graphical transformations

8/10/2019 Graphical Transformations

1/15

Transformations

Transformations

Transformations

Transformations

2.4: Transformations of

Functions and Graphs

We will be looking at simple functions and

seeing how various modifications to the

functions transform them.


2/15

VERTICALTRANSLAT

IONS

Above is the graph of 2xxf

x

y

What wouldf(x) + 1 look like? (This would mean taking all

the function values and adding 1 to them).

x

y

11 2 xxf

What wouldf(x) - 3 look like? (This would mean taking all

the function values and subtracting 3 from them).

x

y

33 2 xxf

2xxf As you can see,a number

added or

subtracted from

a function will

cause a verticalshift or

translationin

the function.


3/15

VERT

ICAL

TR

ANSLAT

IONS

Above is the graph of xxf What wouldf(x) + 2 look like?

22 xxfSo the graph

f(x) + k , wherek

is any real

number is the

graph off(x)

but vertically

shifted byk . Ifk is positive it

will shift up. If

k is negative it

will shift down

x

y

x

y

x

y

44 xxf xxf

What wouldf(x) - 4 look like?


4/15


x

y

What wouldf(x+2) look like? (This would mean taking all thex

values and adding 2 to them before putting them in the function).

As you can see,a number

added or

subtracted from

the xwill cause

a horizontal

shift or

translationin

the function but

opposite way ofthe sign of the

number.

HORIZONTAL TRANSLATIONS

x

y

x

y

2xxf

2

11 xxf

222 xxf

What wouldf(x-1) look like? (This would mean taking all thex

values and subtracting 1 from them before putting them in thefunction).


5/15

HORIZONTAL TRANSLATIONS

Above is the graph of 3xxf What wouldf(x+1) look like?

So the graph

f(x-h), whereh isany real number is

the graph off(x)

but horizontally

shifted byh.Notice the

negative.(If you set the stuff in

parenthesis = 0 & solve

it will tell you how to shiftalongxaxis).

x

y

x

y

x

y

3

11

xxf

3xxf

What wouldf(x-3) look like?

333 xxf

03x

So the graph

f(x-h), whereh isany real number is

the graph off(x)

but horizontally

shifted byh.Notice the

negative.(If you set the stuff in

parenthesis = 0 & solve

it will tell you how to shiftalongxaxis).

3x

So shift along thex-axis by 3

shift right 3


6/15

x

y

x

y

x

y

We could have a function that is transformed or translated

both vertically AND horizontally.

Above is the graph of xxf

What would the graph of look like? 3)2( xxf

up3

left 2


7/15

and

If we multiply a function by a non-zero real number it has the

effect of either stretching or compressing the functionbecause it causes the function value (the yvalue) to be

multiplied by that number.

Let's try some functions multiplied by non-zero real numbers

to see this.

DILATION:


8/15


So the graph

af(x), wherea

is any realnumber

GREATER

THAN 1,is the

graph off(x)

but vertically

stretched or

dilatedby a

factor of a.

x

y

x

y

x

y

xxf

xxf 22 xxf 44

What would2f(x) look like?

What would4f(x) look like?

Notice for anyx on

the graph, the new

(red) graph has a y

value that is 2

times as much as

the original (blue)

graph's yvalue.

Notice for anyx on

the graph, the new

(green) graph has

a yvalue that is 4

times as much as

the original (blue)

graph's yvalue.


9/15


So the graph

af(x), whereais 0 < a< 1, is

the graph of

f(x) but

verticallycompressed

or dilatedby a

factor of a.

x

y

x

y

Notice for anyx on the graph,

the new (red) graph has a y

value that is 1/2 as much as the

original (blue) graph's yvalue.

x

y

Notice for anyx on the graph,

the new (green) graph has a y

value that is 1/4 as much as the

original (blue) graph's yvalue.

xxf 41

4

1

What if the value of awas positive but less than 1?

xxf

xxf2

1

2

1

What would1/4 f(x) look like?

What would1/2f(x) look like?


10/15


So the graph

-f(x) is areflection

about the

x-axis of the

graph off(x).(The new graph

is obtained by

"flipping

or

reflectingthe

function over the

x-axis)

x

y

What if the value of awas negative?

What would-f(x) look like?

x

y

xxf

xxf

Notice anyx on

the new (red)

graph has a y

value that is the

negative of theoriginal (blue)

graph's yvalue.


11/15

x

y


There is one last transformation we want to look at.

Notice anyx on

the new (red)

graph has anx

value that is the

negative of theoriginal (blue)

graph'sxvalue.

x

y

3xxf 3xxf

What wouldf(-x) look like? (This means we are going to

take the negative ofxbefore putting in the function)

So the graph

f(-x) is areflection

about the

y-axis of the

graph of f(x).(The new graph

is obtained by

"flipping

or

reflectingthe

function over the

y-axis)


12/15

Summary of Transformations So Far

khxfa

horizontal translation of h

(opposite sign of number with thex)

If a> 1, then vertical dilation or stretch by a factor ofa

vertical translation of k

If 0 < a< 1, then vertical dilation or compression by a factor of a

f(-x) reflection

abouty-axis

**Do reflections and dilations BEFORE vertical and horizontal translations**

If a< 0, then reflection about thex-axis

(as well as being dilated by a factor of a)


13/15

Graph using transformations 12

1

x

xf

We know what the graph would look like if it was

from our library of functions.

x

xf 1

x

y

moves up 1

moves right 2

reflects

about thex-axis

x

y

x

y

x

y

x

y


14/15

There is one more Transformation we need to know.

kb

hxfa

)(

horizontal translation of h

(opposite sign of number with thex)

If a> 1, then vertical dilation or stretch by a factor ofa

vertical translation of k

If 0 < a< 1, then vertical dilation or compression by a factor of a

f(-x) reflection

abouty-axis

Do reflections and dilations BEFORE vertical and horizontal translations

If a< 0, then reflection about thex-axis

(as well as being dilated by a factor of a)

horizontal dilation by a

factor of b


15/15

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, UtahUSA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded

from www.mathxtc.comand for it to be modified.
http://www.slcc.edu/http://www.mathxtc.com/http://www.mathxtc.com/http://www.slcc.edu/

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