Question

Explain how the graph of y = 2csc(x - pi/4) - 5 is related to the graph of the basic trigonometric function y = cscx.

Answer 1

Answer: The distance between the min and max increased by a factor of 2, horizontally shifted to the right by $\bold{\dfrac{\pi}{4}}$, shifted down 5 units, and has a phase shift of $\bold{\dfrac{\pi}{4}}$

Step-by-step explanation:

The general form of a csc function is: y = A csc (Bx - C) + D  where

• |A| is the amplitude (vertical stretch)
• -A is a reflection over the x-axis
• |B| is the horizontal stretch
• -B is a reflection over the y-axis
• C is a horizontal shift (left or right)
• D is a vertical shift (up or down)

• Period is $\dfrac{2\pi}{B}$
• Phase Shift is $\dfrac{C}{B}$

In the given transformed graph y = 2 csc (x - $\dfrac{\pi}{4}$) - 5

• A = 2   --> distance between the Min and max is 2(2) = 4
• C = $\dfrac{\pi}{4}$  --> horizontally shifted to the right  $\dfrac{\pi}{4}$ units
• D = -5  --> shifted DOWN 5 units
• Phase shift is $\dfrac{\pi}{4}$