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When we explore the period of the tangent function, we're essentially looking at how long it takes for the function to repeat itself. - For the basic tangent function, written as \(y = \tan x\), the period is \(\pi\), which means after every \(\pi\) units along the x-axis, the pattern repeats itself. - Now, if we place a coefficient 'b' next to 'x' in the form \(y = \tan(bx)\), this changes the period.For our specific example of \(y = \frac{1}{2}\tan(2x)\):- The '2' next to the 'x' effectively compresses the function, making it repeat twice as fast.- As a result, the period is modified from \(\pi\) to \(\frac{\pi}{2}\). This compression means that the roots, peaks, and asymptotes occur more frequently over the x-axis. This is why graphing two periods in this specific case will range from 0 to \(\pi\), encompassing two full repetitions.

Amplitude is a familiar term with functions like sine and cosine, where it refers to the height of the wave's peaks and troughs. However, for the tangent function, the concept of amplitude doesn't quite apply in the traditional sense. - The tangent function extends infinitely in the vertical direction, so it technically doesn't have a maximum height or amplitude.- Instead, when we refer to the coefficient in front of the tangent function, such as \(\frac{1}{2}\) in \(y = \frac{1}{2}\tan(2x)\), we're looking at a vertical scaling factor.This means:- In this case, the function is vertically compressed by a factor of \(\frac{1}{2}\).- So, while it appears flatter, there isn't an actual "amplitude" like there is with sine and cosine. The tangent function's range remains unlimited.

Graphing trigonometric functions, especially the tangent, involves understanding their unique behavior over one period and beyond.### Sketching the Tangent Function- Begin by identifying key characteristics: the vertical asymptotes, zeros, and behavior around these points.- For \(y = \tan(x)\), the function has vertical asymptotes at \(x = \pm\frac{\pi}{2}, \pm\frac{3\pi}{2},\ldots\).### Graphing Transformed Tangent FunctionsIn the case of \(y = \frac{1}{2}\tan(2x)\):- The period is \(\frac{\pi}{2}\), cutting the usual period in half. Vertical asymptotes occur every \(\frac{\pi}{2}\) at points such as \(x = 0, \frac{\pi}{2}, \pi, \ldots\).- The function crosses the x-axis at \(x = 0, \frac{\pi}{4}, \frac{\pi}{2}, \ldots\).- Remember to scale the tangent's steepness by the factor \(\frac{1}{2}\), giving it a flatter appearance.Always sketch two periods for comprehensive understanding. Start from x = 0, mark the vertical asymptotes, and plot the path approaching and receding from these points. Use these basic principles for graphing all trigonometric functions effectively. They will guide you in placing asymptotes, finding intercepts, and sketching accurate graph curves.