parent functions and transformations. transformation of functions recognize graphs of common...

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.