Which of the following is the equation of the function below?

The equation of the graph below is y = 0.5 csc[0.5(x + π/2)] - 1 ⇒ 3rd answer

Step-by-step explanation:

* Lets revise the trigonometry transformation

- If the equation is y = A csc(B(x + C)) + D

# Amplitude is A

- The Amplitude is the height from the center line to the peak .

  Or we can measure the height from highest to lowest points and

  divide that by 2

# period is 2π/B

- The Period goes from one peak to the next

# phase shift is C (positive is to the left)

- The Phase Shift is how far the function is shifted horizontally

  from the usual position.

# vertical shift is D

- The Vertical Shift is how far the function is shifted vertically from

 the usual position.

- If y = csc(x)

∴ A = 1 , B = 1 , C = 0 , D = 0

- That means the amplitude is 1, the period is 2π, no phase shift

 or vertical shift

* Now lets solve the problem

- From the graph

# The amplitude = (-0.5 - -1.5)/2 = 0.5

∴ A = 0.5

# The period is from 2.5π to -1.5π

∴ The period is 4π

∵ The period = 2π/B

∴ 4π = 2π/B ⇒ by cross multiplication

∴ B = 2π/4π = 1/2 = 0.5

* There is only one answer has A = 0.5 and B = 0.5

∴ y = 0.5 csc[0.5(x + π/2)] - 1

* The equation of the graph below is 0.5 csc[0.5(x + π/2)] - 1

* For more understand look to the color graph

- The blue graph is y = csc(x)

- The green graph is y = 0.5 csc[0.5(x + π/2)] - 1

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