1.1_big_ideas_ppt.ppt on polynomial functions

Parent Functions and
Transformations

Transformation of Functions
Recognize graphs of common functions
Use shifts to graph functions
Use reflections to graph functions
Graph 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
)
( x
x
f 

x
x
f 
)
(
The square root function

x
x
f 
)
(
The absolute value function

3
)
( x
x
f 
The cubic function

The rational function
1
( )
f x
x


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 Translation
the graph of y = f(x) + d
is the graph of y = f(x)
shifted up d units;
the graph of y = f(x)  d
shifted down d units.
2
( )
f x x
 2
( ) 3
f x x
 
2
( ) 2
f x x
 

Horizontal Translation
INSIDE LIES!
Horizontal Translation
the graph of y = f(x  c)
shifted right c units;
the graph of y = f(x + c)
shifted left c units.
2
( )
f x x

 
2
2
y x
   
2
2
y 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
)
(
,
)
( 2
2


 x
x
h
x
x
f
2
2
)
3
(
)
(
,
)
( 

 x
x
g
x
x
f
2
x

Use the basic graph to sketch the
following:
( ) 3
f x x
 
2
( ) 5
f x x
 
3
( ) ( 2)
f x x
  ( ) 3
f 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
 

Example
 Write the equation of the graph obtained when the
parent graph is translated 4 units left and 7
units down.
3
y x

3
( 4) 7
y x
  

Example
 Explain the difference in the graphs
2
( 3)
y x
 
2
3
y x
 
Horizontal Shift Left 3 Units
Vertical Shift Up 3 Units

 Describe the differences between the graphs
 Try graphing them…
2
y x
 2
4
y x
 2
1
4
y x


A combination
 If the parent function is
 Describe the graph of

2
y x

2
( 3) 6
y 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

3
y x

3
2 5
y 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 x
 
It would be a cubic function reflected
across the x-axis and horizontally
compressed by a factor of ½.

1.1_big_ideas_ppt.ppt on polynomial functions

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1.1_big_ideas_ppt.ppt on polynomial functions