91Ó°ÊÓ

The arctan function, short for the arctangent function, is part of the family of inverse trigonometric functions. It is represented as \( \arctan(x) \) and is the inverse of the tangent function, \( \tan(x) \). This means that if \( y = \arctan(x) \), then \( x = \tan(y) \).
The range of the arctan function is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), representing the output values of \( y \). This defines the endpoints of the central periods of the trigonometric function.
Key features of the arctan function include:

• It is an odd function, which means it is symmetric about the origin. For any \( x \), \( \arctan(-x) = -\arctan(x) \).
• The derivative of \( \arctan(x) \) is \( \frac{1}{1+x^2} \), indicating how rapidly the function changes with respect to \( x \).
• The function approaches \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \) as \( x \to -\infty \) and \( x \to \infty \), respectively. These are the horizontal asymptotes.

A vertical shift involves adding a constant to the function, which translates the entire graph up or down.
In the function \( f(x) = \frac{\pi}{2} + \arctan(x) \), the term \( \frac{\pi}{2} \) is added to the arctan function, resulting in a vertical shift upwards by \( \frac{\pi}{2} \) units.
This transformation affects the position of the horizontal asymptotes, originally at \( y = -\frac{\pi}{2} \) and \( y = \frac{\pi}{2} \). After the upward shift, these asymptotes move to \( y = 0 \) and \( y = \pi \).
Key aspects of a vertical shift include:

• It does not change the shape or orientation of the graph.
• The entire graph moves vertically, including asymptotes and key points.
• Vertical shifts are a simple arithmetic operation applied to all y-values of the function.

Horizontal asymptotes are lines that a graph approaches but never reaches as \( x \) goes to positive or negative infinity. They are crucial in determining the end-behavior of functions.
For the arctan function, \( \arctan(x) \), the horizontal asymptotes are originally at \( y = -\frac{\pi}{2} \) and \( y = \frac{\pi}{2} \).
These lines guide the behavior of the function, indicating where the function will level off as \( x \) becomes very large in the positive or negative direction.
With the vertical shift applied in the function \( f(x) = \frac{\pi}{2} + \arctan(x) \), the asymptotes are shifted up by \( \frac{\pi}{2} \) units, resulting in new asymptotes at \( y = 0 \) and \( y = \pi \).
Important points about horizontal asymptotes include:

• They describe the \'long-term\' behavior of a function as \( x \to \infty \) or \( x \to -\infty \).
• They indicate the height at which the function \( f(x) \) will hover indefinitely.
• Understanding asymptotes is key for sketching the graph accurately and predicting the function\'s behavior.