Parent Function and Transformation

1. Parent Functions
and
Transformations

2. Graphs of common functions
It is important to be able to sketch
these from memory.

3. PARENT FUNCTIONS
A parent function is the simplest
function that still satisfies the
definition of a certain type of function.
For example, linear functions which
make up a family of functions, the
parent function would be y = x.

4. The identity function/linear function
f(x) = x

5. The quadratic function
2
)
( x
x
f 

6. x
x
f 
)
(
The square root function/radical
function

7. x
x
f 
)
(
The absolute value function

8. 3
)
( x
x
f 
The cubic function

9. The rational function
1
( )
f x
x


10. Transformation of Functions
Transformation of functions will
give us a better idea of how to
quickly sketch the graph of certain
functions.

11. The transformations are
(1) Vertical
(2) Horizonal
(3) Stretch/Compression.
(4) Reflection in the x-axis
(whenever the sign in front is
negative)

12. VALUE of a
For f(x) = x (Linear functions)
It general form is given by: f(x)=ax+d
if |a|>1, vertical stretch (narrow)
makes the graph move closer to y-axis
if |a|<1, vertical shrink (opens wide)
makes the graph move way from y-
axis

13. SIGN of a
for negative in front of a, flip over
(i.e..Reflection in x-axis)

14. Positive k means, move k-units up
Negative k, move k-units down
k represents vertical shift

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

16. In general, a vertical translation
means that every point (x, y) on the
graph of f(x) is transformed to (x, y
+ c) or (x, y – c) on the graphs
of
or respectively.

17. Horizontal Translation
The 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

 
2
2
y x
   
2
2
y x
 

18. 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
  

19. Sketch the following:
( ) 3
f x x
 
2
( ) 5
f x x
 
3
( ) ( 2)
f x x
 
( ) 3
f x x
 

20. 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
 



21. To remember the difference between
vertical and horizontal translations,
think:
“Add to y, go high.”
“Add to x, go left.”
Helpful Hint

22. Use the basic graph to sketch the
following: TRY
( )
f x x
 
( )
f x x
 
2
( )
f x x
 
( )
f x x
 

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

24. 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

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

26. 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

27.  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.

28. 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.

29. If a is positive, then the
graph will go up.
If a is negative, then the
graph will go down.

30. Transformations of graphs
y = a(x - h)²+k
h is the horizontal shift. The
graph will move in the opposite
direction.
K is the vertical shift. The
graph will move in the same
direction.
If a is positive, then the graph
will go up.

31. For example: y = x² + 6
 y = (x – 0)² + 6, so the V.S. is up 6 and the H.S. is
none. Therefore, the vertex is
V (0, 6).
 Since a is positive, the direction of the parabola is
up.
 Since a is 1, then the parabola is neither fat or
skinny. It is a standard parabola.