/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Sketch the graph of the function... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 30

Sketch the graph of the function. (Include two full periods.) $$y=-\frac{1}{2} \tan x$$

Short Answer

Expert verified
To sketch the graph of the function \(y=-\frac{1}{2} \tan x\), reflect the basic tangent function over the x-axis and vertically compress it by a factor of 1/2. The resulting graph will have vertical asymptotes at odd multiples of \(\pi/2\) and will pass through the origin, with peaks/troughs at \(y=-1/2\) and \(y=1/2\).

Step by step solution

01

Identify the Characteristics of the Basic Tangent Function

The basic function \(\tan x\) has a period of \(\pi\), meaning it repeats itself every \(\pi\) units. It has vertical asymptotes at odd multiples of \(\pi/2\), and passes through the origin.
02

Recognize the Transformations

The function \(y=-\frac{1}{2} \tan x\) has been transformed from the basic tangent function in two ways. It has been reflected over the x-axis, due to the negative sign. It's also been vertically compressed by a factor of 1/2, due to the fractional coefficient.
03

Draw the Graph

Begin by drawing the vertical asymptotes at odd multiples of \(\pi/2\) (for two full periods, these would be at \(\pi/2\), \(3\pi/2\), \(-\pi/2\), and \(-3\pi/2\)). Next, plot the key points - these will be at the intercepts and peaks/troughs. For the function \(y=-\frac{1}{2} \tan x\), there is an intercept at \(x=0\), and the peaks/troughs are halved compared to the original tangent function, hence located at \(y=-1/2\) and \(y=1/2\). Finally, connect these points with a smooth curve, making sure to approach, but not touch, the vertical asymptotes.

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