Answer :
To determine which function represents the linear parent function [tex]\( F(x) = x \)[/tex] shifted to the right, we need to understand how horizontal shifts work in function transformations.
A right shift of a function [tex]\( F(x) \)[/tex] by [tex]\( a \)[/tex] units is represented by the function [tex]\( F(x - a) \)[/tex]. This means that if we start with the parent function [tex]\( F(x) = x \)[/tex] and we want to shift it to the right by [tex]\( a \)[/tex] units, the new function will be [tex]\( F(x) = x - a \)[/tex].
Let's examine the given options to see which matches this form:
- Option A: [tex]\( G(x) = 4x \)[/tex]
- This function is a linear function with a different slope, not a horizontal shift of the parent function.
- Option B: [tex]\( G(x) = -x \)[/tex]
- This function reflects the parent function across the y-axis, which is not a horizontal shift to the right.
- Option C: [tex]\( G(x) = x - 9 \)[/tex]
- This fits the form we are looking for. It represents the function [tex]\( F(x) = x \)[/tex] shifted to the right by 9 units because subtracting 9 from [tex]\( x \)[/tex] causes the entire graph to move rightward.
- Option D: [tex]\( G(x) = x + 2 \)[/tex]
- This function represents [tex]\( F(x) = x \)[/tex] shifted to the left by 2 units, not to the right.
Therefore, the function that shows [tex]\( F(x) = x \)[/tex] shifted to the right is:
[tex]\[ \boxed{3} \][/tex]
So, Option C is the correct answer.
A right shift of a function [tex]\( F(x) \)[/tex] by [tex]\( a \)[/tex] units is represented by the function [tex]\( F(x - a) \)[/tex]. This means that if we start with the parent function [tex]\( F(x) = x \)[/tex] and we want to shift it to the right by [tex]\( a \)[/tex] units, the new function will be [tex]\( F(x) = x - a \)[/tex].
Let's examine the given options to see which matches this form:
- Option A: [tex]\( G(x) = 4x \)[/tex]
- This function is a linear function with a different slope, not a horizontal shift of the parent function.
- Option B: [tex]\( G(x) = -x \)[/tex]
- This function reflects the parent function across the y-axis, which is not a horizontal shift to the right.
- Option C: [tex]\( G(x) = x - 9 \)[/tex]
- This fits the form we are looking for. It represents the function [tex]\( F(x) = x \)[/tex] shifted to the right by 9 units because subtracting 9 from [tex]\( x \)[/tex] causes the entire graph to move rightward.
- Option D: [tex]\( G(x) = x + 2 \)[/tex]
- This function represents [tex]\( F(x) = x \)[/tex] shifted to the left by 2 units, not to the right.
Therefore, the function that shows [tex]\( F(x) = x \)[/tex] shifted to the right is:
[tex]\[ \boxed{3} \][/tex]
So, Option C is the correct answer.
