Answer :
To find the equation of a new function when the reciprocal parent function [tex]\( F(x) = \frac{1}{x} \)[/tex] is shifted down by 2 units, we need to understand how vertical shifts affect a function.
1. Vertical Shifts: A vertical shift of a function involves adding or subtracting a constant from the function. Specifically:
- Shifting a function up by [tex]\( k \)[/tex] units means adding [tex]\( k \)[/tex] to the function: [tex]\( F(x) + k \)[/tex].
- Shifting a function down by [tex]\( k \)[/tex] units means subtracting [tex]\( k \)[/tex] from the function: [tex]\( F(x) - k \)[/tex].
2. Applying the Shift: The given question asks us to shift the reciprocal function [tex]\( F(x) = \frac{1}{x} \)[/tex] down by 2 units. That translates to subtracting 2 from [tex]\( F(x) \)[/tex]. So, we do:
[tex]\[ G(x) = \frac{1}{x} - 2 \][/tex]
3. Verifying the Options: Now let's examine the given options to identify the correct new function after shifting:
- Option A: [tex]\( G(x) = \frac{1}{x-2} \)[/tex] (This represents a horizontal shift, not a vertical shift, so this is incorrect.)
- Option B: [tex]\( G(x) = \frac{1}{x} + 2 \)[/tex] (This represents a vertical shift up by 2 units, not down, so this is incorrect.)
- Option C: [tex]\( G(x) = \frac{1}{x} - 2 \)[/tex] (This correctly represents a vertical shift down by 2 units, so this is correct.)
- Option D: [tex]\( G(x) = \frac{2}{x} \)[/tex] (This changes the slope of the function and does not represent a vertical shift, so it is incorrect.)
Thus, the equation of the new function after shifting [tex]\( F(x) = \frac{1}{x} \)[/tex] down by 2 units is:
[tex]\[ G(x) = \frac{1}{x} - 2 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
1. Vertical Shifts: A vertical shift of a function involves adding or subtracting a constant from the function. Specifically:
- Shifting a function up by [tex]\( k \)[/tex] units means adding [tex]\( k \)[/tex] to the function: [tex]\( F(x) + k \)[/tex].
- Shifting a function down by [tex]\( k \)[/tex] units means subtracting [tex]\( k \)[/tex] from the function: [tex]\( F(x) - k \)[/tex].
2. Applying the Shift: The given question asks us to shift the reciprocal function [tex]\( F(x) = \frac{1}{x} \)[/tex] down by 2 units. That translates to subtracting 2 from [tex]\( F(x) \)[/tex]. So, we do:
[tex]\[ G(x) = \frac{1}{x} - 2 \][/tex]
3. Verifying the Options: Now let's examine the given options to identify the correct new function after shifting:
- Option A: [tex]\( G(x) = \frac{1}{x-2} \)[/tex] (This represents a horizontal shift, not a vertical shift, so this is incorrect.)
- Option B: [tex]\( G(x) = \frac{1}{x} + 2 \)[/tex] (This represents a vertical shift up by 2 units, not down, so this is incorrect.)
- Option C: [tex]\( G(x) = \frac{1}{x} - 2 \)[/tex] (This correctly represents a vertical shift down by 2 units, so this is correct.)
- Option D: [tex]\( G(x) = \frac{2}{x} \)[/tex] (This changes the slope of the function and does not represent a vertical shift, so it is incorrect.)
Thus, the equation of the new function after shifting [tex]\( F(x) = \frac{1}{x} \)[/tex] down by 2 units is:
[tex]\[ G(x) = \frac{1}{x} - 2 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
