How is the graph of y = csc(x – 6) transformed from its parent function?
A. It is the graph of y = csc(x) shifted 6 units up.
B. It is the graph of y = csc(x) shifted 6 units right.
C. It is the graph of y = csc(x) shifted 6 units left.
D. It is the graph of y = csc(x) shifted 6 units down.
Answer (2)
The graph of y = csc ( x − 6 ) is derived from its parent function y = csc ( x ) by shifting it 6 units to the right. Therefore, the correct multiple-choice option is B. This transformation occurs due to the horizontal shift indicated by the expression ( x − 6 ) .
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To understand how the graph of y = csc ( x − 6 ) is transformed from its parent function y = csc ( x ) , we need to consider how transformations work in trigonometric functions.
The function y = csc ( x ) is the cosecant function, which is the reciprocal of the sine function: y = s i n ( x ) 1 . Its graph has vertical asymptotes wherever sin ( x ) = 0 , and it exhibits a repeating pattern every 2 π , which aligns with the period of the sine function.
For the equation y = csc ( x − 6 ) , the transformation involves a horizontal shift. The expression ( x − 6 ) indicates a shift to the right by 6 units:
Horizontal Shift: A transformation of the form y = f ( x − a ) shifts the graph of f ( x ) horizontally to the right by a units. Therefore, y = csc ( x − 6 ) means that the graph of y = csc ( x ) is shifted 6 units to the right.
To summarize, the correct transformation from the parent function y = csc ( x ) is a horizontal shift to the right by 6 units. Therefore, the right choice from the given options is:
"It is the graph of y = csc ( x ) shifted 6 units right."
