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Chapter 4: Problem 40

Use a graphing utility to graph the function. (Include two full periods.) $$y=-\tan 2 x$$

Short Answer

Expert verified
The function \(y=-\tan 2x\) has a period of \(\pi / 2\), no shift, and a coefficient of -1 in front of the function causing a vertical reflection. The asymptotes are at \(\pi / 4, 3\pi / 4, ...\). One needs to plot these characteristics on a graph to visualize it.

Step by step solution

01

Identify the period

Since the coefficient of x in the function is 2, the period of the function will be \(\pi / 2\). This is achieved by taking the standard period of the tan function, which is \(\pi\), and dividing it by the absolute value of the coefficient of x.
02

Identify the shift and amplitude

The given function is not shifted horizontally or vertically, so the shift is 0. Considering that the tangent function does not have a maximum or minimum value, it technically doesn't have an amplitude. However, we can consider the coefficient in front of the function, which in this case is -1, as a vertical stretch of the graph. Therefore, the amplitude is considered to be 1.
03

Graph the function

Plot the parent function \(y = \tan x\) first over one period from 0 to \(\pi\). Then, apply the vertical reflection (negative sign) and the horizontal compression (coefficient 2 of x) to graph \(y = -\tan 2x\). Extend the graph to cover two full periods.
04

Mark the asymptotes

Asymptotes of \(y=\tan x\) occur at \(\pi / 2, 3\pi / 2, ...\). Since we have a coefficient of 2 inside the function, the asymptotes will occur half as often, at \(\pi / 4, 3\pi / 4, 7\pi / 4, ...\). Thus, mark these asymptotes on your graph.

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Most popular questions from this chapter

Find a model for simple harmonic motion satisfying the specified conditions. Displacement \((t=0)\) $$0$$ Amplitude 4 centimeters Period 2 seconds

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\). (a) As \(x \rightarrow 0^{+},\) the value of \(f(x) \rightarrow\) (b) As \(x \rightarrow 0^{-},\) the value of \(f(x) \rightarrow\) (c) As \(x \rightarrow \pi^{+},\) the value of \(f(x) \rightarrow\) (d) As \(x \rightarrow \pi^{-},\) the value of \(f(x) \rightarrow\) $$f(x)=\cot x$$

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