the parent cosecant function is shifted 4 units right and 3 units up. which of the following is the graph of…

Explanation:

Step1: Recall Cosecant Transformations

The parent cosecant function is ( y = \csc(x) ). A horizontal shift ( h ) units right is ( y=\csc(x - h) ), and vertical shift ( k ) units up is ( y=\csc(x - h)+k ). Here, ( h = 4 ), ( k = 3 ), so the transformed function is ( y=\csc(x - 4)+3 ).

Step2: Analyze Vertical Shift

The parent ( \csc(x) ) has a range of ( (-\infty, - 1]\cup[1, \infty) ). After shifting up 3 units, the range becomes ( (-\infty, -1 + 3]\cup[1 + 3, \infty)=(-\infty, 2]\cup[4, \infty) ). So the graph should have parts above ( y = 4 ) and below ( y = 2 ), shifted up from the parent's positions.

Step3: Analyze Horizontal Shift

A shift right 4 units means the asymptotes (at ( x = n\pi, n\in\mathbb{Z} ) for parent) shift to ( x = 4 + n\pi ). We check the graphs (from the provided options, though some are partially visible) for the correct vertical range (around ( y = 3 ) shift) and horizontal shift. The graph with vertical positions adjusted by +3 (so mid - range between the two branches around ( y = 3 )) and shifted right 4 units (asymptotes moved right) is the correct one. From the visible graphs, the one with the correct vertical shift (e.g., the middle or relevant graph with the right range and shift) is identified.

Answer:

(Note: Since the full graphs are not fully visible, but based on the transformation, the correct graph should be the one where the cosecant curves are shifted 4 units right (asymptotes moved) and 3 units up (range adjusted to ( (-\infty, 2]\cup[4, \infty) )). If we assume the second graph (lower one) has the correct vertical shift (around ( y = 3 ) - 7 range adjusted) and horizontal shift, the answer would be the lower graph (the one with y - axis labels around 5 - 7). However, due to partial visibility, the key is the transformed function ( y=\csc(x - 4)+3 ) and its graphical properties.)

(If we had to pick from the given (partially shown) graphs, the lower graph (the one with the y - axis around 5 - 7) is likely correct as it shows the vertical shift up 3 (since parent's mid - gap is at ( y = 0 ), shifted up 3 would have mid - gap around ( y = 3 ), and the graph's branches are above and below that shifted mid - line).)