Planar
Transformations
Planar
Transformations
Planar
Transformations
Planar
Planar Transformations
Transformations
of
Graphs
of
Graphs
of
of
of Graphs
Graphs
Let ff : :: D
D→
R be
be aaa function
function with
with D
D⊂
R. Then
Then ff(x)
f(x)
—+ R
C R.
y
= y
Let
(x) =
=
Let
D
be
function
with
D
R.
Then
—+
C
= y
Let
fff : : D
!
RR
be
aa function
with
D
⇢
R.
Then
ff(x)
(x)
Let
D
R
be
function
with
D
R.
Then
—+
C
y
=
in two
two variables
variables
inin
two
variables
in
two
variables
in two variables
Examples.
Examples.
Examples.
Examples.
Examples.
• log
loge(x)
is an
an equation
equation in
in two
two variables.
variables.
= yy is
(x) =
==
loge(x)
is
an
equation
in
two
variables.
e (x)
y
•••• log
y
is
an
equation
in
two
variables.
loge(x)
is
an
equation
in
two
variables.
y
e
=
• x2 =
is an
an equation
equation in
in two
two variables.
variables.
y isis
2 =
an
equation
in
two
variables.
•••• x
==
yyy is
an
equation
in
two
variables.
= y is an equation in two variables.
• 4x
333 −—— 2x
4x
2x ++
is an
an equation
equation in
in two
two variables.
variables.
+ 11 == yyy isis
3
4x
2x
an
equation
in
two
variables.
•••• 4x
2x
+
3 — 2x + xx === yy isis an
4x
an equation
equation in
in two
two variables.
variables.
is an
an
is
is
an
is
is an
an
equation
equation
equation
equation
Graphs are
are sets
sets of
of solutions
solutions
Graphs
Graphs
are
sets
of
solutions
Graphs
are
sets
of
solutions
Graphs
are
sets
of
solutions function
The point (2,4) is in the graph of
222 =
the function
222 because
x
because 222
= 4.
4. Similarly,
Similarly,
The
point
(2,
4) is
isis in
in
the
graph
of
the
xx
The
point
(2,4)
in
the
graph
of
function
the
2
because
= 4.
4.
Similarly,
222 =
The
point
(2,
4)
the
graph
of
the
function
x
because
2
Similarly,
The
point
(2,4)
is
in
the
graph
of
function
the
2
x
because
=
4.
Similarly,
2
(3,9),
(5,25),
and (−1,
(—1,1)
1) are
are points
points in
in the
the graph
graph of
of xx
.
2
x
(3,
9),
(5,
25),
and
2.
(3,9),
(5,25),
and
(—1,
1)
are
points
in
the
graph
of
.
2
(3,
9), (5,
25), and
((—1,
1, 1) are
in the
of
(3,9),
(5,25),
and
are points
points
the graph
graph point
of xx
.. in the graph of x
2
2 is
Notice
that each
each point
point1)listed
listed
above,inand
and
every
,
2
Notice
that
above,
every
point
in
the
graph
of
xx
2 , is
Notice
that
each
point
listed
above,
and
every
point
in
the
graph
of
,
2
is
Notice
that
each
point
listed
above,
and
every
point
in
the
graph
of
x
Notice that
each
point listed
and every point in the graph of x
,, is
2
2 = above,
a
solution
of
the
equation
2
x
y.
solution
of the
the equation
equation x
x2
y.
2 =
solution
= y.
aaaasolution
solution ofof
of the
the equation
equation x
2=
x
= y.
y.
This
R with D ⊂ R, which is
This observation
observation applies
applies to
any function
f:: D
to any
function ff:
D→
—* R with D C R.
This
observation
applies
any
to
function
D
R
with
D
R.
—*
C
This
observation
applies
to
This
observation
applies
any function
the
content
of the fact
below.
to any
function ff:: D
D!
R with
with D
D⇢
R.
—* R
C R.
Important Fact:
Fact: The
The graph
graph of
of f(x)
f(x)
is the
the set
set of
of solutions
solutions of
of the
the equation
equation
Important
is
Important
Fact:
The
graph
of
fff(x)
(x)
is
the
set
of
solutions
of
the
equation
Important
Fact:
The
graph
Important
Fact:
The
graph
of
(x)
is
the
set
of
solutions
of
the
equation
of
is
the
set
of
solutions
of
the
equation
f(x)=y.
f(x)=y.
fff(x)=y.
(x)
(x) =
= y.
y.
Jilt
Jilt
Jilt
249
279
249
249
249
4(x’)i
4(x’)i
Examples.
• The graph of 1og(x) is the set of solutions of the equation 1og(x) = y.
Examples.
Examples.
• The graph of 1og(x) is the set of solutions of the equation 1og(x) = y.
• The graph of loge (x) is the set of solutions of the equation loge (x) = y.
1Oe (X
1Oe (X
• The set of solutions of the equation ex + 3x
• Theexset
of solutions of the equation ex + 3x −
the function
+ 3x
• The set
of 3x
solutions
of the equation ex + 3x
the function
ex +
− x1
the function ex + 3x
—
—
y is the graph of
= y is the graph of
= y is the graph of
=
1
x
—
—
*
*
*
*
*
*
*
*
*
*
//
// II
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
Common
of
graphs
*
*transformations
*
*
*
*
*
*
*
*
*
*
*
Common
transformations
The next three
propositions — 18,of19,graphs
and 20 — explain how some common
planar
transformations
can transform
thesome
graph
of a function
f (x) into the
The next
three propositions
explain how
planar transforma
common
Common
transformations
of graphs
graph
a different,
closely
function.
tions
canoftransform
thebut
graph
of arelated
function
f(x) into the graph of a different,
nextreading
three propositions
explain howit some
transforma
Before
the
three propositions,
mightcommon
be worthplanar
skipping
over them
butThe
closely
related
function.
250
tions
transform
canchart
the 283
graph
of asummarizes
function f(x)
intocontent,
the graph
a different,
to the
on page
that
their
andofto
the list of
but
closely that
related
function.
examples
follow.
You can come back to the proofs of the propositions
at the conclusion of the chapter. 250
280
2
Propostion (18).
(17). IfIfSS ⊆ RR
the graph
is the
Propostion
isis the
(x), then
2
graph ofof ff(x),
then AA(a,b)(S)
the
(a,b) (S) is
graphofofff(x—a)+b.
graph
(x − a) + b.
Proof:
(x), the
Proof: The
Thegraph
graphofofff(x),
setS,
theset
set ofofsolutions
solutions of
ofthe
the equation
equation
theset
5, isisthe
f f(x)=y.
(x) = y.
5
A
The
(x−a)a)==yy−bbisisequivalent
(x−a)+b
Theequation
equationff(x
equivalentto
tothe
theequation
equationff(x
a) + b =
= y,
so50AA(ab)(S)
(x − a)
theset
setofofsolutions
solutions ofofff(x
a) ++bb == y.y. That
That is,
is, AA(a,b)(S)
is
(a,b) (S) isisthe
(a,b) (S) is
the
(x − a)a)++b.b.
thegraph
graphofofthe
thefunction
functionff(x
—
—
—
—
—
2
Propostion
that
isis the
Propostion(19).
(18).
Suppose
that cc6= 00 and
and that
that dd 6=4 0.0. IfIf SS ⊆C RR
2
the
Suppose
c 0
graph
(x), then
(S)
( xc ).
graphofofff(x),
then
thegraph
graphofofdfdf().
(5) isisthe
0 d
o\
a) (s
/c.
II
S
on
‘Co
on
Proof:
Proof:
p-o
()
)
c 0
The
(S)
( xc ) == yd , or
Theset
set
(5) isis the
the set
set ofofsolutions
solutions ofof the
the equation
equation ff()
or
0 d
c
0
equivalently,
( xc ) == y.y. Thus,
(S)
equivalently,ofofthe
theequation
equationdfdf()
Thus,
(5) isis the
the graph
graph of
of
0 d
dfdf().
( xc ).
(g
,
()
281
251
Propostion (20). Suppose
that
f (x) is an invertible function. If S ⊆ R2
Propostion (19). Suppose
function. If S ç R
0 1that f(x) is an invertible
−1
is the graph of f (x), then
(S) is the graph of f (x).
is the graph of f(x), then 1 0 (5) is the graph of f’(x).
()
Proof:
Proof:
S
—--
()(s)
‘\1o
The equation f (y) = x is equivalent to y = f −1 (x), so
The equation f(y) = x is equivalent to y = f’(x), so
graph of f −1 (x).
graph of f’(x).
**
**
**
**
**
**
**
282
252
**
**
**
**
0 1
(S) is the
1 0 (5) is the
()
**
**
Seven most basic graph transformations
The chart below breaks up Propositions, 18, 19, and 20, into the seven most
basic graph transformations. In the chart, S ⊆ R2 is the graph of a function
f (x). The numbers c and d that appear in the third and fourth row are not
equal to 0.
function
function’s graph
how to derive the function’s graph
from the graph of f (x)
f (x − a)
A(a,0) (S)
add a to the x-coordinate
f (x) + b
A(0,b) (S)
add b to the y-coordinate
c 0
(S)
0 1
scale the x-coordinate by c
df (x)
1 0
(S)
0 d
scale the y-coordinate by d
f (−x)
−1 0
(S)
0 1
flip over the y-axis
−f (x)
1 0
(S)
0 −1
flip over the x-axis
f −1 (x)
0 1
(S)
1 0
flip over the y = x line
f ( xc )
283
Examples.
Examples.
Examples.
2
We know
the graph
is a parabola.
The row
first of
row
row
of
theof the
parabola.
The first
first
graph
of x
is xaa parabola.
know
2 ofis
x
that that
the graph
We• know
the
The
of
2
that
the
•• We
2)2
on previous
the previous
tells
us that
wederive
can derive
the graph
(x − 2)2
(x of 2)2
the graph
graph
of (x
that
we can
can
tells us
us that
the
page page
chartchart
on the
of
we
derive the
tells
previous page
chart
on
2
by adding
2 toxcoordinate
the x-coordinate
of every
the graph
of x
.of x .
2
x
pointpoint
in the
theingraph
graph
of every
every
to the
the
xcoordinate
adding
by adding
of
.
2
point
in
of
22 to
by
—
—
—
xa
xa
I
A
o)
o)
2
2)2+is2)the
• The
is graph
the graph
x2 shifted
left2. byThat’s
2. That’s
shifted
left by
by
of x
2 ofshifted
x
of (x
(xof+
The
graphgraph
2)2
+ (x
left
2. That’s
the
graph
of
2
of
is
•• The
graph
2
2
2
(_2))2
2)2+=2)(x = (x
(_2))2,
because
− (−2))
and
to graph
(x − (−2))
we need
add −2
addto—2
—2
we need
need
to add
graph
because
(x (_2))2
and, to
to
(x +
2)2
(_2))2,
+ (x
we
to
graph
because
(x
and
(x
= (x
=
2
(or subtract
2) from
the x-coordinates
the points
the graph
graph
of x
.of x .
2
x
theofpoints
points
in the
theingraph
x-coordinates
of the
2) from
from
the x-coordinates
(or subtract
subtract
of
.
2
in
of
2)
the
(or
x2\
x2\
—
—
—
—
—
—
I
aa
(x÷2’)
(x÷2’)
o’)
AA a00a o’)
2
2
The graph
the graph
x with
1 added
the y-coordinate
graph
of x
added
to the
thetoy-coordinate
y-coordinate
The• graph
of x
2 of
x
with
2 of
x
is+1
theisgraph
graph
+ 11xis
of
to
•• The
of
2
with
11 added
2
the
+
2
of every
is, graph
the graph
1 isgraph
the graph
x2 shift
graph
of x
shift
of x
is+the
the
is, the
the
graph
2 of
x
up by
byup11 by 1
of every
every
point.point.
ThatThat
2 of
x
+ x11 is
of
shift
of
is,
2
up
of
point.
That
2
+
unit unit.
unit
X2\
X2\
I
AA
254
254
254
Jxl÷1
Jxl÷1
284
• The graph of ()2 is the graph of x
2 scaled by 3 in the x-coordinate.
()2
x 2 is the graph of x
The graph
ofgraph
2=
(4x)
22 scaled
scaled by
by 33 in
•• The
of
(()2
) is
the
graph
of
x
in the
the x-coordinate.
x-coordinate.
The
graph
of
is
graph
the
of
2
x
scaled
by
in
the
x-coordinate.
()2
3
• The graph2 of= ()2
graph
the
of
2
x
scaled
by
3
in
the
x-coordinate.
1
x is
2
the
graph
x
scaled
by
in the
the x-coordinate.
x-coordinate.
The
graph
of
(4x)
( 1/4 )2 is
Thegraph
graphof
of(4x)
2== ()2
(4x)
thegraph
graphofof
ofx
2scaled
scaledby
by 4inin
The
2
isisthe
2x
the x-coordinate.
k
k
,130
(*0
\0l
-I
-I
-I
,130
(*0
,130
(*0
\0l
• The graph of loge(—x) is the graph of loge(x)
\0lflipped over the y-axis.
• The
graph
of
log
the
graph
log
flipped over
over the
the y-axis.
y-axis.
e (−x) is
e (x) flipped
Thegraph
graphof
ofloge(—x)
loge(—x)
graphofof
thegraph
ofloge(x)
loge(x)
••The
isisthe
flipped over
the y-axis.
f-i
0
‘oI
f-i 00 >
f-i
‘oI
‘oI
>>
• The graph of loge(x) is the graph of loge(x) flipped over the x-axis.
• The
graph
of
log
the
graph
log
over the
the x-axis.
x-axis.
e (x) is
e (x) flipped
Thegraph
graphof
of −loge(x)
loge(x)
thegraph
graphofof
ofloge(x)
loge(x)
flippedover
over
••The
isisthe
flipped
the
x-axis.
0
—
—
—
f
i’oI
f 00
fi’oI
i’oI
I
II
255
285
255
255
• The
graph
of the
inverse
function
of 1og
which
is the
function
(x),(x),
• The
graph
of the
inverse
function
of 1og
which
is the
function
which is the function
ex , is the graph of loge (x) flipped over the y = x line.
The
graph
the
inverse
function
log
(x),
ex, ex,
is the
graph
of loge(x)
flipped
over
the the
y =of
is •the
graph
of of
flipped
over
loge(x)
x eline.
y x=line.
7?
7?
()
()
Multiple
transformations
Multiple
transformations
Multiple
transformations
As As
longlong
as there
is only
oneone
typetype
of operation
involved
“inside
thethe
function”
as there
is only
of operation
involved
“inside
function”
Aseither
long
as
there is only
one
type ofand
operation
theinvolved
function”
either
multiplication
or addition
onlyonly
oneinvolved
typetype
of “inside
operation
multiplication
or
addition
and
one
of operation
involved
—“outside
either
or
addition
—
and
only
one
type
of
operation
involved
of multiplication
“outside
the
function”
either
multiplication
or
addition
then
you
cancan
of the function”
either multiplication or addition
then you
“outside
the function”
— rules
either
multiplication
addition
— basic
then
yougraph
can
easily
combine
different
rules
from
the the
chart
of the
seven
most
graph
easily
combine
different
from
chart
oforthe
seven
most
basic
easily
combine For
different
rulesyou
from
chart
of
the the
seven
most
basic
graph
transformations.
example,
canthe
easily
combine
rules
for for
functions
transformations.
For
example,
you
can
easily
combine
the
rules
functions
For
example,
youf(x
can
the
rules
for combining
functions
oftransformations.
the
form
—2f(x
3f(2x),
5),
f(x
4) easily
f(—7x)
but
combining
+
+4)3,+or
+ 2;+
of the
form
—2f(x
3f(2x),
or f(—7x)
2; but
3,combine
+ 5),
ofrules
the
−2f (x
3fform
(2x),
f (x+−3 4)
+or3,f(4x
or f5)(−7x)
+ 2. more
The
next
three
rules
for form
functions
of+the
form
2f(x)
requires
care.
for
functions
of5),the
2f(x)
5) requires
more
care.
+or3 f(4x
examples illustrate how this can be done.
Examples.
Examples.
Examples.
• The
graph
of e
graph
of ex
down
2 units
ex moved
by by
andand
• The
graph
of −x
e2 is2 the
is the
graph
of moved
down
2 units
x
•
The
graph
of
e
−
2
is
the
graph
of
e
moved
down
by
2
units
and
flipped
overover
the the
y-axis.
flipped
y-axis.
flipped over the y-axis.
—
—
—
—
—
—
—
—
—
—
—
—
—
—
‘C.
e e ‘C.
x
x
e-2
e-2
256 256
286
e-2
e-2
The graph Qf —(x + 2)2 2is the graph of 2 shifted
left by two units
22
•
The
graph
of
−(x
+
2)
is
the
graph
of
x
shifted
leftleftbybytwo
2)2
The
graph
Qf
—(x
is
the
graph
of
shifted
twounits
units
+
and flipped over the’-axis.
andand
flipped
over
thethe’-axis.
x-axis.
flipped
over
1’ 1’
jj
()
a
a
The graph of 1og
−x is the graph of 1og(x) flipped over the
•
The
graph
of
−
log
is istheby
of of
log1og(x)
e
e (x) flipped
The
of 1og
thegraph
flippedover
overthe
the
x-axis, flipped
overgraph
the y-axis,
and2 scaled
2 graph
in the
x-coordinate.
x-axis,
flipped
over
the
y-axis,
and
scaled
by
2
in
the
x-coordinate.
x-axis, flipped over the y-axis, and scaled by 2 in the x-coordinate.
—
—
()
Graph transformations become a little more complicated if we both mul
Graph
transformations
become
asuch
little
complicated
if if
weweboth
mulGraph
transformations
become
a little
more
complicated
mul
tiply
and
“inside the function”
add
with
f(2x
asmore
weboth
have
+ 5), where
tiply
and
add
“inside
the
function”
such
as
with
f
(2x
+
5),
where
we
have
tiply and
withcomplicated
f(2x + 5), where
we have
multiplied
2 and“inside
by add
addedthe
It’s also asuch
littleasmore
5. function”
if we both
multiplied
by
2
and
added
5.
It’s
also
a
little
more
complicated
if
we
multiplied
by 2“outside
and added
more3f(x)
complicated
5. It’s alsosuch
a little
if weboth
multiply
and add
the function”
as with
weboth
+ 6, where
multiply
and
add
“outside
the
function”
such
as
with
3f
(x)
+
6,
where
andbyadd
“outside
as with 3f(x)
+ 6, where
havemultiply
multiplied
addedthe
The extra such
complication
3 and
6. function”
with these
sortswewe
multiplied
by by
and
added
6. 6.The
extra
complication
with these
sorts
have
multiplied
added
The
extra
complication
3 and
ofhave
functions
is that
it3 matters
which
graph
transformation
you with
applythese
first.sorts
of of
functions
is that
it matters
which
graph
transformation
functions
is that
itx-coordinate
matters
which
graph
transformationyou
youapply
applyfirst.
first.
For
example,
scaling
the
by
1and then shifting the graph left
For
example,
scaling
the
x-coordinate
by
and
then
shifting
the
graph
left
example,
scaling the
x-coordinate
and
shifting
graph
2
by 5For
results
in a different
picture
then if weby
shift
left then
thenthe
by 5 and
scale
the left
by by
5 results
in in
a different
picture
then
if we
shift
leftleft
byby
5 and
then
scale
thethe
5 results
a different
picture
then
if we
shift
then
5 and
scale
x-coordinate
by
1
x-coordinate
by
2.
Forx-coordinate
all of thebyhomework
exercises in this text, you’ll only be given graph
For
all
of
the
homework
exercises
in in
this
text,
you’ll
only
bebe
given
graph
For
all
of
the
homework
exercises
text,
you’ll
only
given
graph
transformation problems where
the order
ofthis
transformations
that
you
need
transformation
problems
where
the
order
of
transformations
that
you
need
transformation
problems
where
the order of transformations that you need
to apply
does not affect
the final
result.
to to
apply
does
notnot
affect
thethe
final
result.
257
apply
does
affect
final
result.
.
.
287 257
/
Exercises
/
Use the chart on page 283 to help you match the numbered functions with
their lettered graphs.
1.) ex
4.) ex + 2
7.) e x
A.)
2.) ex+2
5.) ex 2
8.) ex
/
D.)
3.) ex 2
6.) 2ex
9.) loge (x)
B.)
C.)
E.)
F.)
H.)
I.)
\i
(i
(i
(i
\i
(i
288
\i
(i
/
\i
\i
\i
\i
G.)
/
I
10.) x2
11.) (x + 1)2
13.) x2 + 1
14.) x2 − 1
/
7
12.) (x − 1)2
15.) (−x)2
16.) −x2
I/
A.)
/I
J
D.)
I
J
7
7
B.)
E.)
/
F.)
I
\J
\J
—1—1
289
I
I
\J
—1
I
I
II
C.)
7/
I
\J
I
—1
I
17.) x3
18.) −x3
20.) (x − 2)3
21.) (x + 2)3
23.)
√
3
7
19.) x3 − 2
22.) (−x)3
I
24.) x3 + 2
x
A.)
B.)
II
I
D.)
C.)
7
E.)
F.)
I
G.)
I
290
7
II
77
25.)
1
x
28.)
1
x
31.)
1
x−1
−1
26.)
1
x
29.)
1
x+1
27.) − x1
+1
30.)
32.) The inverse of
1
x
A.)
B.)
C.)
D.)
E.)
F.)
291
1
−x
N
33.) loge (x)
36.) loge (x − 1)
a
38.) loge
x
2
a
37.) loge (x − 1) + 2
39.) loge −
A.)
D.)
−1
C-a,
F.)
N
G.)
x
2
a
E.)
C-a,
N
40.) loge −
C.)
C-a,
C-a,
N
x
2
B.)
N
a
35.) − loge (x + 1)
34.) loge (x + 1)
C-a,
H.)
a
a
a,
N
292
a
C-a,
N
C-a,
In mathematics, angles are measured by lengths of segments of the unit
circle, as we’ve done in this text. In most contexts though, angles are measured in degrees. In mathematics, a full revolution is an angle that measures
2π, and using degrees, the same angle would be measured as 360◦ . That is,
360◦
2π
when measuring angles, 2π = 360◦ , or equivalently, 360
◦ = 1, or 2π = 1.
To convert degrees into the mathematical measurement of angles, multiply
2π
◦
by 360
◦ . For example, an angle of 180 is the same as an angle of
2π 180◦ 1
◦
180
= 2π
= 2π
=π
360◦
360◦
2
Conversely, to change a mathematical measurement of an angle into de◦
π
grees, multiply by 360
2π . For example, a right angle of 2 can be written in
degrees as an angle of
π 360◦ π2 ◦ 1 ◦
=
360 =
360 = 90◦
2 2π
2π
4
Convert the following angles in degrees into mathematical angles.
41.) 45◦
42.) 120◦
43.) 30◦
44.) 300◦
Convert the following mathematical angles into degrees.
45.)
π
6
46.)
4π
3
47.)
π
12
48.)
Find the solutions of the following equations in one variable.
49.) 5x2 − 3x = −2
50.) ex = e2−x
13
51.) loge (x − 3)
= 26
293
3π
2