Download x - Dalton State

Graphing
Transformations
It is the supreme art of the teacher to
awaken joy in creative expression and
knowledge.
Albert Einstein
Basic Functions
Given some fairly simple graphs, we can easily
graph more complicated functions by utilizing
some specific methods commonly referred to as
transformations.
Some of the basic functions we will concentrate
on are:
f(x) = x
f(x) = |x|
f(x) = x2
f(x) = x3
f(x) =√x
IDENTITY FUNCTION (x) = x
Domain: (–, )
Range: (–, ) y
x f(x)
-2 -2
0 0
2 2
x
SQUARING FUNCTION (x) = x2
x f(x)
Domain: (–, )
Range: [0, )
y
-2 4
-1 1
0 0
1 1
2 4
x
CUBING FUNCTION (x) = x3
x f(x)
Domain: (–, )
Range: (–, )
y
-2 -8
-1 -1
0 0
1 1
2 8
x
SQUARE ROOT
x f(x)
(x) = √x
Domain: [0, )
Range: [0, )
y
0 0
1 1
4 2
x
ABSOLUTE VALUE (x) = |x|
x f(x)
Domain: (–, )
Range: [0, )
y
-2 2
-1 1
0 0
1 1
2 2
x
Transformations
Let’s investigate what happens when we make
slight changes to the basic function.
For convenience, we will look at the squaring
function (parabola); but the logic will apply to
any function.
Graph: (x) = x2 – 3
Recall our initial function
x
x2
f(x)
-2 4
1
-1 1
-2
0 0
-3
1 1
-2
2 4
1
Now let’s graph our
new functiony
x
Graph: (x) = x2 – 3
Notice our new
function is the same
as our initial function,
it is just moved down
3 units
y
Down 3
units
x
Graph: (x) = x2 + 2
Recall our initial function
x
x2
f(x)
-2 4
6
-1 1
3
0 0
2
1 1
3
2 4
6
Now let’s graph our
new functiony
x
Graph: (x) = x2 + 2
Notice our new
function is the same
as our initial function,
it is just moved up 2
units
y
x
Up 2
units
Vertical Shift
(x) + c
Represents a shift in the graph of (x) up or
down c units
If c > 0, vertical shift up
If c < 0, vertical shift down
Graph: (x) = (x – 1)2
Recall our initial function
x
x2
f(x)
-2 4
9
-1 1
4
0 0
1
1 1
0
2 4
1
Now let’s graph our
new functiony
x
Graph: (x) = (x – 1)2
Right
1 unit
Notice our new
function is the same
as our initial function,
it is just moved to the
right 1 unit
y
x
Notice the 1 to
the right is the
opposite of -1
applied to the x
Graph: (x) = (x + 2)2
Recall our initial function
x
x2
f(x)
-2 4
0
-1 1
1
0 0
4
1 1
9
2 4
16
Now let’s graph our
new functiony
x
Graph: (x) = (x + 2)2
Left 2
units
Notice our new
function is the same
as our initial function,
it is just moved to the
right 1 unit
y
x
Notice the 2 to
the left is the
opposite of +2
applied to the x
Horizontal Shift
(x + c)
Represents a shift in the graph of (x) left or
right c units
If c > 0, shift to the left
If c < 0, shift to the right
Graph: (x) = -x2
Recall our initial function
x
x2
f(x)
-2 4
-4
-1 1
-1
0 0
0
1 1
-1
2 4
-4
Now let’s graph our
new functiony
x
Graph: (x) = -x2
flipped
Notice our new
function is the same
as our initial function,
it is just flipped across
the x-axis.
y
x
Reflection
–(x)
Represents a reflection in the graph of
(x) across the x-axis
(– x)
Represents a reflection in the graph of
(x) across the y-axis
Graph: (x) = ½x2
Recall our initial function
x
x2
f(x)
-2 4
2
-1 1
½
0 0
0
1 1
½
2 4
2
Now let’s graph our
new functiony
x
Graph is wider!
Graph: (x) = 2x2
Recall our initial function
x
x2
f(x)
-2 4
8
-1 1
2
0 0
0
1 1
2
2 4
8
Now let’s graph our
new functiony
x
Graph is thinner!
Vertical Stretching
a(x)
Represents a stretch in the graph of (x) either
toward or away from the y-axis
y=4f(x)
If |a| < 0
Graph is wider
(stretch away form y-axis)
If |a| > 0
Graph is thinner
(stretch toward form y-axis)
y=½f(x)
Operations of Functions
Given two functions  and g, then for all values
of x for which both (x) and g(x) are defined, the
functions  + g,  – g, g, and /g are defined as
follows.
 f  g  x   f ( x )  g( x ) Sum
 f  g  x   f ( x )  g( x ) Difference
 fg  x   f ( x ) g( x ) Product
 f 
f (x)
 g   x   g( x ) , g( x )  0
 
Quotient
 f  g x  f x  g x
This just says that to find the sum of two functions, add
them together. You should simplify by finding like terms.
f x   2 x  3
g x   4 x  1
2
3
f  g  2x  3  4x 1
2
3
 4x  2x  4
3
2
Combine like
terms & put in
descending
order
 f  g x  f x  g x
To find the difference between two functions, subtract
the first from the second. CAUTION: Make sure you
distribute the – to each term of the second function. You
should simplify by combining like terms.
f x   2 x  3
2

g x   4 x  1
3

f  g  2x  3  4x  1
2
3
Distribute
negative
 2 x  3  4 x  1  4 x  2 x  2
2
3
3
2
 f  g x  f x gx
To find the product of two functions, put parenthesis
around them and multiply each term from the first
function to each term of the second function.
f x   2 x  3
g x   4 x  1
2

3


f  g  2x  3 4x 1
2
3
 8 x  2 x  12 x  3
5
2
3
FOIL
Good idea to put in
descending order but not
required.
f
f x 
 x  
g x 
g
To find the quotient of two functions, put the first one
over the second.
f x   2 x  3
2
f 2x  3
 3
g 4x 1
2
g x   4 x  1
3
Nothing more you could do
here. (If you can reduce
these you should).
So the first 4 operations on functions are
pretty straight forward.
The rules for the domain of functions would
apply to these combinations of functions as
well. The domain of the sum, difference or
product would be the numbers x in the
domains of both f and g.
For the quotient, you would also need to
exclude any numbers x that would make the
resulting denominator 0.
Composition of Functions
If  and g are functions, then the composite
function, or composition, of g and  is
defined by
 g f  x   g  f ( x ) .
The domain of g f is the set of all
numbers x in the domain of  such that (x)
is in the domain of g.
Composition of Functions
Composition is simply taking the result of
one function and sticking it into the other
function
x
g  x
g
Function
Machine
f
Function
Machine
f  g  x 
 f  g x  f gx
This is read “f composition g” and means to copy the f
function down but where ever you see an x, substitute in
the g function.
f x   2 x  3
2

g x   4 x  1
3

2
f  g  2 4x 1  3
3
FOIL first and
then distribute
the 2
 32 x  16 x  2  3  32 x  16 x  5
6
3
6
3
g  f x  g f x
This is read “g composition f” and means to copy the g
function down but where ever you see an x, substitute in
the f function.
f x   2 x  3
g x   4 x  1
2

3

3
g  f  4 2x  3 1
2
You could multiply
this out but since it’s
to the 3rd power we
won’t