Transformations can change how functions behave, especially their domain and range. Here’s a simpler breakdown:

1. Moving Up or Down (Vertical Translations):

   • When you shift a function up or down, it changes the range but not the domain.
   • For example, if we look at $f(x) = x^2$, its range is $[0, \infty)$. If we add 3 to it, like in $g(x) = x^2 + 3$, the range now starts at 3: $[3, \infty)$.

2. Moving Left or Right (Horizontal Translations):

   • If you move a function to the left or right, it changes the domain but not the range.
   • For instance, $f(x) = x^2$ and $h(x) = (x - 2)^2$ have the same range of $[0, \infty)$, but they are positioned differently.

3. Stretching and Shrinking:

   • When you stretch a function vertically, the range gets larger, while stretching it horizontally affects the domain.
   • An example is $j(x) = 2x^2$. Its range is still $[0, \infty)$, just like $f(x)$.

So, remember to check both the domain and range when looking at transformations!