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Chapter 4: Problem 16

Sketch the graph of the function. (Include two full periods.) $$y=\tan 4 x$$

Short Answer

Expert verified
To graph \(y=\tan 4x\), calculate the period as \(\pi/4\). Find x-values where the function is undefined (vertical asymptotes) as \(x=\pi/8 + n\pi/4\) for any integer n. The function's value can be calculated at midpoints between these asymptotes for determining y-coordinates between the asymptotes. Repeat this process across multiple periods to sketch a complete graph.

Step by step solution

01

Step - Identify the period

The period for the tangent function is \(\pi\). However, for \(\tan 4x\), the period is altered. The period can be calculated as \(\pi/4\) due to the factor '4' applied on 'x'.
02

Step - Mark key points

The graph of any \(\tan kx\) function has vertical asymptotes (where the function is undefined) at the points \(x=\pi/2k + n\pi\) for all integers 'n'. For \(\tan 4x\), the vertical asymptotes occur at \(x=\pi/8 + n\pi/4\). Calculate these key points by plugging in for 'n' different integer values for at least two periods.
03

Step - Draw vertical asymptotes

Draw the vertical asymptotes at the points defined by \(x=\pi/8 + n\pi/4\).
04

Step - Determine y-values between asymptotes

Because tangent is periodic and odd (meaning, symmetric about the origin), you only need to find the y-values between two asymptotes and repeat this pattern for other periods. For \(\tan 4x\), calculate the function value at midpoints between the asymptotes.
05

Step - Sketch the graph

With the calculated points and asymptotes, sketch the graph. Remember, the graph of the tangent function passes through the origin and is symmetrical, it starts from negative infinity, crosses the x-axis at the origin, and goes to positive infinity.

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