If you shift the linear parent function, [tex]f(x) = x[/tex], down 6 units, what is the equation of the new function?

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]



Answer :

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]