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.