Function Transformations

Horizontal and Vertical Translations A __________________ of a function alters the equation and any combination of the ___________________, _____________, and ___________________ of the graph. Translation Vertical translations shift _____ or ____________.

Or rearranged:

Horizontal translations shift _________ or ____________. Together the equation looks like: or Let’s use a parabola to compare vertical and horizontal translations. Example 1: Sketch each of the following: a.) 2y x= b.) 22y x− =

c.) ( )23y x= − Note: This can also be expressed as:

Example 2: Sketch the graph of 4 3y x= − + .

Example 3: Describe the translation that has been applied to the graph of ( )f x to obtain the graph of ( )g x . Determine the equation of the translated function in the form ( )y k f x h− = − .

Example 4: Given the following graph of ( )f x , sketch each of the following transformations and write each using mapping notation:

a.) ( ) ( ) 2h x f x= − b.) ( ) ( 1)t x f x= + c.) ( ) ( 2) 5v x f x= − −

Reflections and Stretches A _________________ of a graph creates a ______________________ in a line called the line of reflection. The ______________ of the graph does not change. Reflections: ( )f x

y

x

( )f x−

( )f x− Example 1: Given the graph of ( )y f x= , graph ( )y f x= − and ( )y f x= − .

Invariant points:

Vertical and Horizontal Stretches A ____________, unlike a translation or a reflection, changes the ____________ of the graph. They do not change the ___________________ of the graph. Vertical stretches

( )y af x= i.e.: 2 ( )f x

1 ( )2

f x

Example 1: Given the graph of ( )y f x= , sketch: a.) ( ) 2 ( )g x f x=

b.) 1( ) ( )2

h x f x=

a.) b.)

Horizontal Stretches

( )y f bx= i.e.: (2 )f x

12

f x

Example 2: Given the graph of ( )y f x= , sketch: a.) ( ) (2 )g x f x=

b.) 1( )2

h x f x =

a.) b.)

Example 3: The graph of the function ( )y f x= has been transformed by either a stretch or reflection. Write the equation of the transformed graph, ( )g x .

Example 4: Given: ( ) 3 4f x x= − , a.) Determine the equation ( )g x if ( ) ( )g x f x= − . b.) Determine the equation ( )h x if ( ) ( )h x f x= − .

Combining Transformations The order in which we apply transformations does matter when transforming a graph. The equation that models all the transformations is: or When applying transformations, you must ______________ and ________________ before you ______________________________________. ** ** Example 1: Given the graph of ( )y f x= , sketch each of the following: a.) ( )3 2y f x=

b.) ( )3 6y f x= + Example 2: Given the graph 2( )f x x= , describe the transformations in order

that must be applied to sketch 1( ) (2( 4)) 12

g x f x= − − + . Write the corresponding

equation and sketch the graph of ( )g x .

Example 3: The graph of ( )y g x= represents a transformation of the graph of ( )y f x= . Write an equation of ( )g x in the form ( )( )y af b x h k= − + .

Example 4: The graph of ( )y g x= represents a transformation of the graph of ( )y f x= . Write an equation of ( )g x in the form ( )( )y af b x h k= − + .

Inverse of a Relation Inverse functions are denoted by ________________. To find the inverse of a function _______________ the x- and y- coordinates. The inverse of a relation is a reflection over the line _________. Example 1: Given the following graph:

a.) Sketch the graph of the inverse relation. b.) State the domain and range of the relation and its inverse. c.) Determine whether the relation and its inverse are functions.

a.) b.) c.) Vertical line test:

Horizontal line test:

Restricting the Domain Example 2: Given the function 2( ) 2f x x= − :

a.) Sketch the function and determine whether its inverse is a function. b.) Sketch the inverse of ( )f x on the same coordinate axes. c.) Describe how the domain of ( )f x could be restricted so that the inverse

of ( )f x is a function. Show the results on a graph. a.) and b.)

c.)

Equations of Inverses Example 3: Determine the equation of the inverse of each of the following and verify by sketching the relation and its inverse: a.) ( ) 3 6f x x= + b.) 2( ) 4f x x= −