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The tangent function is one of the basic trigonometric functions and is often represented as \(\tan(x)\). It is defined in terms of sine and cosine functions, where \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). This function has some unique characteristics:

• It is periodic, meaning it repeats its values at regular intervals.
• The tangent function has an infinite range, covering all real numbers.
• It has vertical asymptotes where the cosine function is zero, leading to undefined values.

In the context of the given function, \(y = 7 \tan(2 \pi x)\), the "7" is a scaling factor that modifies the amplitude of the tangent wave, but does not affect its period or frequency. The core component to understand here is \(\tan(2\pi x)\), which adjusts how often the function repeats its pattern.

The period of a function in trigonometry refers to the interval at which the function repeats its values. For the tangent function \(\tan(x)\), the standard period is \(\pi\), as the function repeats every \(\pi\) units along the x-axis. Typically, when a tangent function includes a frequency factor \(b\) in the form of \(\tan(bx)\), the period changes. The period can be calculated using the formula:

• Period of tangent function: \(\pi / b\)

In our specific problem, \(b = 2\pi\). Applying the formula, the period becomes \(\pi / (2\pi) = 1/2\). This means the function \(y = 7 \tan(2 \pi x)\) repeats every \(1/2\) units. This characteristic is crucial for predicting the behavior of the function over different domains.

The frequency factor in trigonometric functions determines how often oscillations occur within a given interval. In the form of the tangent function \(\tan(bx)\), \(b\) is the frequency factor. A larger \(b\) compresses the waveform horizontally, leading to a higher frequency and a smaller period. Conversely, a smaller \(b\) stretches the waveform, resulting in a lower frequency and a longer period.
For the function \(y = 7 \tan(2 \pi x)\), the frequency factor is \(2 \pi\). It signifies that within one complete cycle of \(2 \pi\) radians of a regular tangent function, this function completes two full cycles. This is why the period reduces to \(1/2\) instead of the typical \(\pi\). Understanding the frequency factor helps in visualizing how the function behaves and how quickly it oscillates, which is essential for applications in physics, engineering, and signal processing.