Chapter 3mm34 CAS

8/12/2019 Chapter 3mm34 CAS

1/17

1

Chapter 3 Graphing with Transformations

Function

Form

Conditions Domain Range Type of

function

Asymptotes Examples Graph

q

p

xxf

xxf n

)(

)(

n, p & q are +ve

odd integers

R\{0} R\{0} 1-1

odd

x=0

y=0533

1,

1,

1

xxx

q

p

xxf

xxf n

)(

)(

n & p +ve even

integers

q +ve odd integer

R\{0} R+ many 1

even

x=0

y=05

4

1,

1,

162

xxx

q

p

xxf )(

p & q +ve odd

integers

R R 1 1

odd

none3

5

3

1

,,3 xxx

y

x

(1, 1)

(-1, -1)

y=0

x=0

y

x

(-1, 1) (1, 1)

y=0

x=0

(0, 0) (0, 0)

y

x

y

x

(1, 1)

(-1, -1)

(1, 1)

(-1, -1)

0 1


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2

q

p

xxf )(

p +ve even

integer

q +ve odd integer

R [0, ) many 1

even

none34

32

,,2 xxx

q

p

xxf )(

p +ve odd integer

q +ve even integer

[0, ) [0, ) 1 1

neither

none23

41

21

,, xxx

q

p

xxf )(

p +ve odd integer

q +ve even integer

R+ R+ 1 1

neither

x=0

y=02

3

4

3

2

1

1,

1,

1

xxx

Eg 1 pg 57

Ex 3A all

y

x

y

x(0, 0) (0, 0)

0 1

y

x

y

x(0, 0) (0, 0)

0 1

y

x

1 2 3 4 5 1 2 3 4 5

1

2

3

4

5

1

2

3

4

5

(1, 1)

y=0

x=0


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We will concentrate on the following functions in this chapter, although these

transformations are transferable to other graphs.

2)(,: xxfRRf x

xfRRf 1

)(,}0{\:

Asymptotes

x=0

y=0

xxfRRf )(,: 2

1)(,}0{\:

xxfRRf

Asymptotes

x=0

y=0

These graphs can be transformed in these ways.

Translated moved vertically or horizontally.

Dilated stretched vertically or horizontally.

Reflected in the x or y axes.

You can apply the following transformations to 2)(,: xxfRRf .

y

x

y

x

y

x

y

x

where a,b,c,d>0f(x) = a(b(xc) )2d

Reflection in

the x-axis if -

reflection in

the y-axis if -

dilation of factor

a from x or

parallel to y

dilation of factor

1/b from y or

parallel to x

horizontal

translation of

-+c units

vertical

translation of

+-d units


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4

Similarly you can apply the same transformations to the following:

xxfRRf

1)(,}0{\: you will get d

cxb

axf

)()(

xxfRRf )(,: you will get dcxbaxf )()(

2

1)(,}0{\:

xxfRRf you will get d

cxb

axf

2))(()(

Mapping notation

Translations

),(),( yhxyx moves the ),(),( kyxyx moves the

graph h units horizontally. graph k units vertically.

y

x

(x, y)

(x+h, y)

y

x

(x, y)

(x, y+k)

Dilations

),(),( kyxyx dilates the graph ),(),( yhxyx dilates the graph

by a factor of k from the x-axis. by a factor of h from the y-axis.y

x

(x, y)

(x, ky)

y

x

(x, y)

(hx, y)

Reflections

),(),( yxyx is a reflection in ),(),( yxyx is a reflection in

the y-axis. the x-axis.

y

x

(x, y)(-x, y)

y

x

(x, y)

(x, -y)


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In general:

Translations),(),( kyhxyx moves the graph h units horizontally and k units vertically.

),(),( kyhxyx is the graph )( hxfky

Dilations),(),( kyxyx dilates the graph by a factor of k from the x-axis (or a factor of k parallel to

the y-axis or in the y).

),(),( kyxyx is the graph )()( xkfyxfk

y

),(),( yhxyx dilates the graph by a factor of h from the y-axis (or a factor of h parallel to

the x-axis or in the x)

),(),( yhxyx is the graph

h

xfy

Reflections),(),( yxyx is a reflection in the y-axis.

),(),( yxyx is the graph )( xfy

),(),( yxyx is a reflection in the x-axis.

),(),( yxyx is the graph )(xfy

Eg2. State a transformation which maps the graphs of )(xfy to )(1 xfy for each of the

following. Sketch the graphs of ).(1 xf

a) 2

12 2

1)(,

1)(

xxf

xxf b) 3)(,)( 1 xxfxxf

Eg3. If2

1)(

xxf , sketch

4

xf


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3

2

)(

1

)( xxfxxf

Eg4. State a transformation which maps the graphs of )(xfy to )(1 xfy for each of the

following.

a) xxfxxf )(,)( 1 b) 2122

)(,1

)(x

xfx

xf

c) 41

)(,1

)(212

x

xfx

xf d)212 2

1)(,

1)(

xxf

xxf

Eg5. State transformations for

Eg6. Find the rule of the function when the function with equation xy is transformed by:

a) a reflection in the y-axis b) a dilation of factor 2 from the x-axis

c) a dilation of factor2

1from the y-axis

Eg7. Find the image of the curve with equation ),(xfy wherex

xf 1)( under a translation

3 units in the positive direction of the x -axis and 2 units in the negative direction of the y -

axis.


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Ex 3B: 1-3 (1/2), 4, 5 (1/2), 6; Ex 3C: 1-4 ; Ex 3D: 1-2 (1/3), 3 (1/2), 4

Combinations of transformationsRecommended order of transformations when sketching or describing:

1.Dilations 2.Reflections 3.Translations

NB: always ensure that the coefficient of x is taken out as a factor first beforedescribing the sequence of transformations or sketching.

Eg8. Identify the sequence of transformations that maps the graph of the functionx

xf 1)(

onto the graph of the function123

1)(

xxf .

Eg9. Identify the sequence of transformations that maps the graph of the function

2

1)(

xxf onto the graph of the function

1

1

2)(

2

xxf , and use this to sketch the graph

of

11

2)(

2

xxf , stating the equations of asymptotes and the coordinates of axes

intercepts.


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When given a sequence of transformations and the resulting image equation is required,

the following rules apply:

1. If there is a translation parallel to the x-axis after any transformations, it must be putinside a bracket with any coefficient of x placed in front of the bracket. Any

transformation after this translation is placed directly.

Eg10. Consider the following sequences of transformations for the

functionx

y 1 :

Given,

D= dilation of factor2

1parallel to the

x-axis

R= reflection in the y-axis

T= translation of 3 units in the positive

direction of the x-axis

In general given,

D= dilation of factor b parallel to the

x-axis

R= reflection in the y-axis

T= translation of c units in the positive

direction of the x-axis

DRT

)3(2

1

2

1

2

11

xxxx

TRD

RDT

)3(2

1

2

111

xxxx

TDR

DTR

)3(2

1

)3(2

1

2

11

xxxx

RTD

RTD

)32(

1

)3(

111

xxxx

DTR

TDR

321

321

311

xxxx

RDT

TRD

32

1

3

1

3

11

xxxx

TRD

DRT

)(1

1

1

1

1

11

cxb

xb

xb

x

TRD

RDT

)(1

1

1

111

cxb

xb

xx

TDR

DTR

)(1

1

)(1

1

1

11

cxb

cxb

xb

x

RTD

RTD

)1(

1

)(

111

cxb

cxxx

DTR

TDR

cxb

cxb

cxx

RDT

1

1

1

111

TRD

cxb

cxcxx

TRD

1

1111

2. A dilation parallel to the y-axis (from the x-axis), will result in the wholeequation beingmultiplied by the dilation factor when it occurs. A reflection in the x-axis will also result

in the wholeequation being multiplied by -1 when it occurs. Any vertical translation is

placed directly when it occurs.


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Eg11.

a) Find the rule of the image when the graph of the function xy

undergoes a reflection in the x-axis, followed by a translation of 3

units in the positive direction of the y-axis, followed by a dilation of

factor 2 from the x-axis.

DTR

x

b) Find the rule of the image when the graph of the function xy

undergoes a reflection in the x-axis, followed by a dilation of

factor 2 from the x-axis, followed by a translation of 3 units in the

positive direction of the y-axis, .

TDR

x

c) Find the rule of the image when the graph of the function xy

undergoes a translation of 3 units in the positive direction of the

y-axis, followed by a reflection in the x-axis, followed by a dilation of

factor 2 from the x-axis.

DRT

x

Ex 3E: 1acd, 3, 4

Eg12. Show that1

13123

xxx and hence sketch

123)(,}1{\:

xxxfRRf

Ex 3F: 1-4 (1/2), 5b, 6 (1/2)


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TRANSFORMATIONS AND MATRICES

Under any transformation, we say a point is mappedonto an imagepoint .i.e = image of point

Eg13.Find the image of the point (1, 5) under the following transformations:

a) Translation of 2 units right

(image of point

(image of point (1, 5) ( , )= image point of (1, 5)

b) Dilation of factor 2 parallel to the y-axis.

(image of point (image of point

(1, 5) ( , )= image point of (1, 5)

We can use matrices to find the image of a point under different transformations.

1. For reflections and dilationsMapping-(Linear

Transformation)

Rule Transformation Matrix (T)

Reflection in the x-axis x=x (x=1x+0y)

y=-y (y=0x-1y)

Reflection in the y-axis x=-x (x=-1x+0y)

y=y (y=0x+1y) Dilation by a factor k from the

y-axis(parallel to x-axis)

x=kx (x=kx+0y)

y=y (y=0x+1y)

Dilation by a factor k from the

x-axis(parallel to y-axis)

x=x (x=1x+0y)

y=ky (y=0x+ky)

Reflection in the line y=x (inverse) x=y (x=0x+1y)

y=x (y=1x+0y)

The image point (x, y) is found by finding: [

]

2. For translationsMapping- (Non-linear

transformation)

Rule Image point found by:

Translation by aunits in the x-

axis and bunits in the y-axis

x= x + a

y=y + b[]

These matrices will allow us to find the image points (x, y) under different transformations.


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1

0

0

-1

1

0

0

2

1

0

0

21

0

0

-1

1

4

Eg14. Find the image of the point (1, 5) under a:

a) reflection in the y-axis

b) a dilation of factor 3 from the x-axis

Composition of mappings

More than one transformation can occur. If a point undergoes a linear transformation given

by a matrixA, followed by another linear transformationgiven by matrix B, then BA is theresulting transformation matrix.

Eg15.

a) Find the image of the point (3, 2) under a reflection in the x-axis followed by a dilation offactor 2 parallel to the y-axis(from the x-axis).

Let A = (reflection in x-axis) B= (dilation by factor 2 from x-axis)

Then, BA= = is the transformation matrix (T)

The image point is:

b) Find the image of the point (3, 2) under a dilation of factor 2 from the x-axis followed bya translation given by the matrix B = .


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a

c

b

d

a

c

b

d1

-2

-4

5

a

c

b

d3

4

18

5

a 2b

c 2d

-4

5

3a+ 4b

3c+ 4d

18

5

a

b

c

d

-4

5

18

5

a

b

c

d

1

0

3

0

-2

0

4

0

0

1

0

3

0

-2

0

4

-1

-4

5

18

5

a

b

c

d

2

3

3

-1

1

0

3

0

-2

0

4

0

0

1

0

3

0

-2

0

4

Eg16.Consider the linear transformation such that (1, -2) (-4, 5) and (3, 4) (18, 5).

Find the image of the point (2, 7) under the same transformation.

Let T = be the linear transformation matrix. Then,

= and =

= =

We need to solve the following system of equations:

=

=

=

a= b= c= d=

Transformation Matrix, T=


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1

0

0

2

1

0

0

2x

y

x'

y'

x

y 1

0

0

2

-1

x'

y'

x

y

x'

12y'

0

2

-3

0

1

2

x

y

0

2

-3

0

-1

x'

y'

1

2

Transformation of Graphs of Functions with Matrices

Matrices can be used to find the equationof the image of a graph under a given

transformation.

Eg17. Find the equation of the image of the graph of the following quadratic equations under

the transformation defined by the matrix .

i) ii) Solution:

( ) = image of point

=

=

=

i) ii)

Eg18. A transformation is described by the equation T(X+B) = X, where T= and B=

Find the image of the straight line with equation under this transformation.Solution:

T(X+B) = X X + B= T-1X

X= T-1

X- B

= -

=

x= y=

becomes


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Ex 3G:1, 2, 3, 4, 5,6,7,10,11,12

Determining the rule for a function of a graph

This requires the same number of points as there are unknowns in the function.

Eg19. Find the equation of the graph with equation caxy 2 .

Eg20. Find the equation of the graph with equation cbxaxy 2

Eg21. The points (2,1) and (10,6) lie on the curve bxay 1 .Find a & b.

Ex 3H: 1-6

Addition of Ordinates

Remember that )()( gdomfdomgfdomandxgxfxgf .

Addition of ordinates is most useful when )(xgf is not a function whose graph you would

familiar with.

The process is to add corresponding y values for both )(and)( xgxf for given x values, to

produce the graph of )(xgf .

y

x

3

(-3,1)

y

x-1

2

1


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Eg22. Sketch f+g

Eg23. Sketch f+g

y

x

f

g

y

x

f

g


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Eg24. Sketch f + g

Eg25. Sketch g - f

Ex 3I: 1, 4, 5

Graphing Inverse Functions

Remember that )(xf has to be a 1-1 function for )(1 xf to be a function.

Eg26. Find )(1 xf if 423)( xxf and sketch both on the same set of axis.

y

x

fg

y

x

f

g


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Eg27. Find the inverse relation of 214)( xxf and sketch both on the same set of axis.

Eg28. We can restrict the domain of 214)( xxf so that its inverse will be a function. Find

the maximum possible domain for this to happen.

Ex 3J: 1-4 (1/2)

Review: (1/2)

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Chapter 3mm34 CAS