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Chapter 4: Problem 18

Sketch the graph of the function. (Include two full periods.) $$y=-3 \tan \pi x$$

Short Answer

Expert verified
The graph of the function \(y=-3 \tan \pi x\) is a vertically stretched, upside down version of the standard tangent function with two full periods between -0.5 to 0.5 and 0.5 to 1.5 respectively.

Step by step solution

01

Identify key features of the function

This function \(y = -3 \tan \pi x\) is a transformation of the standard tangent function \(\tan x\). The coefficient of -3 in front of the tangent function will flip the graph upside down and stretch it vertically by a factor of three, and multiplying the variable \(x\) by \(\pi\) compresses the graph horizontally by a factor of \(\pi\). The period of the function is 1 as opposed to \(\pi\) for a standard tangent function.
02

Plot the key points

Plot key points on the graph. Begin by sketching out the first period from \(x = -0.5\) to \(x = 0.5\). At \(x = 0\), \(y = -3 \tan(0) = 0\). At \(x = -0.25\) and \(x = 0.25\), the tangent function reaches its extremes, so these are our vertical asymptotes. Because of the negative sign in front of the function, the graph will start from the lower bound at \(x=-0.5\) and end at lower bound at \(x=0.5\) while passing through (0,0).
03

Repeat for second period

To complete the two full periods, simply repeat the previous process for the second period from \(x = 0.5\) to \(x = 1.5\). Draw the vertical asymptotes at \(x = 0.75\) and \(x = 1.25\), with the curve passing through (1,0), starting at the lower bound at \(x=0.5\) and ending at the lower bound at \(x=1.5\).

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