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To analyze the properties of the graph of \( y = \sec x \), let's go through each option given:

1. **Range: \( (-\infty, \infty) \)** - This is **incorrect**. The range of \( \sec x \) is actually \( (-\infty, -1] \cup [1, \infty) \).

2. **Period = \( 2\pi \)** - This is **correct**. The secant function \( \sec x \) has a period of \( 2\pi \) because \( \sec x = \frac{1}{\cos x} \) and the cosine function repeats every \( 2\pi \).

3. **Even function** - This is **correct**. The function \( \sec x \) is even because \( \sec(-x) = \sec x \).

4. **Domain: \( (-\infty, \infty) \)** - This is **incorrect**. The function \( \sec x \) is undefined where \( \cos x = 0 \), which occurs at \( x = \frac{\pi}{2} + k\pi \) for integer \( k \). So, the domain excludes these points.

5. **Range: \( (-\infty, -1] \cup [1, \infty) \)** - This is **correct**. As discussed, this is the correct range for \( \sec x \).

6. **Domain: All real numbers except \( k\pi \), where \( k \) is an integer** - This is **incorrect**. \( \sec x \) is undefined where \( \cos x = 0 \), which is at \( x = \frac{\pi}{2} + k\pi \), not at multiples of \( \pi \).

7. **Domain: All real numbers except \( \frac{\pi}{2} k \), where \( k \) is an odd integer** - This is **correct**. This matches the points where \( \sec x \) is undefined.

8. **Odd function** - This is **incorrect**. \( \sec x \) is an even function, not odd.

9. **Period = \( \pi \)** - This is **incorrect**. The period of \( \sec x \) is \( 2\pi \), not \( \pi \).

### Summary of Correct Answers:
- **Period = \( 2\pi \)**
- **Even function**
- **Range: \( (-\infty, -1] \cup [1, \infty) \)**
- **Domain: All real numbers except \( \frac{\pi}{2} k \), where \( k \) is an odd integer**

Would you like more details on any of these properties or a specific explanation?

Here are 5 related questions to expand on this:
1. What is the range of the cosine function, and how does it relate to the secant function?
2. How can we determine the period of a trigonometric function like \( \sec x \)?
3. Why is the secant function even, and what symmetry property does it exhibit?
4. What other trigonometric functions have a similar domain restriction as \( \sec x \)?
5. How would the graph of \( \sec x \) change if we modified the period?

**Tip:** Remember, the secant function \( \sec x \) has vertical asymptotes wherever \( \cos x = 0 \), which helps identify where the function is undefined.

Identify the characteristics of the graph of y = sec x. Check all that applies.

Explore the key properties of the secant function y = sec x, including its domain, range, period, and whether it is an even function. This question helps high school students understand the characteristics of trigonometric functions.

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