Summary of Graphing Techniques

To Graph: Draw graph of f and: Functional Change to f(x)

Vertical shifts

( ) , 0y f x k k Raise graph of f by k units Add k to each y-coordinate.

( ) , 0y f x k k Lower graph of f by k units Subtract k from each y-

coordinate.

Horizontal shifts

( ), 0y f x c c Shift graph of f left c

units

Subtract c from each x-

coordinate

( ), 0y f x c c Shift graph of f right c

units Add c to each x-coordinate

Compress or stretch

( ), 0y af x a Stretch vertically if 1a ,

or compress if 0 1a

Multiply each y-coordinate

by a

( ), 0y f ax a

Stretch horizontally if

0 1a , compress if

1a

Multiply each x-coordinate

by 1 a

Reflection about the x-axis

( )y f x Reflect graph of f about

x-axis

Multiply each y-coordinate

by 1 .

Reflection about the y-axis

( )y f x Reflect graph of f about

the y-axis

Multiply each x-coordinate

by 1 .