Transformations
Warm Up – Evaluate the function
and simplify.
1. f x    x  x  3
a. f 4
b. f x  2
2
1
2. f x    x x  1
2
a. f  4
 2
b. f   
 3
Solution to Warm Up #1
f x    x  x  3
2
a. f 4   17
b.
2


f x  2   x  5x  1
Solution to Warm Up #2
1
f x    x x  1
2
a. f  4  6
 2 1
b. f    
 3 9
Given the graph of f(x), find f(x)+3
x
f(x)
f(x) +
3
0
6
9
(0, 9)
2
3
6
(2, 6)
7
3
6
(7, 6)
9
6
9
(9,9)
Point
Given the graph of f(x), find f(x) - 2
x
f(x)
f(x) 2
0
5
3
(0, 3)
2
2
0
(2, 0)
7
2
0
(7, 0)
9
5
3
(9,3)
Point
Given the graph of f(x), find - f(x)
x
f(x)
-f(x)
Point
0
5
-5
(0, -5)
2
2
-2
(2, -2)
7
2
-2
(7, -2)
9
5
-5
(9,-5)
 The following graphs represent the
most commonly used functions in
algebra.
Parent Graphs
CONSTANT : f x   c
Cubic : f x   x
Linear : f ( x)  x
Quadratic : f ( x)  x
2
Absolute Value : f ( x)  x
Square Root : f ( x)  x
3
Linear : f ( x)  x
Quadratic : f ( x)  x
2
Absolute Value : f ( x)  x
Square Root : y  x
CONSTANT : y  c
Cubic : f x   x
3
 Many complicated graphs are derived by
shifting, stretching, shrinking, or reflecting
the common graphs listed.
 Shifts, stretches, shrinks, and reflections
are called transformations.
 Many graphs of functions can be created
from a combination of these
transformations.
Translations
Vertical and Horizontal Shifts
Shift Upward
Vertical
Shift Up
“k” units.
Shift Downward
Vertical
Shift Down
“k” units.
Shift to the Right
Horizontal
Shift Right
“k” units.
Shift to the Left
Horizontal
Shift Left
“k” units.
Sketch:
Parent Graphs:
f  x  x
Up 3
f  x  x  3
Sketch:
Parent Graphs:
f  x  x
2
f  x  x  2
2
Down 2
Sketch:
Parent Graphs:
f  x  x
f  x  x  2
Right 2
Sketch:
Parent Graphs:
f  x  x
Left 1
f  x  x 1  2
Down 2
Sketch the graph:
f  x  x  4
2
Down 4
Sketch the graph:
f  x    x  4  2
2
Right 4
Down 2
Sketch the graph:
f  x  x  3 1
Right 3
Up 1
Sketch the graph and find domain
and range:
f  x  x  4
Up 4
D : [0,  )
R : [ 4,  )
Interesting to Note…….
Reflections
Reflections in the x-axis
A mirror image across
the x-axis.
Reflections in the y-axis
A mirror image across
the y-axis.
Sketch the graph:
f  x  x
2
Flip
Flip across the
x-axis.
Sketch the graph and state the
domain and range:
f  x   x  2  3
Flip
Left 2
D : (, )
R : (,3]
Up 3
Sketch the graph:
f  x   x 1  3
Flip
Down 3
Left 1
D :  ,  
R :  ,3
Stretches
Horizontal Stretch (Compress)
Vertical Stretch (Compress)
Solution
Solution
Solution
Solution
Solution
Finding Equations from
Graphs
Ex: Find the equation of the graph.
g x    x  2
4
Ex: Find the equation of the graph.
g x   x  3
4
Nonrigid Transformations
 Horizontal shifts, vertical shifts, and
reflections are called rigid
transformations because the basic
shape of the graph is unchanged.
 These transformations change only
the position of the graph in the
coordinate plane.
 Nonrigid Transformations are
those that cause a distortion – a
change in the shape of the original
graph.
Examples