When we talk about translations in graphs, it's really cool to see how they move the whole function around on a graph! Let's break down how translations change a graph's points:

1. Vertical Translations:

   • When we add or take away a number ( k ) from the function, like changing ( f(x) ) to ( f(x) + k ), the graph goes up or down.
   • Example: If ( k = 3 ), then every point on the graph of ( f(x) ) goes up by 3. So, if ( f(2) = 5 ), then after the move, it becomes ( f(2) + 3 = 8 ).

2. Horizontal Translations:

   • This happens when we add or take away a number ( h ) from ( x ), which means we change ( f(x) ) to ( f(x - h) ).
   • Example: If ( h = 2 ), the graph shifts to the right by 2 spaces. So, if ( f(2) = 5 ), it changes to ( f(4) = 5 ).

3. Overall Effect:

   • The whole graph moves. When we move it up or down, only the ( y )-coordinates change. When we move it left or right, only the ( x )-coordinates change.
   • This means that the shape of the graph stays the same; it just moves to a different spot on the graph!

Understanding how translations work helps us see how functions change, and it's exciting to notice that a simple change can create new graphs!