To determine which of the following functions translates the graph of the parent function [tex]\( f(x) = x^2 \)[/tex] vertically up 6 units, let’s analyze each of the given options:
Option A: [tex]\( g(x) = x^2 + 6 \)[/tex]
Adding a constant to the parent function [tex]\( f(x) = x^2 \)[/tex] results in a vertical translation. Specifically, [tex]\( g(x) = x^2 + 6 \)[/tex] means that for every [tex]\( x \)[/tex], the value of [tex]\( g(x) \)[/tex] will be 6 units greater than the corresponding value of [tex]\( f(x) \)[/tex]. This translates the graph of [tex]\( x^2 \)[/tex] upwards by 6 units.
Option B: [tex]\( g(x) = (x-6)^2 \)[/tex]
This represents a horizontal shift. Specifically, [tex]\( g(x) = (x-6)^2 \)[/tex] shifts the graph of [tex]\( x^2 \)[/tex] to the right by 6 units, not vertically.
Option C: [tex]\( g(x) = x^2 - 6 \)[/tex]
Subtracting a constant translates the graph of [tex]\( x^2 \)[/tex] downwards by that constant. Hence, [tex]\( g(x) = x^2 - 6 \)[/tex] results in a vertical shift downward 6 units.
Option D: [tex]\( g(x) = (x+6)^2 \)[/tex]
This represents a horizontal shift to the left by 6 units.
After analyzing the choices, it’s clear that the function that translates the graph of [tex]\( x^2 \)[/tex] vertically up 6 units is:
Option A: [tex]\( g(x) = x^2 + 6 \)[/tex]
### Illustration:
- For [tex]\( f(x) = x^2 \)[/tex], if [tex]\( x = 0 \)[/tex], then [tex]\( f(0) = 0 \)[/tex]
- For [tex]\( g(x) = x^2 + 6 \)[/tex], if [tex]\( x = 0 \)[/tex], then [tex]\( g(0) = 0^2 + 6 = 6 \)[/tex]
Therefore, every point on the graph of [tex]\( f(x) \)[/tex] has been moved up by 6 units to form the graph of [tex]\( g(x) \)[/tex]. This confirms that the vertical shift upward is achieved by adding 6 to the entire function [tex]\( x^2 \)[/tex]. Thus, the correct answer is Option A, which is:
[tex]\[ g(x) = x^2 + 6 \][/tex]
