The equation \( y = \frac{1}{2} + \cos x \) represents a vertical shift of the cosine function. Here's a detailed breakdown of this function:

### General form of the cosine function:
\[ y = A \cos(Bx + C) + D \]

In this case:
- Amplitude \( A = 1 \), because the coefficient of \( \cos x \) is 1.
- Period \( \frac{2\pi}{B} \), where \( B = 1 \), so the period is \( 2\pi \).
- Phase shift \( C = 0 \), because there's no horizontal shift.
- Vertical shift \( D = \frac{1}{2} \), shifting the entire graph up by 0.5 units.

Thus, the graph of \( y = \frac{1}{2} + \cos x \) is the standard cosine wave, but raised by 0.5 units along the y-axis.

Would you like to see more details or have any questions about this function?

Here are some related questions:
1. How does the amplitude affect the cosine graph?
2. What happens if there is a horizontal phase shift?
3. How would the graph change if there was a coefficient in front of \( x \) in \( \cos x \)?
4. What is the range and domain of this function?
5. How does the vertical shift impact the maximum and minimum points of the graph?

**Tip:** When analyzing trigonometric functions, always consider how changes to the amplitude, period, and phase shift affect the graph.

Explore the cosine function y = 1/2 + cos x, which includes a vertical shift of 0.5 units. This problem covers trigonometric transformations such as amplitude, period, and vertical shifts, with a detailed explanation for students in Grades 10-12.

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