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

y

x

x

-2

0

2

f(x)

-2

0

2

Domain: (– , ) Range: (– , )

Domain: (– , ) Range: [0, )

SQUARING FUNCTION (x) = x2

y

x

x

-2

-1

0

1

2

f(x)

4

1

0

1

4

CUBING FUNCTION (x) = x3

y

x

x

-2

-1

0

1

2

f(x)

-8

-1

0

1

8

Domain: (– , ) Range: (– , )

SQUARE ROOT (x) = √x

y

x

x

0

1

4

f(x)

0

1

2

Domain: [0, )Range: [0, )

Domain: (– , ) Range: [0, )

ABSOLUTE VALUE (x) = |x|

y

x

x

-2

-1

0

1

2

f(x)

2

1

0

1

2

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

xfgxfg 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.

1432 32 xxgxxf

132432 xfg

You could multiply this out but since it’s to the 3rd power we won’t