Question

Does \(y=\) tan \(x\) have an amplitude? Explain.

Short answer

The function \(y = \tan x\) does not have an amplitude because it does not have a fixed maximum or minimum value.

Step-by-step solution

Understand the Concept of Amplitude

Amplitude refers to the maximum value a periodic function reaches above or below its central axis. It is commonly associated with functions like sine and cosine.

Analyze the Tangent Function

The function given is the tangent function, defined as \(y = \tan x\). Unlike sine and cosine functions, which oscillate between maximum and minimum values, the tangent function does not have fixed maximum or minimum values.

Examine the Behavior of Tangent

The function \(y = \tan x\) has vertical asymptotes where it approaches positive or negative infinity. Specifically, it becomes undefined at \(x = \frac{\frac{\text{π}}{2}} and \frac{3\text{π}}{2}\), causing the function to extend infinitely in both positive and negative directions near these points.

Conclusion

Since amplitude refers to the maximum value of a periodic function and \(y = \tan x\) does not have fixed maximum or minimum values, it does not have an amplitude.

Key concepts

Amplitude

Amplitude is a concept often associated with periodic functions like sine (\( \text{sin} \theta \, \text{or} \, \text{cos} \theta \)) and cosine (\( \text{cos} \theta \)). These functions oscillate between fixed maximum and minimum values. For instance, sine and cosine functions have an amplitude of 1 because their maximum value is 1 and their minimum value is -1.
The amplitude represents the height from the centerline to the peak of the wave.
Unlike sine and cosine, the tangent function (\( y = \tan x \)) does not have an amplitude. This is because the tangent function extends infinitely and does not have fixed maximum and minimum values. This lack of a fixed peak or trough means there is no amplitude for the tangent function.
Thus, we conclude that the tangent function does not have an amplitude.

Periodic Functions

A periodic function is a function that repeats its values at regular intervals or periods.
Classic examples include sine and cosine functions, which have a period of \( 2\text{π} \).
The tangent function (\( y = \tan x \)) is also a periodic function but with some differences.
While sine and cosine have a period of \{2\text{π} \}, the tangent function repeats every \( \text{π} \).
This means that for any value of x, \( \tan(x + \text{π}) = \tan x \).
Being aware of the periodic nature helps in understanding the behavior and evaluation of the function over its entire domain.

Vertical Asymptotes

Vertical asymptotes are lines where the function grows without bound, approaching infinity or negative infinity. For the tangent function, vertical asymptotes occur where the function is undefined.
These points occur at \( \frac{\text{π}}{2}+\text{π}n \) for all integers n, because the function approaches infinity as it gets close to these points.
The existence of these asymptotes makes the tangent function unique compared to sine and cosine.
This feature is crucial for understanding the graph and behavior of the tangent function.

Tangent Function Behavior

The behavior of the tangent function sets it apart from other trigonometric functions. It exhibits unique characteristics, including:

• Undefined at Asymptotes: The function is undefined at \( \frac{\text{π}}{2}+\text{π}n \). Here, it approaches infinity or negative infinity.
• Periodic Nature: It repeats every \(\text{π}\).
• Infinite Range: Unlike sine and cosine, which are bounded, the tangent function's values extend from negative to positive infinity.

Understanding these properties helps in comprehensively grasping the behavior and applications of the tangent function.