Question

select the correct answer. each statement describes a transformation of the graph of ( f(x) = x ). which statement correctly describes the graph of ( g(x) = f(x - 11) )? a. it is the graph of ( f(x) ) where the slope is increased by 11. b. it is the graph of ( f(x) ) translated 11 units to the left. c. it is the graph of ( f(x) ) translated 11 units up. d. it is the graph of ( f(x) ) translated 11 units to the right.

Explanation:

To determine the transformation of \( f(x) = x \) to \( g(x) = f(x - 11) \), we use the rule for horizontal translations of functions. For a function \( y = f(x - h) \), the graph of \( f(x) \) is translated \( h \) units to the right when \( h>0 \). Here, \( h = 11 \), so \( g(x)=f(x - 11) \) means the graph of \( f(x) \) is translated 11 units to the right.

• Option A: The slope of \( f(x)=x \) is 1, and \( g(x)=x - 11 \) also has a slope of 1, so the slope is not increased by 11. Eliminate A.
• Option B: A horizontal translation \( f(x - h) \) for \( h>0 \) is a shift to the right, not left. Eliminate B.
• Option C: The transformation \( g(x)=f(x - 11) \) is a horizontal translation, not a vertical translation (which would be \( f(x)+k \) for vertical shift). Eliminate C.
• Option D: Matches the horizontal translation rule for \( f(x - 11) \) with \( h = 11>0 \), so it is a shift 11 units to the right.

Answer:

D. It is the graph of \( f(x) \) translated 11 units to the right.