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Chapter 4: Problem 78

Use a graphing utility to graph each function. Use a viewing rectangle that shows the graph for at least two periods. $$y=\tan 4 x$$

Short Answer

Expert verified
First, consider that for \(y = \tan x\) one period is from \(-蟺/2\) to \(蟺/2\). However, \(y = \tan(4x)\) changes the period of the function to \(蟺/4\). Graph the function in an interval from 0 to \(蟺/2\), marking points for every \(蟺/8\) in this interval and drawing asymptotes at multiples of \(蟺/4\). The process can be repeated for any further periods.

Step by step solution

01

Understand the period of the function

Normally, for the tangent function \(y = \tan x\), one period is from 鈭捪�/2 to 蟺/2. However, multiplying x by 4, as in \(y = \tan(4x)\), changes the period of the function to 蟺/4. This means the tangent function completes a cycle every 蟺/4 along the x-axis.
02

Identify the intervals

From step 1 we know that the function completes a cycle every 蟺/4. To graph two periods, choose intervals that cover two periods. So an appropriate interval on x-axis could be from 0 to 蟺/2.
03

Graph the function

Now plot the function using the chosen interval. Mark points for every 蟺/8 (since 蟺/8 is halfway between the asymptotes for y = tan(4x)). Connect these points with smooth curve making sure the function goes to positive or negative infinity at the asymptotes which are at multiples of 蟺/4 during the interval [0, 蟺/2]. Do not forget to draw the asymptotes at x = 蟺/4 and x = 蟺/2 which are vertical lines in the given interval for two periods. The process can be repeated for any further periods.

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