Question

select the correct answer. each statement describes a transformation of the graph of ( f(x)=sqrt3{x} ). which statement correctly describes the graph of ( y = f(x - 7)+3 )? a. it is the graph of ( f ) translated 3 units up and 7 units to the left. b. it is the graph of ( f ) translated 7 units down and 3 units to the right. c. it is the graph of ( f ) translated 7 units up and 3 units to the right. d. it is the graph of ( f ) translated 3 units up and 7 units to the right.

Explanation:

To determine the transformation of the graph \( y = f(x - 7) + 3 \) from the original graph \( f(x)=\sqrt[3]{x} \), we use the rules of function transformations:

• For horizontal shifts: If we have \( f(x - h) \), the graph shifts \( h \) units to the right (when \( h>0 \)). Here, \( h = 7 \), so the graph shifts 7 units to the right.
• For vertical shifts: If we have \( f(x)+k \), the graph shifts \( k \) units up (when \( k>0 \)). Here, \( k = 3 \), so the graph shifts 3 units up.

So the graph of \( y = f(x - 7)+3 \) is the graph of \( f(x) \) translated 3 units up and 7 units to the right.

Answer:

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