graphing transformations. it is the supreme art of the teacher to awaken joy in creative expression...
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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
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
y
x
x
-2
-1
0
1
2
x2
4
1
0
1
4
f(x)
1
-2
-3
-2
1
Recall our initial function
Now let’s graph our new function
Graph: (x) = x2 – 3
y
x
Notice our new function is the same as our initial function, it is just moved down 3 units
Down 3 units
Graph: (x) = x2 + 2
y
x
x
-2
-1
0
1
2
x2
4
1
0
1
4
f(x)
6
3
2
3
6
Recall our initial function
Now let’s graph our new function
Graph: (x) = x2 + 2
y
x
Notice our new function is the same as our initial function, it is just moved up 2 units
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
y
x
x
-2
-1
0
1
2
x2
4
1
0
1
4
f(x)
9
4
1
0
1
Recall our initial function
Now let’s graph our new function
Graph: (x) = (x – 1)2
y
x
Notice our new function is the same as our initial function, it is just moved to the right 1 unit
Right 1 unit
Notice the 1 to the right is the opposite of -1 applied to the x
Graph: (x) = (x + 2)2
y
x
x
-2
-1
0
1
2
x2
4
1
0
1
4
f(x)
0
1
4
9
16
Recall our initial function
Now let’s graph our new function
Graph: (x) = (x + 2)2
y
x
Notice our new function is the same as our initial function, it is just moved to the right 1 unit
Left 2 units
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
y
x
x
-2
-1
0
1
2
x2
4
1
0
1
4
f(x)
-4
-1
0
-1
-4
Recall our initial function
Now let’s graph our new function
Graph: (x) = -x2
y
x
Notice our new function is the same as our initial function, it is just flipped across the x-axis.
flipped
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
y
x
x
-2
-1
0
1
2
x2
4
1
0
1
4
f(x)
2
½
0
½
2
Recall our initial function
Now let’s graph our new function
Graph is wider!
Graph: (x) = 2x2
y
x
x
-2
-1
0
1
2
x2
4
1
0
1
4
f(x)
8
2
0
2
8
Recall our initial function
Now let’s graph our new function
Graph is thinner!
Vertical Stretching
a(x)
Represents a stretch in the graph of (x) either toward or away from the y-axis
If |a| < 0 Graph is wider
(stretch away form y-axis)
If |a| > 0 Graph is thinner
(stretch toward form y-axis)
y=4f(x)
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.
( ) ( )x x x f g f g Sum
( ) ( )x x x f g f g Difference
( ) ( )x x x fg f g Product
( ), ( ) 0
( )
xx x
x
f fg
g gQuotient
xgxfxgf This just says that to find the sum of two functions, add them together. You should simplify by finding like terms.
1432 32 xxgxxf
1432 32 xxgf
424 23 xx
Combine like terms & put in descending order
xgxfxgf 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.
1432 32 xxgxxf
1432 32 xxgf
1432 32 xx
Distribute negative
224 23 xx
xgxfxgf 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.
1432 32 xxgxxf
1432 32 xxgf
31228 325 xxx
FOIL
Good idea to put in descending order but not required.
xg
xfx
g
f
To find the quotient of two functions, put the first one over the second.
1432 32 xxgxxf
14
323
2
x
x
g
f 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
( ) .x xg f g f
The domain of is the set of all numbers x in the domain of such that (x) is in the domain of g.
g f
Composition of FunctionsComposition is simply taking the result of one function and sticking it into the other function
g x
FunctionMachine
x
f g x
FunctionMachine
gf
xgfxgf 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.
1432 32 xxgxxf
314223 xgf
51632321632 3636 xxxx
FOIL first and then distribute the 2
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