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Chapter 4: Problem 36

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

Short Answer

Expert verified
The graph of the function y=-3 sec(2x/3) is a secant graph with a vertical stretch of 3, a reflection about the x-axis, and a period of \(3\pi\). It passes through the points (0, -3), \((4.5\pi, 3)\), \((9\pi, -3)\) and has vertical asymptotes at \(x = \pi/2 + k\pi\).

Step by step solution

01

Identify the amplitude, vertical stretch and reflection

From the form \(y=A \sec{Bx}\), A represents the vertical stretch and reflection. Here, A is -3 and since A<0, there will be a vertical stretch of 3 and the graph will be reflected about the x-axis.
02

Identify the period

The period of the secant function is normally \(2\pi\), but here it will be adjusted by B. The value of B is the coefficient within the sec function which is \(2/3\). So, the new period will be \(2\pi / B = 2\pi / (2/3) = 3\pi\).
03

Plot the key points

The secant function has vertical asymptotes at \(x = \pi/2 + k\pi\), and can be plotted at the key points \(x = k\pi\). For \(y = \sec{x}\), the key points are (0, 1), \((\pi, -1)\), \((2\pi, 1)\). For the function \(-3 \sec{ (2/3)x }\), adjust for the period change to get the new key points (0, -3), \((4.5\pi, 3)\), \((9\pi, -3)\).
04

Draw the graph

Now, plot these points and sketch the graph. The graph will be reflected across the x-axis, and will have a pattern repeating every \(3\pi\).

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