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Chapter 4: Problem 33

Sketch one full period of the graph of each function. $$y=-2 \csc \frac{x}{3}$$

Short Answer

Expert verified
The graph of \(y = -2 \csc(x/3)\) is a graph of the cosecant function reflected in the x-axis, horizontally stretched by a factor of three, and vertically stretched by a factor of two. It includes vertical asymptotes at each integer multiple of \(6\pi\) and reaches a minimum of -2 at each odd integer multiple of \(3\pi\).

Step by step solution

01

Understanding the Function

Cosecant (csc) is the reciprocal of sine, defined as \(csc(x) = 1/\sin(x)\). Its crucial features include undefined values at each integer multiple of \(\pi\) for the basic function, \(csc(x)\), consequent vertical asymptotes, and a period of \(2\pi\). It has two types of transformations: horizontal \(csc(bx)\) and vertical \(acsc(x)\). The period changes by a factor of \(1/b\) and the amplitude is affected by the absolute value of \(a\).
02

Apply Transformations

The given function \(y=-2 \csc(x/3)\) has a vertical stretch by a factor of 2 and a horizontal stretch by a factor of 3. It's also reflected in the x-axis due to the negative coefficient. So, the period is \(2\pi*3 = 6\pi\) and the amplitude is 2.
03

Sketch the Graph

Sketch one full period of \(y = -2 \csc(x/3)\) starting at \(x=0\). First, draw vertical asymptotes at \(x=0\) and \(x=6\pi\), the start and end of one period. Next, as the function is negative, it takes on extreme values at half-period intervals, reaching -2 at \(\pi*3 = 3\pi\). Also, due to the reflection, the branches of the graph open downwards instead of upwards. Sketch the branches of the graph in each half-period, remembering that the graph approaches but never reaches the asymptotes.

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