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 y=csc(x)y = \csc(x) behaves, and then examine which graph matches its properties.

Properties of the Cosecant Function

  1. Cosecant Definition: csc(x)=1sin(x)\csc(x) = \frac{1}{\sin(x)}, so the function is undefined when sin(x)=0\sin(x) = 0. This happens at x=nπx = n\pi, where nn is any integer. These points correspond to vertical asymptotes.

  2. Period: The period of csc(x)\csc(x) is the same as sin(x)\sin(x), which is 2π2\pi.

  3. Shape:

    • Between the asymptotes (at x=nπx = n\pi), the function has U-shaped branches.
    • When sin(x)\sin(x) is positive, csc(x)\csc(x) has upward U-shaped curves.
    • When sin(x)\sin(x) is negative, csc(x)\csc(x) has downward U-shaped curves.
  4. Range: The values of y=csc(x)y = \csc(x) never lie between -1 and 1. The range is (,1][1,)(-\infty, -1] \cup [1, \infty).

Now, let's identify the correct graph:

  • Option A: This graph has alternating U-shaped curves, and the curve between 00 and π\pi is a downward U-shaped curve with its minimum at π2\frac{\pi}{2}. This matches the behavior of the cosecant function, as the cosecant is negative between 0 and π\pi.

  • Option B: The graph also shows alternating U-shaped curves, but the minimum occurs at π2\frac{\pi}{2} instead of a maximum. This is inconsistent with the behavior of the cosecant function.

  • Option C: The curves are shifted, with the downward U-shaped curve between π2-\frac{\pi}{2} and π2\frac{\pi}{2}, making the minimum occur at 0. This graph is not a match because the cosecant function has vertical asymptotes at integer multiples of π\pi, not at fractions of π\pi.

  • 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:

  1. What is the relationship between the sine and cosecant functions?
  2. How do the vertical asymptotes in csc(x)\csc(x) compare to those in sec(x)\sec(x)?
  3. How would the graph of y=csc(x)y = \csc(x) change if the function is transformed, e.g., y=2csc(x)y = 2\csc(x)?
  4. How can you determine the period of a transformed cosecant function?
  5. What happens to the graph of y=csc(x)y = \csc(x) as xx \to \infty?

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