Question

drag the tiles to the correct boxes to complete the pairs. match each transformation of the function y = csc x with the description of the resultant shift in the original cosecant functions graph. -1 + csc(x + π) -1 + csc(x − π) 1 + csc(x + π) 1 + csc(x − π) resultant shift in the functions graph transformation of the function the graph of csc x shifts one unit up and π radians to the right. the graph of csc x shifts one unit up and π radians to the left. the graph of csc x shifts one unit down and π radians to the right. the graph of csc x shifts one unit down and π radians to the left.

Explanation:

Step1: Recall Transformations

For a function \( y = f(x) \), the transformation \( y = f(x - h) + k \) shifts the graph \( h \) units horizontally (right if \( h>0 \), left if \( h<0 \)) and \( k \) units vertically (up if \( k>0 \), down if \( k<0 \)).

Step2: Analyze Each Transformation

• Shift up 1, right \( \pi \): \( k = 1 \) (up), \( h=\pi \) (right). So \( y = 1 + \csc(x - \pi) \).
• Shift up 1, left \( \pi \): \( k = 1 \) (up), \( h = -\pi \) (left, since \( x + \pi = x - (-\pi) \)). So \( y = 1 + \csc(x + \pi) \).
• Shift down 1, right \( \pi \): \( k = -1 \) (down), \( h=\pi \) (right). So \( y = -1 + \csc(x - \pi) \).
• Shift down 1, left \( \pi \): \( k = -1 \) (down), \( h = -\pi \) (left). So \( y = -1 + \csc(x + \pi) \).

Answer:

• The graph of \( \csc x \) shifts one unit up and \( \pi \) radians to the right: \( 1 + \csc(x - \pi) \)
• The graph of \( \csc x \) shifts one unit up and \( \pi \) radians to the left: \( 1 + \csc(x + \pi) \)
• The graph of \( \csc x \) shifts one unit down and \( \pi \) radians to the right: \( -1 + \csc(x - \pi) \)
• The graph of \( \csc x \) shifts one unit down and \( \pi \) radians to the left: \( -1 + \csc(x + \pi) \)