Determine the period of the function

y equals cosine left parenthesis 2 x right parenthesis
 
and sketch its graph.

The given function is \( y = \cos(2x) \).

### Determining the Period:

1. The general form of the cosine function is \( y = \cos(bx) \), where \( b \) affects the period of the function.
2. The period of the basic cosine function \( \cos(x) \) is \( 2\pi \).
3. When there is a coefficient \( b \) inside the cosine function (i.e., \( \cos(bx) \)), the period of the function becomes \( \frac{2\pi}{|b|} \).

In this case, \( b = 2 \), so the period of \( \cos(2x) \) is:
\[
\text{Period} = \frac{2\pi}{2} = \pi
\]

Thus, the period of the function \( y = \cos(2x) \) is \( \pi \).

### Sketching the Graph:

To sketch the graph of \( y = \cos(2x) \), we note the following:
- The period is \( \pi \), so one full cycle of the cosine function will repeat every \( \pi \) units along the x-axis.
- The amplitude remains 1 (since there is no vertical scaling).
- The function still oscillates between -1 and 1, but completes one cycle over an interval of length \( \pi \).

The key points for one period (from \( 0 \) to \( \pi \)) are:
- At \( x = 0 \), \( y = \cos(0) = 1 \)
- At \( x = \frac{\pi}{2} \), \( y = \cos\left( 2 \times \frac{\pi}{2} \right) = \cos(\pi) = -1 \)
- At \( x = \pi \), \( y = \cos(2\pi) = 1 \)

The pattern repeats every \( \pi \) units. The graph is a cosine wave that oscillates with a shorter wavelength compared to \( \cos(x) \).

Would you like a graph to visualize this, or do you have any specific questions about the period? 

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Here are 5 related questions you might explore:

1. What is the amplitude of the function \( y = 3 \cos(2x) \)?
2. How does the graph of \( y = \cos(2x + \frac{\pi}{4}) \) compare to \( y = \cos(2x) \)?
3. What is the phase shift of the function \( y = \cos(2x - \frac{\pi}{2}) \)?
4. How would the period change if the function were \( y = \cos(4x) \)?
5. What is the general formula for the period of the function \( y = \cos(bx) \)?

**Tip:** The period of any trigonometric function can be found by examining the coefficient of \( x \) inside the function.

Determine the period of the function y = cos(2x) and sketch its graph.

Learn how to determine the period of the function y = cos(2x) and sketch its graph. This trigonometry problem covers the calculation of the period using the formula for cosine functions and guides you through graphing the function with its key points. Suitable for students in Grades 10-12.

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