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) if g(x) = f(x - 11)?
a. it is the graph of f(x) translated 11 units to the left.
b. it is the graph of f(x) translated 11 units up.
c. it is the graph of f(x) where the slope is increased by 11.
d. it is the graph of f(x) translated 11 units to the right.

Explanation:

To determine the transformation of the graph of \( f(x) = x \) to \( g(x) = f(x - 11) \), we use the rules of function transformations. For a function \( y = f(x - h) \), the graph is a horizontal translation of \( y = f(x) \). If \( h>0 \), the graph shifts \( h \) units to the right; if \( h < 0 \), it shifts \( |h| \) units to the left. Here, \( h = 11 \), so \( g(x)=f(x - 11) \) is the graph of \( f(x) \) translated 11 units to the right. Option A is incorrect (left translation would be \( f(x+11) \)), option B is incorrect (vertical translation is \( f(x)+11 \)), and option C is incorrect as 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.

Answer:

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