1. graph y = 2tan(x)-1

Explanation:

Step1: Recall properties of tangent function

The parent - function of (y = 2\tan(x)-1) is (y=\tan(x)) which has vertical asymptotes at (x=\frac{\pi}{2}+n\pi), (n\in\mathbb{Z}), period (\pi), and passes through the origin ((0,0)).

Step2: Consider the vertical stretch

The coefficient 2 in front of (\tan(x)) vertically stretches the graph of (y = \tan(x)) by a factor of 2. That is, for each (x) - value, the (y) - values of (y = 2\tan(x)) are twice the (y) - values of (y=\tan(x)).

Step3: Consider the vertical shift

The (- 1) in the function (y = 2\tan(x)-1) shifts the graph of (y = 2\tan(x)) down 1 unit.

Step4: Find key points

For (y=\tan(x)), key points in the interval ((-\frac{\pi}{2},\frac{\pi}{2})) are ((-\frac{\pi}{4}, - 1)), ((0,0)), ((\frac{\pi}{4},1)). For (y = 2\tan(x)), the corresponding points are ((-\frac{\pi}{4},-2)), ((0,0)), ((\frac{\pi}{4},2)). For (y = 2\tan(x)-1), the corresponding points are ((-\frac{\pi}{4},-3)), ((0, - 1)), ((\frac{\pi}{4},1)).

Step5: Sketch the graph

Sketch the vertical asymptotes at (x=\frac{\pi}{2}+n\pi), (n\in\mathbb{Z}), plot the key - points, and draw the graph of the tangent - type function passing through the key - points and approaching the asymptotes.

Answer:

Sketch the graph with vertical asymptotes at (x=\frac{\pi}{2}+n\pi), (n\in\mathbb{Z}), passing through key - points ((-\frac{\pi}{4},-3)), ((0, - 1)), ((\frac{\pi}{4},1)) and having a period of (\pi) and being vertically stretched by a factor of 2 and shifted down 1 unit from the graph of (y = \tan(x)).