To find the equation of a linear function that has been shifted down, we need to understand how vertical shifts affect the function.
1. Start with the Parent Function:
The linear parent function is given by [tex]\( f(x) = x \)[/tex]. This is the simplest form of a linear function, representing a straight line passing through the origin with a slope of 1.
2. Understand the Shift:
Shifting a function vertically simply involves adding or subtracting a constant value to the function. Specifically, if we want to shift the function down by a certain number of units, we subtract that number from the function.
3. Apply the Shift:
Here, we are instructed to shift the function down by 6 units. To achieve this, we subtract 6 from the original function [tex]\( f(x) = x \)[/tex].
[tex]\[ g(x) = f(x) - 6 \][/tex]
4. Substitute the Parent Function:
Replace [tex]\( f(x) \)[/tex] with [tex]\( x \)[/tex]:
[tex]\[ g(x) = x - 6 \][/tex]
Thus, the new function after shifting the parent function [tex]\( f(x) = x \)[/tex] down by 6 units is:
[tex]\[ g(x) = x - 6 \][/tex]
5. Identify the Correct Option:
Comparing this to the given options:
- A. [tex]\( g(x) = x + 6 \)[/tex]
- B. [tex]\( g(x) = \frac{1}{6} x \)[/tex]
- C. [tex]\( g(x) = x - 6 \)[/tex]
- D. [tex]\( g(x) = 6x \)[/tex]
The correct option is clearly:
[tex]\[ \boxed{C. \ g(x) = x - 6} \][/tex]
