Graphing Techniques Summary

7/30/2019 Graphing Techniques Summary

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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:


= )(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:


= |)(| 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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