To understand the transformation given by [tex]\(g(x) = f(x-11)\)[/tex], we first need to understand how it affects the graph of the original function [tex]\(f(x) = x\)[/tex].
Here's a detailed, step-by-step analysis:
1. Understanding the transformation relation: The equation [tex]\( g(x) = f(x-11) \)[/tex] implies that for each value of [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex], we look at the value of [tex]\( f \)[/tex] at [tex]\( x-11 \)[/tex]. This is known as a horizontal shift.
2. Horizontal shift details:
- A transformation [tex]\( f(x - a) \)[/tex] shifts the graph of [tex]\( f(x) \)[/tex] horizontally.
- When [tex]\( a \)[/tex] is positive, [tex]\( f(x - a) \)[/tex] shifts the graph of [tex]\( f(x) \)[/tex] to the right by [tex]\( a \)[/tex] units.
- When [tex]\( a \)[/tex] is negative, [tex]\( f(x - a) \)[/tex] shifts the graph of [tex]\( f(x) \)[/tex] to the left by [tex]\( |a| \)[/tex] units.
3. Applying the transformation to [tex]\( f(x) \)[/tex]:
- In this problem, [tex]\( g(x) = f(x - 11) \)[/tex]. The term [tex]\(-11\)[/tex] indicates a horizontal shift to the right.
- Therefore, [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] translated 11 units to the right.
4. Eliminating other options:
- Option A suggests a translation 11 units to the left, which is incorrect.
- Option C suggests a translation 11 units up, which is incorrect because we are dealing with a horizontal shift, not a vertical one.
- Option D suggests the slope of [tex]\( f(x) \)[/tex] is increased by 11, which is irrelevant since the transformation [tex]\( g(x)=f(x-11) \)[/tex] only affects the horizontal position, not the slope.
By careful analysis of the given transformation, the correct description of [tex]\( g(x) = f(x-11) \)[/tex] is:
B. It is the graph of [tex]\( f(x) \)[/tex] translated 11 units to the right.
