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Chapter 4: Problem 29

Sketch one full period of the graph of each function. $$y=2 \tan \frac{x}{2}$$

Short Answer

Expert verified
The graph starts from about -2 at \(x = - \pi\), crosses the x-axis at \(x = 0\), and increases to 2 at \(x = \pi\). It has vertical asymptotes at \(x= - \pi\) and \(x = \pi\).

Step by step solution

01

Identify key features

The function \(y=2 \tan \frac{x}{2}\) is a modified tangent function (base function being \(y=\tan(x)\)). First, notice that there is a '2' outside the tan function - this is a vertical stretch and forms the amplitude of the function, so the amplitude is 2. Next, identify the period of the function which for tan is \(\pi\), but here it is multiplied by \(2\) in the denominator, so the period of the function will be \(2\pi\).
02

Draw the vertical asymptotes and x-intercept

Now that we have identified the amplitude and period, we can start graphing. The graph shall have two vertical asymptotes for one complete period, which will be at \(x= - \pi\) and \(x = \pi\) since a tangent function has vertical asymptotes at the ends of each period. The x-intercept will be at \(x = 0\) as the tangent function always passes through the origin.
03

Sketch the curve

Finally, we need to sketch the curve according to its unique shape for a tangent function. The graph starts from negative infinity at \(x = - \pi\), crosses the x-axis at \(x = 0\) (origin), and goes to positive infinity at \(x = \pi\). It shows the property of the tangent function that as it approaches its asymptotes, the value increases/decreases without bound.

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