Question

find the graph of (y = \cot(x)).

Explanation:

⚡ Using what you learned: Graphing Trigonometric Functions

Step 1: Identify key features of \( y = \cot(x) \)

The cotangent function is defined as:

$$ y = \cot(x) = \frac{\cos(x)}{\sin(x)} $$

• Vertical Asymptotes: Occur where \( \sin(x) = 0 \), which means at \( x = \pi n \) for any integer \( n \) (e.g., \( x = 0, \pi, -\pi \)).
• Intercepts (\(x\)-intercepts): Occur where \( \cos(x) = 0 \), which means at \( x = \frac{\pi}{2} + \pi n \) (e.g., \( x = \frac{\pi}{2}, -\frac{\pi}{2} \)).
• Behavior: Since \( \cot(x) \) is the reciprocal of \( \tan(x) \), it decreases on each interval between its asymptotes (sloping downwards from left to right).

Step 2: Compare with the given options

Let's analyze the three vertical graphs (oriented sideways in the image):

1. Top Graph: Shows curves that increase from left to right. This represents the tangent function, \( y = \tan(x) \).
2. Middle Graph: Shows curves that decrease from left to right, with a vertical asymptote at the vertical axis (\( x = 0 \)) and crossing the horizontal axis at half-intervals. This matches \( y = \cot(x) \).
3. Bottom Graph: Shows curves shifted horizontally, where the vertical axis (\( x = 0 \)) is not an asymptote but rather an intercept or a shifted point.

Therefore, the middle graph correctly represents \( y = \cot(x) \).

Answer:

The correct graph is the middle option.