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Graphing Techniques Summary
A. Fundamental knowledge of the following graphs is highly essential:
1. Graphs of the form dcx
bax
y +
+=
Example:1
1
+=x
xy
The asymptotes for this equation are 1=x and 1=y ; this is achieved by having
y and respectively.
The workings are as follows:
y
xx
11
+= ; as , 101 = xx is a vertical asymptote
x
x
x
x
x
xy
11
11
1
1
1
1
+=
+= ; as , 11
01
01==
+ yy is a horizontal
asymptote
Using this information, coupled with the relevant and intercepts, allows the
graph to be realised rather easily:
1
-1
0 1 x
-1
1
1
+=
xy
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2. Graphs of the formfex
cbxaxy
+
++=
2
Example:12
2
+=xy
To find the asymptotes:
y
xx
21
2 += ; as y , 101 = xx is a vertical asymptote
;1
31
1
3)1()1(
1
22
++=
++=
+=
xx
x
xxx
x
xy as x ,
101 +=++ xyxy is an oblique asymptote.
The graph is presented below:
y
1+= xy
1
-1
0 1 x
1
22
+=
x
y 1=
x
Note: The turning points ,31( + 322 + ) and ,31( )322 can be
obtained by setting 0=dx
dy. This graph has no intersection with the axes.
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3. Graphs of the form 2axy = or axy =2 (Parabolas)
2
2xay =
2
1xay =
0 x
213 0 aaa
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4. Graphs of the form 1)()(
2
2
2
2
=
+
B
by
A
ax(Ellipses)
y
B
A
( ),ba
0 x
Note that if BA = , then the ellipse becomes a circle centered at ),( ba with radius A
units. When an ellipse is presented in a quadratic form, completing the square is
required to fashion the equation into the structure above for extraction of its relevant
characteristics.
Example: 0109324549 22 =+++ yyxx
0109)8(4)6(9 22 =+++ yyxx
010964)4(481)3(9 22 =+++ yx
036)4(4)3(9 22 =++ yx
36)4(4)3(9 22 =++ yx
19
)4(
4
)3( 22=
++
yx(Divide both sides by 36)
13
)4(
2
)3(2
2
2
2
=+
+ yx
5. Graphs of the form 222 )()( rbyax =+ (Circles)
r
),( ba
0 x
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6. Graphs of the form 12
2
2
2
=b
y
a
x(Hyperbolas)
Example: 1
94
22
=yx
When ,0=x = 92y there are no -intercepts.
When ,0=y 242 == xx
Hence, the graph has 2 x -intercepts at (2,0) and (-2,0)
149
194
2222
==xyyx
When , xyxy
2
3
49
22
= are oblique asymptotes.
y
-2 2
0 x
194
22
= yx
xy2
3= xy
2
3=
B. Transformation of graphs
1. Operations on y -coordinates:
Considering the original graph )(xfy = ,
= )(xafy Scaling of graph )(xfy = parallel to the -axis by a factor ofa .
+= bxfy )( Translation of graph )(xfy = parallel to the y -axis by b units.
y bxfy += )( )0( >b
)(xafy = )1( >a
)(xfy =
x
0
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2. Operations on the x -coordinates:
Considering the original graph )(xfy = ,
= )(axfy Scaling of graph )(xfy = parallel to the -axis by a factor ofa
1.
= )( bxfy Translation of graph )(xfy = parallel to the x -axis by b units
y
)(xfy =
)(axfy = )1( b
0 x
Note: for (1) and (2), both a and b can assume the set of real values , althoughspecific instances of the various graph transformations (eg for )(xafy = ,
only 1>a was considered ) were illustrated due to space constraints. A combinationof transformations can exist as well, for example, [ ])( bxafy = implies the graph is
scaled parallel to the x -axis by a factor ofa
1, and subsequently translated horizontally
along the x -axis by b units.
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3. Miscellaneous transformations:
Considering the original graph )(xfy = ,
= |)(| xfy Reflection of all graph segments below the x -axis about the
x -axis, keeping all other segments unchanged.
= )( xfy Erasure of graph segment for 0x about the y -axis.
To obtain )(2 xfy = ,
(i) Erase all graph segments below the y -axis.
(ii) For the graph segment above the y -axis, draw a guiding line 1=y .
(iii) All points with y -coordinates =0 or 1 will remain invariant (unchanged).(iv) The new graph will exist above the original for 1y
(v) Reflect the resulting graph about the x -axis.
Example: =
xy =2
1
graph above the original graph below the original
1
0x
To obtain)(
1
xfy = ,
(i) All x intercepts will become vertical asymptotes, and vice-versa.(ii) All maximum points will become minimum points, and vice-versa.
(iii) Graph segments which were decreasing with x will now increase
with , and vice versa.
(iv) All values shall be inverted, with the exception of intercepts.
(v) Graph segments that were originally above the -axis shall remain
in the same region; this applies to graph segments below the y -axis as well.
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Example:y
0 x
)(xfy =
)(
1
xfy =
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