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Each statement describes a transformation of the graph of [tex]f(x)=\sqrt[3]{x}[/tex]. Which statement correctly describes the graph of [tex]y=f(x-7)+3[/tex]?

A. It is the graph of [tex]f[/tex] translated 3 units up and 7 units to the left.
B. It is the graph of [tex]f[/tex] translated 7 units down and 3 units to the right.
C. It is the graph of [tex]f[/tex] translated 7 units up and 3 units to the right.
D. It is the graph of [tex]f[/tex] translated 3 units up and 7 units to the right.



Answer :

GinnyAnswer
Let's analyze the transformation of the graph of the function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] with the given expression [tex]\( y = f(x - 7) + 3 \)[/tex].

1. Horizontal Transformation:
- The expression inside the function, [tex]\( (x - 7) \)[/tex], indicates a horizontal shift of the graph. For [tex]\( f(x - a) \)[/tex], the graph of [tex]\( f(x) \)[/tex] is shifted to the right by [tex]\( a \)[/tex] units if [tex]\( a \)[/tex] is positive, and to the left if [tex]\( a \)[/tex] is negative.
- In this case, [tex]\( x - 7 \)[/tex] means the graph is shifted 7 units to the right.

2. Vertical Transformation:
- The [tex]\( + 3 \)[/tex] outside the function indicates a vertical shift. For [tex]\( f(x) + b \)[/tex], the graph of [tex]\( f(x) \)[/tex] is shifted up by [tex]\( b \)[/tex] units if [tex]\( b \)[/tex] is positive and down if [tex]\( b \)[/tex] is negative.
- In this case, [tex]\( +3 \)[/tex] means the graph is shifted 3 units up.

Combining both transformations:
- The graph of [tex]\( f(x) = \sqrt[3]{x} \)[/tex] is shifted 7 units to the right.
- It is also shifted 3 units up.

Thus, the correct statement describing the transformation is:

D. It is the graph of [tex]\( f \)[/tex] translated 3 units up and 7 units to the right.

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