Question

graph two periods of the given cosecant function.
y = 7 csc x

choose the correct graph of two periods of y = 7 csc x below.
a.
10
0
-10
4π
b.
10
0
-10
4π
c.
10
0
-10
4π
d.
10
0
-10
4π

Explanation:

Step1: Recall Cosecant Function Properties

The cosecant function \( y = \csc x=\frac{1}{\sin x} \). The period of \( \csc x \) is \( 2\pi \), so two periods would be \( 4\pi \). The amplitude (vertical stretch) for \( y = 7\csc x \) is 7, meaning the peaks and troughs are at \( y = 7 \) and \( y=-7 \) (since \( \csc x \) has vertical asymptotes where \( \sin x = 0 \), i.e., \( x = n\pi, n\in\mathbb{Z} \), and the graph is composed of "U" and "inverted U" shapes with vertical stretch by 7). Also, \( \csc x \) has the same sign as \( \sin x \) in each interval between asymptotes. For \( x\in(0,\pi) \), \( \sin x>0 \), so \( \csc x>0 \) (upper "U" shape), and for \( x\in(\pi,2\pi) \), \( \sin x<0 \), so \( \csc x<0 \) (lower "inverted U" or "n" shape). The vertical stretch by 7 means the maximum and minimum values (excluding asymptotes) are \( \pm7 \), but looking at the graph options, the key is the shape and asymptotes. The correct graph should have vertical asymptotes at \( x = 0,\pi,2\pi,3\pi,4\pi \) (for two periods \( 0 \) to \( 4\pi \)), and the "U" shapes (for positive \( \csc x \)) above the x - axis and "n" shapes (for negative \( \csc x \)) below the x - axis, with vertical stretch. Also, the reference graph for \( \csc x \) has the first "U" in \( (0,\pi) \) (above x - axis), then "n" in \( (\pi,2\pi) \) (below x - axis), then "U" in \( (2\pi,3\pi) \), "n" in \( (3\pi,4\pi) \) for two periods (0 to \( 4\pi \)). The function \( y = 7\csc x \) will have the same shape as \( \csc x \) but stretched vertically. Looking at the options, we need to check the position of the "U" and "n" shapes and the asymptotes.

Step2: Analyze Graph Options

• Option A: Incorrect sign in some intervals (mix of upper and lower shapes in wrong intervals).
• Option B: The "U" and "n" shapes are in the correct intervals (upper in \( (0,\pi),(2\pi,3\pi) \), lower in \( (\pi,2\pi),(3\pi,4\pi) \)) and vertical stretch is consistent (peaks/troughs around \( \pm7 \) range as per the y - axis scale of 10). The asymptotes are at \( x = 0,\pi,2\pi,3\pi,4\pi \), which matches the period and vertical asymptote locations of \( \csc x \).
• Option C: Incorrect shape (wrong direction of "U" and "n" in some intervals).
• Option D: Incorrect sign and shape alignment.

Answer:

B. The graph with upper "U" shapes in \( (0,\pi) \) and \( (2\pi,3\pi) \), lower "n" shapes in \( (\pi,2\pi) \) and \( (3\pi,4\pi) \), vertical asymptotes at \( x = 0,\pi,2\pi,3\pi,4\pi \), and consistent with the vertical stretch of \( y = 7\csc x \).