To determine which function translates the graph of the parent function [tex]\( f(x) = x^2 \)[/tex] horizontally to the left by 8 units, we need to understand how horizontal translations work.
### Horizontal Translations
For a given function [tex]\( f(x) \)[/tex]:
- To translate it to the left by [tex]\( h \)[/tex] units, you transform [tex]\( f(x) \)[/tex] to [tex]\( f(x + h) \)[/tex].
- To translate it to the right by [tex]\( h \)[/tex] units, you transform [tex]\( f(x) \)[/tex] to [tex]\( f(x - h) \)[/tex].
Given that we want to translate the function [tex]\( f(x) = x^2 \)[/tex] to the left by 8 units, we replace [tex]\( x \)[/tex] with [tex]\( x + 8 \)[/tex]:
[tex]\[ h(x) = (x + 8)^2 \][/tex]
Now, let's verify the given options against our transformation:
- Option A: [tex]\( h(x) = x^2 + 8 \)[/tex]
This shifts the graph vertically up by 8 units, not horizontally. So, this is incorrect.
- Option B: [tex]\( h(x) = (x - 8)^2 \)[/tex]
This shifts the graph horizontally to the right by 8 units, not to the left. So, this is also incorrect.
- Option C: [tex]\( h(x) = x^2 - 8 \)[/tex]
This shifts the graph vertically down by 8 units, not horizontally. Thus, this is incorrect as well.
- Option D: [tex]\( h(x) = (x + 8)^2 \)[/tex]
This shifts the graph horizontally to the left by 8 units, which is exactly what we need.
Thus, the correct option is:
[tex]\[ \boxed{D} \][/tex]
