Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Graphing Two Periods of a Cosecant Function
**Objective:**
Graph two periods of the given cosecant function:
\[ y = 3 \csc x \]
**Task:**
Choose the correct graph of two periods of \( y = 3 \csc x \) from the options below.
**Options:**
- **Option A:**
- The graph displays a series of repeating U-shaped curves extending upwards and downwards, with vertical asymptotes where the sine function is zero. The cycles are positioned between x = 0 and 4π, with the curves reaching a minimum and maximum at y = ±3.
- **Option B:**
- Similar U-shaped patterns extend into both positive and negative y-directions, with vertical asymptotes indicating undefined points at multiples of π. The periods extend from x = 0 to 4π, consistent with two cycles.
- **Option C:**
- Again, the U-shaped curve pattern is evident, with asymptotes where the sine function passes zero. This option also covers two periods, ranging from x = 0 to 4π.
- **Option D:**
- Displays similar characteristics, featuring repeating U-shaped patterns and vertical asymptotes at regular intervals. The range is consistent with two full cycles of the cosecant function.
**Note:**
Each graph includes labeled axes with the y-axis extending from -6 to 6 and the x-axis extending from 0 to 4π. Vertical asymptotes are marked with dashed lines where the cosecant function is undefined.
**Instructions:**
To make an informed selection, observe the position and frequency of the asymptotes and the amplitude of the curves, which should reach ±3 in accordance with the function \( y = 3 \csc x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9fe330d6-ed92-43b3-9547-c6e2b1ed5b0a%2F0f8b421c-c9e8-4b31-84e7-790ef4d9cc3b%2Fkytfhaa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Graphing Two Periods of a Cosecant Function
**Objective:**
Graph two periods of the given cosecant function:
\[ y = 3 \csc x \]
**Task:**
Choose the correct graph of two periods of \( y = 3 \csc x \) from the options below.
**Options:**
- **Option A:**
- The graph displays a series of repeating U-shaped curves extending upwards and downwards, with vertical asymptotes where the sine function is zero. The cycles are positioned between x = 0 and 4π, with the curves reaching a minimum and maximum at y = ±3.
- **Option B:**
- Similar U-shaped patterns extend into both positive and negative y-directions, with vertical asymptotes indicating undefined points at multiples of π. The periods extend from x = 0 to 4π, consistent with two cycles.
- **Option C:**
- Again, the U-shaped curve pattern is evident, with asymptotes where the sine function passes zero. This option also covers two periods, ranging from x = 0 to 4π.
- **Option D:**
- Displays similar characteristics, featuring repeating U-shaped patterns and vertical asymptotes at regular intervals. The range is consistent with two full cycles of the cosecant function.
**Note:**
Each graph includes labeled axes with the y-axis extending from -6 to 6 and the x-axis extending from 0 to 4π. Vertical asymptotes are marked with dashed lines where the cosecant function is undefined.
**Instructions:**
To make an informed selection, observe the position and frequency of the asymptotes and the amplitude of the curves, which should reach ±3 in accordance with the function \( y = 3 \csc x \).
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