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Graph Transformations

There are many times when you'll know very well what the graph of a particular function looks like, and you'll want to know what the graph of a very similar function looks like. In this chapter, we'll discuss some ways to draw graphs in these circumstances.

Transformations "after" the original function

Suppose you know what the graph of a function ( ) looks like. Suppose fx

d 2 R is some number that is greater than 0, and you are asked to graph the

function f (x) + d. The graph of the new function is easy to describe: just

take every point in the graph of ( ), and move it up a distance of . That

fx

d

is, if ( ) is a point in the graph of ( ), then ( + ) is a point in the

a, b

fx

a, b d

graph of f (x) + d.

(9')

g

As an explanation for what's written above: If ( ) is a point in the graph a, b

of ( ), then that means ( ) = . Hence, ( ) + = + , which is to say

fx

fa b

fa d b d

that ( + ) is a point in the graph of ( ) + .

a, b d

fx d

The chart on the next page describes how to use the graph of f (x) to create

the graph of some similar functions. Throughout the chart, 0, 1, and d> c>

(a, b) is a point in the graph of f (x).

Notice that all of the "new functions" in the chart dier from ( ) by some fx

algebraic manipulation that happens after plays its part as a function. For f

example, first you put x into the function, then f (x) is what comes out. The

function has done its job. Only after has done its job do you add to get

f

d

the new function ( ) + . fx d

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Because all of the algebraic transformations occur after the function does its job, all of the changes to points in the second column of the chart occur in the second coordinate. Thus, all the changes in the graphs occur in the vertical measurements of the graph.

New function

How points in graph of f (x) become points of new graph

visual eect

( )+ fx d

() fx d

() cf x

1 () fx

c

() fx

( ) 7! ( + ) a, b a, b d

( ) 7! (

)

a, b a, b d

( ) 7! ( ) a, b a, cb

( ) 7! ( 1 ) a, b a, b

c

( ) 7! ( ) a, b a, b

shift up by d

shift down by d

stretch vertically by c

shrink vertically by 1

c

flip over the -axis x

Examples.

? The graph of ( ) = 2 is a graph that we know how to draw. It's fx x

drawn on page 59.

We can use this graph that we know and the chart above to draw ( ) + 2, fx

( ) 2, 2 ( ), 1 ( ), and ( ). Or to write the previous five functions

fx

fx fx

fx

2

without the name of the function , these are the five functions 2 + 2, 2 2,

f

xx

2

2,

2

x

,

and

2. These graphs are drawn on the next page.

x

x

2

68

}

2.

69

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zx- (-

z

Urv\Of'

Z_

N

c1'l 4LLS

c3\

Transformations "before" the original function

We could also make simple algebraic adjustments to ( )

the func-

f x before

tion gets a chance to do its job. For example, ( + ) is the function where

f

fx d

you first add d to a number x, and only after that do you feed a number into

the function . f

The chart below is similar to the chart on page 68. The dierence in the

chart below is that the algebraic manipulations occur before you feed a num-

ber into , and thus all of the changes occur in the first coordinates of points f

in the graph. All of the visual changes aect the horizontal measurements of the graph.

In the chart below, just as in the previous chart, 0, 1, and ( ) is

d> c>

a, b

a point in the graph of f (x).

New function

How points in graph of f (x) become points of new graph

visual eect

f (x + d)

f (x d)

() f cx

(1 ) fx

c

() fx

(a, b) 7! (a d, b)

(a, b) 7! (a + d, b)

( ) 7! (1 ) a, b a, b

c

( ) 7! ( ) a, b ca, b

( ) 7! ( )

a, b

a, b

shift left by d

shift right by d

shrink horizontally by 1

c

stretch horizontally by c

flip over the -axis y

One important point of caution to keep in mind is that most of the visual horizontal changes described in the chart above are the exact opposite of the eect that most people anticipate after having seen the chart on page 68. To

70

get an idea for why that's true let's work through one example. We'll see

why the first row of the previous chart is true, that is we'll see why the graph

of ( + ) is the graph of ( ) shifted left by :

fx d

fx

d

Suppose that 0. If ( ) is a point that is contained in the graph of

d>

a, b

( ), then ( ) = . Hence, (( ) + ) = ( ) = , which is to say that

fx

fa b

f a d d fa b

(a d, b) is a point in the graph of f (x + d). The visual change between the

point ( ) and the point (

) is a shift to the left a distance of .

a, b

a d, b

d

Examples.

? Beginning with the graph ( ) = 2, we can use the chart on the fx x

previous page to draw the graphs of ( + 2), ( 2), (2 ), (1 ), and fx fx f x f x

2

( ). We could alternatively write these functions as ( + 2)2, ( 2)2,

fx

x

x

(2 )2, (x)2, and ( )2. The graphs of these functions are drawn on the next

x

x

page. 2

Notice on the next page that the graph of ( )2 is the same as the graph x

of our original function 2. That's because when you flip the graph of 2

x

x

over the -axis, you'll get the same graph that you started with. That 2 and

y

x

( )2 have the same graph means that they are the same function. We know x

this as well from their algebra: because ( 1)2 = 1, we know that ( )2 = 2. xx

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