Understanding Horizontal and Vertical Shifts in Graphs

When we look at graphs of functions, we can change their position on the grid. This is done through two main types of shifts: horizontal shifts and vertical shifts.

Horizontal Shifts:

• A horizontal shift happens when we change the function to look like this: ( f(x - c) ), where ( c ) is a positive number.
• This move pushes the graph to the right by ( c ) units.
• If we have ( f(x + c) ), then the graph shifts to the left instead.
• Example: If we start with the graph of ( f(x) = x^2 ) and change it to ( f(x - 3) = (x - 3)^2 ), the graph moves to the right by 3 units.

Vertical Shifts:

• Vertical shifts happen when we change the function to look like this: ( f(x) + d ), where ( d ) is also a positive number.
• This change lifts the graph up by ( d ) units.
• If we write ( f(x) - d ), the graph moves down.
• Example: Starting with ( f(x) = x^2 ), if we change it to ( f(x) + 2 = x^2 + 2 ), the graph moves up by 2 units.

Important Points:

• Horizontal shifts affect the x-values (side to side), while vertical shifts affect the y-values (up and down).
• You can think of it this way: for horizontal shifts, the new coordinates change from ( (x, y) ) to ( (x + c, y) ). For vertical shifts, they change to ( (x, y + d) ).

These shifts help us move the graph around on the coordinate grid while keeping its shape the same.