Chapter l0 : Transformations olGraphs ln this chapter, students will: (a) use a graphic calculator to investigale the relationships betvveen the graphs of and i(r + a), where 17 is a constant, and J .= f(r), .f -- f(-r) + express the transforrnations involved in terms of transrations;
i
r
(b) use a graphic calculator to investigate the relationships betlveen the graphs of f(ar), where d is a constant, and .y: 1 a f(x.) and express the transformalions involved in lerms of sfreaches;
(r). :
Notei
s
y:
.l
t:
Explore cases when a js posilive and when a is negalive. When a 1 . the transformations are reflections in the axes.
= :
/Lf(r + ll), 1., two transformations of graphs;
(c) interpret e.g.
),
Note: For example,
tf(a-{) and
y=
t:
f(Lr + a) ;5 a composition of-l' = f(,i. r d) followed by a cornpocilion of
A.
J
lih)'ollowed by-A r
(tr+a)
y:
iI/cI); or
t(,
d.1
lntroduction
Given a graph o1
recap on Function 1 (-r) = -r+ 4and r e llt ,
wltat is thl: valuc of f(3)?
What is lhe expression lbr l(2:c)? Ans:2Y+,1 ln
r errcn llnclion l{ r}. I (ay t b)
me.rns r eplg9t11g ' wirh
What is lhe differerce between f (-t+ b) and f(x)+b
f(r+b)
refers
to
replacing
r with (r+b)'
ar-
i'
?
whereas f(,r)+ll refers to
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1
each as a composition of
B.
Single Transformafion oI Graphs
Consider y = f(-r) =;er. Using the GC, sketch the lbllowing graphs:
(i) r=f(x) (ii) y:f(:r) + 2 andy=f(x) 2 (iii) y=f(:r + 2) and.l'=f(r 2) (i") y:f(-;r) andy= f(;)
Compare each oflhe above, what do you notice?
we will study
each of the following transfomations in more detail:
a)
b) c)
L
Translation
Translating a cu e in the direction of an axis is to move the curve io th€ direction of the axis without chansine its shape or size.
1.1
TranslationinJr-direction
A
-7*,
Log onto hltp: \\
\
w.anal) /cm,rlh.convven icalShifl Vcd icalshi tl.hl ml
The function to be analyzed is ofthe form g(:r) = f(r) + d where
f0,):
L{
f(;) is any ofthe functions:
2l ( a "W" shapod graph)
fft)=v'. How does the addition ofa constant to
a
function (ie f(;r) + d) affect the gaph ofthis flmction?
Ans:' Given a gaph
oft : f(r) and a e 1R ,
G)
the graph of is a translation of the graph positiv€ /-direchon. (add d uruts to all the .),-coordinates).
of y
:
f(r) by a units in the
(b)
is a tanslation of the graph of .y negative/-direchon. (subtract a units from all they-coordinates).
:
f(r) by a units in
the glaph
of
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2
the
Example
l:
The diagram shows the gaph
ofy
:
f(.)r). Sketch, on the diagram, the graph
of/:
f(jt)
5
t
Solution:
1.2
Translation in.x direction
B Fo*
L
hllp: www .rnal1 uem,rlh.com Horr,,onlal Shi H.html The function to be anallzed is ofthe fom g(:r): f(r+ d) where f(*) is any ofthe functions: og onto:
f(;):l|vl
2l ( a
"W"
shaped graph)
(.r)=r2 or f(r)::r3. How does setting tho constant to negative and positive values affect the graph? Ans: (D
(iD
- f(r) and a e lR, (a) the gaph ofciven
a graph
ofl,
negative r. - direction. (subtract
O)
the graph
of-
is .z
a translation
of the curve
oft :
(.r) by d units in the
units from all the i-coordinates). is a translation ol the curve ol'
positive .;r-direction. (add a units to all the r-coordinates).
crl:)
3
y
t1xlby a unitsinlhe
Examplc 2: The diagmDl shows the graph
ofy
: f(r). Sketoh, on the same diagjam, the graph of / = f(jr
5).
Solution:
2.
Stretch
2l
Stretch along y-axis
a{
*
+-,"' Log onto: http://wwu. analyzemath.contverticalscaling/verticalscaling.html
The function to be anallzed is ofthe fbtm g(x) = af(x) where (-lr) is any ofthe functions:
f(x)-llrl
(;)=-{
2l ( a "W" shaped graph)
or
f(r)=r'. How does the multiplication ofa function by Ans:
Given a graph
oll
=
l(.rJ
2.2
A
.F*
ofy
:
a
positive constant affect the graph of this imction?
f(x) and a e lR, the graph ofy
is a strctch of the graph
wilh scale factor a along they-axis. {mulliply d 1o ally-coordinales}.
Stretchalong.r-axis httpv'rv**.-ulyzemath.com,/honzonlalscaling,4lori/onlalscaling.html
The functions to be explored are ofthe form f(a*;) where
f(.r):lkl f(;)=l f(")
l.
2l ( a "W" shaped graph)
or
ctO
4
{x)
is any ofthe functions:
How does the muitiplication ofthe independent variable r ofa function by a positive constant affect the gaph ofthe functjon?
a
Ans:
The graph
ofj
the -y-axis
(multiply
Example 3: The graph ofy =
is a st.etch of the graph of
1 a
(r)
:
(ii)
/ = f(1")
y
:
fb) with
scale
factor
l
along
is as shown.
./: f(r) of
f(2tr) E(2,0)
D(l,
Solution: (1)
:
to all .r-coordinates).
Skeich, on sepa.ate diagrams, the graphs
(i)r
y
f(2x)
,,,,
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5
, = ,[],.J
1)
3.
Reflection
3.1
Reflection in y-axis
EI
.Fn*' http:
www. slu.ed u classes/mayrk-/A pplels SliderCrdph.hlml
The graph below shows the function f(x) = ;gr 1 * , " Draw the graph ofeach on the same diagran provided. Observe how the graph changes for
f(-;).
Observation:
3.2
a
f(
r) = (-r)r +(-r)+2is
of
f1r.1 =
1' 1;r 12
.
Reflection in i:-axis
.'F'*, http:/ w\l'w.slu.edu/classesha),rnkApplets Sl iderGraph.html The graph below shotr's the function f(x)= v3 1t*2. Observe how the graph changes for
Observation: f(r)
=
-f(:r).
Draw the graph ofeach on the same diagram provided.
of f(x) = ;sr -l- ir 12
(r3 +.r+ 2) is
.
Given a graph ofy = f(r),
(a)
The graph of
(b)
The graph
.1
ofl
is a reflection ofthe graph
of/ : f(r) in the r-axis-
is a reflection.of the graph of7 = f(:r) in the y-axis.
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6
Example 4: The graph of y =
f(;)is
given as shown. Sketch, on a separate diagram, the graph
ol
(i) 1 - f(-;r) (ii)
Y=
f(:r)
Solution:
c.
Transformation of Asymptotes Note that all aslanptotes will need to be tansfomed.
ExamDle 5: The graph ofy: f(x) is as shown. Sketch sepamtely, the graphs of
(i)
r=
fOr)
(ii)
y=
2f(x)
c10-7
Sp!.c!4!,
(ii)y: -2f(x)
(i) y = f(;r)
D.
Finding the original function f(r) A function f(r) undergoes hansformation A followed by B where A and B are given transformations. The resulting equation g(x) is given, find f(:r) .
Example 6: The graph
ofy = f(x)
,4: B. C:
undergoes in succession, the following three hansfomations:
A translation of I unit in the negative r-direction. A reflectron aboul ther-axrs. A stretch parallel to the.n-a,\is (with y-axis invariant) with a scale factor of2.
Theequationoftheresultrngcunc Express (.jr) in terms
of:r.
isl
gtr)
where
' g(r) -2r-r ,ff
(Modi/ied from RJC Prelim 2007)
Solution: To express f(r) in terms of*, we
will 'work backwards' liom g(r) but in the reverse transformation:
C' B'
Step 1 : When g(r) go through C':
.
g(x) =
t'-2^-3 " -_-:- --_+. clo-8
Step 2: When
g'(r)
gr) through
*1"=1''l\+)4' r
Stcp 3: When
g"(i)
2(2
"
through,{':
+4t I
r"{,1-4t
Ans: t(r.) =
go
ll':
2r.+2
'
,
- r)
Allcmatjvcly, to avoid naking carcless mistakcs, you could do the follor.ving:
C'.
A strelch parallel to thc.:r-axis (with y-axis invariant) with Replaccr in g(;) with
B': '.
factot of
'l
)
A rellection about thcJl-axjs (no change in the reverse) Replace.r in g(2r)
,,1
2'lc
a scale
with
-r
t
A translation of I unit in thc positivc Replacer in g( 2r) with Thus,
(r) :
r
dircction.
(r 1))
g(-2r+2), rvhere
a:
-2,
b=2
EIAl]IplqL (Do it yourselo A graph with equation y=f(,r) undergoes transfom.rtion where ,4 and B ale described as lbilows:
A: B:
a translation
of 2
followed
units in the negative direction oI
a scaling parallel to
the n-axis by
f(x)=8r+s+(2r L. I l)" worklng' clcarl) Ans: l ( \) -'1r (-r ' lf
The resulting equation
I
is
a
lactor
by
transformation
t-axis
]
Find the equarion
.1,
= f(;), showing
@dapted ftom IICI Pretim 2007)
D.
Sketch the graph of af(r+cr)+d 'fo sk?tch .t(b+.x)+d when the graphbf y=11t.1 isgi\,en, there are lwomethodsto approach this question:
clo
your
9
Example ll:
The diagram shows the gaph of l, = f().). On separate diagrams, sketch the graphs of : f (2-r 1) showing clearly in each case the axial intercepts, the as),rnptotes and fhe coordinates of _y the point cor€sponding to ,4 and B. (Adapted from C/C P,"omo 2007)
Solution: Step 1: Identify the points and as)4nptotes: Method
I
v=
y=2,( 1,0),(3,0),A(0, 3),B(1, 4)
f{2r- 1l=f.2lx-:D Step 2:
Steps 1:
Step 3:
Identify all points and as)rmPtotes
As)4nptote
Y=2
(
r, 0)
(3,0) A(0, -3)
80,
4)
ct0
t0
Method 2: l, = f(2jr -1) Steps 1:
Step 2:
Step 3:
Identify all points
Aslrnptote
!=2
(
1,
0)
(3, 0)
A(0,
3)
B0, -4) Final Answer:
B' (1, -4)
Example 9: (Do it yoursel0 The diagram shows the graph of y : f (.r). The curve passes through the origin O, the point .4(2, 0) and the point .R(-4, 0).
on
separate clearly labelled diagrams, the graphs of l, : f (3 - Lx) showing, in each case, the corresponding coordinates of O, A a']'d B. (Adapted from
Sketch,
SRJC Promo 2007)
clo-
11
Solution: Step
l:
Identify the points and as)4nptotes: A(2,0),
Method
B(4,
O) ,
O(0,0)
l: ?
f (3
- 2x) = f( 2(-t-;)) Step 2:
Steps 1:
Step 3:
Step 4:
Identify all points
A(2,
0)
B(-4,
0)
o(0,
0)
Method 2: f (3 - 2x) = f (-)1 1'31 Steps
l:
Step 2:
St€p
3
|
Identify all points
A(2,
0)
B(-4,
0)
o(0,
0)
Final Answer:
cto
12
Step 4:
Example l0: Thc diagran shows the graph of-! = l(-r + I). 'fhe curve passes through the points,.,l (-3,0),,a (0,3), C (1. 0) and D (7, 0). Thc point, (4, -l) is a minimum point and the lines r 2 auld '}) 3 arc asynptotes lo the curv".
=
:
u
1J
(0,l)
D (7,
Sketch, on separate clearly labelled diagrarns, the graphs of
Solution: To skctch l'(-r)
to
f(r) lion1
f(-r+ l),
Step 1: ldentify the poiits and as)4nptotes: ,4( 3, 0), r(0, 3) , c(1,0), D(7, 0), E(4, 3),
A(-3,
3)
c(r,
0)
D(7,
0)
E
the transfomation requircd are in reve$e order:
r: 2nnd.y:3 Step 3:
2:
0)
B(0,
f(jr).(Adapted lrom -LlC Proru) 2007)
f(-r+ l), the trarsfo.'Dations rcquired are:
Thus, in order to obtain
Stql
.y:
O)
(4, i) x: 2 v:3 clo
13
r=fG)
r'(5,
3)
Other online refer€nces:
a) b) c) d)
http://orion.math.iastate.edu/algebra/sp/xcurrenYapplets4rorizontalstretch.html http://www.members.shaw.ca./ron.blond/QFA.CSF.APPLET/index.html
http://www.youtube.conl/watch?v:Auic4R dAjY http://www.cet.ac.illmath,/i nction/englisltline/tmnsfomations/index.htm (click on'Translations and Refl ection')
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