Math Problem Statement
Question content area top Part 1 Sketch the graph of yequals cosecant x , and then identify as many of the following properties that apply. Answer (a) through (e). Question content area bottom Part 1 Choose the correct graph of the function below. font size decreased by 15 A.
-44xy 2 pi negative pi pi negative 2 pi
A coordinate system has a horizontal x-axis labeled from negative 2 pi to 2 pi in increments of StartFraction pi over 4 EndFraction and a vertical y-axis labeled from negative 4 to 4 in increments of 1. The graph shows a series of alternating and unconnected downward U-shaped and upward U-shaped curves separated by vertical asymptotes with period 2 pi. One of the curves is between the asymptotes at x = 0 and x = pi and has its maximum at (StartFraction pi over 2 EndFraction, negative 1). The curve next to it is facing upwards with a maximum at (StartFraction 3 pi over 2 EndFraction, 1). B.
-44xy 2 pi negative pinegative 2 pi pi
A coordinate system has a horizontal x-axis labeled from negative 2 pi to 2 pi in increments of StartFraction pi over 4 EndFraction and a vertical y-axis labeled from negative 4 to 4 in increments of 1. The graph shows a series of alternating and unconnected downward U-shaped and upward U-shaped curves separated by vertical asymptotes with period 2 pi. One of the curves is between the asymptotes at x = 0 and x = pi and has its minimum at (StartFraction pi over 2 EndFraction, 1). The curve next to it is facing downwards with a maximum at (StartFraction 3 pi over 2 EndFraction, negative 1). C.
-44xy 2 pi negative pi pi negative 2 pi
A coordinate system has a horizontal x-axis labeled from negative 2 pi to 2 pi in increments of StartFraction pi over 4 EndFraction and a vertical y-axis labeled from negative 4 to 4 in increments of 1. The graph shows a series of alternating and unconnected downward U-shaped and upward U-shaped curves separated by vertical asymptotes with period 2 pi. One of the curves is between the asymptotes at x = negative StartFraction pi over 2 EndFraction and x = StartFraction pi over 2 EndFraction and has its maximum at (0, negative 1). The curve next to it is facing upwards with a minimum at (pi, 1). D.
-44xy 2 pi negative pinegative 2 pi pi
Solution
Let's first review how the graph of the cosecant function behaves, and then examine which graph matches its properties.
Properties of the Cosecant Function
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Cosecant Definition: , so the function is undefined when . This happens at , where is any integer. These points correspond to vertical asymptotes.
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Period: The period of is the same as , which is .
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Shape:
- Between the asymptotes (at ), the function has U-shaped branches.
- When is positive, has upward U-shaped curves.
- When is negative, has downward U-shaped curves.
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Range: The values of never lie between -1 and 1. The range is .
Now, let's identify the correct graph:
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Option A: This graph has alternating U-shaped curves, and the curve between and is a downward U-shaped curve with its minimum at . This matches the behavior of the cosecant function, as the cosecant is negative between 0 and .
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Option B: The graph also shows alternating U-shaped curves, but the minimum occurs at instead of a maximum. This is inconsistent with the behavior of the cosecant function.
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Option C: The curves are shifted, with the downward U-shaped curve between and , making the minimum occur at 0. This graph is not a match because the cosecant function has vertical asymptotes at integer multiples of , not at fractions of .
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Option D: This option doesn't match the behavior of the cosecant function either, as it is similarly shifted.
Conclusion:
The correct graph is Option A.
Do you want further details or explanations about the properties of the cosecant function?
Here are five related questions to explore further:
- What is the relationship between the sine and cosecant functions?
- How do the vertical asymptotes in compare to those in ?
- How would the graph of change if the function is transformed, e.g., ?
- How can you determine the period of a transformed cosecant function?
- What happens to the graph of as ?
Tip: Always keep in mind that cosecant is undefined wherever sine is zero, so vertical asymptotes appear at these points.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Graphs of Trigonometric Functions
Formulas
y = csc(x)
csc(x) = 1/sin(x)
Theorems
Properties of Cosecant Function
Vertical Asymptotes of Trigonometric Functions
Suitable Grade Level
Grades 10-12