is dictated by the phase angle in the expression \( \cos(kx + c) \).
In the given function \( f(x) = \cos 2x \), there is no additional term with \( x \) inside the cosine function, meaning there is no horizontal shift. This concludes that the phase shift is zero.
What you observe is:

• The graph begins at (0, 1) – no displacement from the origin.
• The entire pattern starts and ends in the expected interval with no distortion.

Phase shift plays a significant role when comparing or combining different trigonometric functions. The absence of phase shift in this scenario implies that \( \cos 2x \)arises in its standard form, showcasing its inherent repetition without any offsets.