Problem 8 - Precalculus Essentials

Question

Graph two periods of the given tangent function. $$y=2 \tan 2 x$$

Step-by-Step Solution

Answer

The graph of y=2 tan(2x) is a repeated wave stretching from -Infinity to Infinity on the y-axis, with asymptotes at \(x = n * pi/2\) where n is an integer, and passing through the points \((n * pi/2, 0)\). Two periods of this function extend from \(x = -pi\) to \(x = pi\).

Step 1: Identify the basic function

First, understand that this function is a variation of the basic tangent function \(y = tan(x)\). The graph of this base function is a repeated wave with asymptotes at \(x = (n + 0.5) * pi\), where n is an integer. The period of this base function is \(pi\). The graph passes through the points \((n*pi, 0)\), where n is an integer.

Step 2: Identify the variations

Sketch the alterations to the base function. The 2 in front of the function stretches the graph vertically, changing the maximum and minimum points from 1 and -1 to 2 and -2. The 2 inside the parenthesis compresses the graph horizontally, cutting the period of the function in half, from \(pi\) to \(pi/2\). This also means the asymptotes will be at \(x = n * pi/2\) and the graph will pass through the points \((n * pi/2, 0)\).

Step 3: Sketch the graph

Sketch the graph, keeping in mind the areas of the graph close to the asymptotes, where the function approaches infinity or negative infinity. For two periods of the function, sketch the graph from \(x= -pi\) to \(x= pi\). The graph will have asymptotes at \(x = -pi, -pi/2, 0, pi/2, pi\) and will pass through the points \((-pi, 0), (-pi/2, 0), (0, 0), (pi/2, 0), (pi, 0)\).