Parent Functions and Transformations

Transformation of FunctionsRecognize graphs of common functions

Use shifts to graph functionsUse reflections to graph functionsGraph functions w/ sequence of transformations

The following basic graphs will be used extensively in this section. It is important to be able to sketch these from memory.

The identity function f(x) = x

The quadratic function

2)( xxf

xxf )(

The square root function

xxf )(The absolute value function

3)( xxf

The cubic function

The rational function1

( )f xx

We will now see how certain transformations (operations) of a function change its graph. This will give us a better idea of how to quickly sketch the graph of certain functions. The transformations are (1) translations, (2) reflections, and (3) stretching.

Vertical Translation

OUTSIDE IS TRUE!Vertical Translationthe graph of y = f(x) + d is the graph of y = f(x) shifted up d units;

the graph of y = f(x) d is the graph of y = f(x) shifted down d units.

2( )f x x 2( ) 3f x x

2( ) 2f x x

Horizontal Translation

INSIDE LIES!Horizontal Translationthe graph of y = f(x c) is the graph of y = f(x) shifted right c units;

the graph of y = f(x + c) is the graph of y = f(x) shifted left c units.

2( )f x x

22y x 2

2y x

The values that translate the graph of a function will occur as a number added or subtracted either inside or outside a function.

Numbers added or subtracted inside translate left or right, while numbers added or subtracted outside translate up or down.

( )y f x c d

Recognizing the shift from the equation, examples of shifting the function f(x) = Vertical shift of 3 units up

Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)

3)(,)( 22 xxhxxf

22 )3()(,)( xxgxxf

2x

Use the basic graph to sketch the following:

( ) 3f x x 2( ) 5f x x 3( ) ( 2)f x x ( ) 3f x x

Combining a vertical & horizontal shift

Example of function that is shifted down 4 units and right 6 units from the original function.

( ) 6

)

4

( ,

g x x

f x x

Use the basic graph to sketch the following:

( )f x x

( )f x x 2( )f x x

( )f x x

The big picture…

Example

Write the equation of the graph obtained when the parent graph is translated 4 units left and 7 units down.3y x

3( 4) 7y x

ExampleExplain the difference in the graphs

2( 3)y x 2 3y x

Horizontal Shift Left 3 Units

Vertical Shift Up 3 Units

Describe the differences between the graphs

Try graphing them…

2y x 24y x 21

4y x

A combinationIf the parent function is

Describe the graph of

2y x

2( 3) 6y x The parent would be horizontally shifted right 3 units and vertically shifted up 6 units

If the parent function is

What do we know about

3y x32 5y x

The graph would be vertically shifted down 5 units and vertically stretched two times as much.

What can we tell about this graph?

3(2 )y xIt would be a cubic function reflected across the x-axis and horizontally compressed by a factor of ½.