transformation of functions

1. 1
Transformations of Functions
SECTION 2.7
1
2
3
4
Learn the meaning of transformations.
Use vertical or horizontal shifts to graph
functions.
Use reflections to graph functions.
Use stretching or compressing to graph
functions.

2. 2
TRANSFORMATIONS
If a new function is formed by performing
certain operations on a given function f , then
the graph of the new function is called a
transformation of the graph of f.

5. Parent Functions – The simplest function of its kind. All
other functions of its kind are Transformations of the
parent.

8. 8
EXAMPLE 1 Graphing Vertical Shifts
Let      
, 2, and 3.
f x x g x x h x x
    
Sketch the graphs of these functions on the
same coordinate plane. Describe how the
graphs of g and h relate to the graph of f.

9. 9
EXAMPLE 1 Graphing Vertical Shifts
Solution Make a table of values.

10. 10
EXAMPLE 1 Graphing Vertical Shifts
Solution continued
Graph the equations.
The graph of
y = |x| + 2 is the
graph of y = |x|
shifted two units
up. The graph of
y = |x| – 3 is the
graph of y = |x|
shifted three units
down.

11. 11
VERTICAL SHIFT
Let d > 0. The graph of y = f(x) + d is the graph
of y = f(x) shifted d units up, and the graph of
y = f(x) – d is the graph of y = f(x) shifted d units
down.

12. 12
EXAMPLE 2 Writing Functions for Horizontal Shifts
Let f(x) = x2, g(x) = (x – 2)2, and h(x) = (x + 3)2.
A table of values for f, g, and h is given on the
next slide. The graphs of the three functions f,
g, and h are shown on the following slide.
Describe how the graphs of g and h relate to the
graph of f.

13. 13
EXAMPLE 2 Writing Functions for Horizontal Shifts

14. 14
EXAMPLE 2 Writing Functions for Horizontal Shifts

15. 15
EXAMPLE 2 Writing Functions for Horizontal Shifts
All three functions are squaring functions.
Solution
The x-intercept of f is 0.
The x-intercept of g is 2.
a. g is obtained by replacing x with x – 2 in f .
 
   
2
2
2
f x
g x
x
x
 

For each point (x, y) on the graph of f , there
will be a corresponding point (x + 2, y) on
the graph of g. The graph of g is the graph of
f shifted 2 units to the right.

16. 16
EXAMPLE 2 Writing Functions for Horizontal Shifts
Solution continued
The x-intercept of f is 0.
The x-intercept of h is –3.
b. h is obtained by replacing x with x + 3 in f .
 
   
2
2
3
f x
h x
x
x
 

For each point (x, y) on the graph of f , there
will be a corresponding point (x – 3, y) on the
graph of h. The graph of h is the graph of f
shifted 3 units to the left.
The tables confirm both these considerations.

17. 17
HORIZONTAL SHIFT
The graph of y = f(x – c) is the graph of
y = f(x) shifted |c| units to the right, if c > 0,
to the left if c < 0.

19. 19
EXAMPLE 3
Sketch the graph of the function
  2 3.
g x x
  
Solution
Identify and graph the parent function
  .
f x x

Graphing Combined Vertical and
Horizontal Shifts

20. 20
EXAMPLE 3
Solution continued
Graphing Combined Vertical and
Horizontal Shifts
  2 3.
g x x
  
Translate 2 units to the left
Translate 3 units down

21. 21
REFLECTION IN THE x-AXIS
The graph of y = – f(x) is a reflection of
the graph of y = f(x) in the x-axis.
If a point (x, y) is on the graph of y = f(x),
then the point (x, –y) is on the graph of
y = – f(x).

22. 22
REFLECTION IN THE x-AXIS

23. 23
REFLECTION IN THE y-AXIS
The graph of y = f(–x) is a reflection of
the graph of y = f(x) in the y-axis.
If a point (x, y) is on the graph of y = f(x),
then the point (–x, y) is on the graph of
y = f(–x).

24. 24
REFLECTION IN THE y-AXIS

25. 25
EXAMPLE 4 Combining Transformations
Explain how the graph of y = –|x – 2| + 3 can be
obtained from the graph of y = |x|.
Solution
Step 1 Shift the graph of y = |x| two units right
to obtain the graph of y = |x – 2|.

26. 26
EXAMPLE 4 Combining Transformations
Solution continued
Step 2 Reflect the graph of y = |x – 2| in the
x–axis to obtain the graph of
y = –|x – 2|.

27. 27
EXAMPLE 4 Combining Transformations
Solution continued
Step 3 Shift the graph of y = –|x – 2| three
units up to obtain the graph of
y = –|x – 2| + 3.

28. 28
EXAMPLE 5
Stretching or Compressing a Function
Vertically
Solution
Sketch the graphs of f, g, and h on the same
coordinate plane, and describe how the graphs
of g and h are related to the graph of f.
f x
  x , g x
  2 x , and h x
 
1
2
x .
Let
x –2 –1 0 1 2
f(x) 2 1 0 1 2
g(x) 4 2 0 2 4
h(x) 1 1/2 0 1/2 1

29. 29
EXAMPLE 5
Stretching or Compressing a Function
Vertically
Solution continued

30. 30
EXAMPLE 5
Stretching or Compressing a Function
Vertically
Solution continued
The graph of y = 2|x| is the graph of y = |x|
vertically stretched (expanded) by multiplying
each of its y–coordinates by 2.
The graph of |x| is the graph of y = |x|
vertically compressed (shrunk) by multiplying
each of its y–coordinates by .
1
2
y 
1
2

31. 31
VERTICAL STRETCHING OR
COMPRESSING
The graph of y = af(x) is obtained from the graph
of y = f(x) by multiplying the y-coordinate of each
point on the graph of y = f(x) by a and leaving the
x-coordinate unchanged. The result is
1. A vertical stretch away from the x-axis if
a > 1;
2. A vertical compression toward the x-axis if
0 < a < 1.
If a < 0, the graph of f is first reflected in the
x-axis, then vertically stretched or compressed.