question 11\nthe graph above is a graph of what function?\n$y = \\cot(x)$\n$y = \\sin(x)$\n$y =…

question 11\nthe graph above is a graph of what function?\n$y = \\cot(x)$\n$y = \\sin(x)$\n$y = \\tan(x)$\n$y = \\cos(x)$\n$y = \\csc(x)$\n$y = \\sec(x)$

question 11\nthe graph above is a graph of what function?\n$y = \\cot(x)$\n$y = \\sin(x)$\n$y = \\tan(x)$\n$y = \\cos(x)$\n$y = \\csc(x)$\n$y = \\sec(x)$

Answer

Explanation:

Step1: Analyze the properties of trigonometric functions

  • The function (y = \cot(x)=\frac{\cos(x)}{\sin(x)}) has vertical asymptotes at (x = n\pi), (n\in\mathbb{Z}) and is a decreasing function in each of its intervals.
  • The function (y=\sin(x)) has a range of ([- 1,1]) and is a periodic function with period (2\pi) that oscillates between (-1) and (1).
  • The function (y = \tan(x)=\frac{\sin(x)}{\cos(x)}) has vertical asymptotes at (x=(n +\frac{1}{2})\pi), (n\in\mathbb{Z}) and is an increasing function in each of its intervals.
  • The function (y=\cos(x)) has a range of ([-1,1]) and is a periodic function with period (2\pi) that oscillates between (-1) and (1).
  • The function (y=\csc(x)=\frac{1}{\sin(x)}) has vertical asymptotes at (x = n\pi), (n\in\mathbb{Z}) and its graph consists of two - part curves (upper and lower) in each period.
  • The function (y=\sec(x)=\frac{1}{\cos(x)}) has vertical asymptotes at (x=(n+\frac{1}{2})\pi), (n\in\mathbb{Z}).

Step2: Match the graph with the function

The given graph has vertical asymptotes at (x=(n+\frac{1}{2})\pi) ((n\in\mathbb{Z})) (since the graph has breaks at these (x) - values) and the shape of the graph (the U - shaped and inverted U - shaped curves) matches the general form of (y = \sec(x)).

Answer:

(y=\sec(x))