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Translations of Graphs
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Objective:
=
*  Vertical stretches and
   compressions of graphs =
=
* Vertical and horizontal translations
   of graphs
Translations of Graphs
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Translations of Graphs
        = ;            &n= bsp; Stretch and Compress
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Translations of Graphs
        = ;            &n= bsp; Stretch and Compress
   
The graph of y=3Dc*f(x) i= s a
vertical str= etch or compression &#= 13;
of the graph y =3D f(x).<= /span>
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Translations of Graphs
        = ;            &n= bsp; Stretch and Compress
   
The graph of y=3Dc*f(x) i= s a
vertical str= etch or compression
of the graph y =3D f(x).<= /span>
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y=3D2(2x-x<= span lang=3DEN-US style=3D'font-family:Tahoma;mso-ascii-font-family:Tahoma;mso-= fareast-font-family: \5B8B\4F53;mso-hansi-font-family:Tahoma;font-size:67%;color:black;position: relative;top:-.45em;mso-text-raise:30%;mso-fareast-language:ZH-CN'>2= )
Stretch (2x-x2) vertically by a factor of 2=
Ó=  F= r<= span style=3D'position:absolute;top:76.25%;left:98.5%;width:1.12%;height:2.25%'= >a<= span style=3D'position:absolute;top:78.5%;left:98.5%;width:1.12%;height:2.25%'>= n<= span style=3D'position:absolute;top:80.75%;left:98.5%;width:1.12%;height:2.25%'= >k<= span style=3D'position:absolute;top:83.0%;left:98.5%;width:.18%;height:2.25%'><= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:56%;color:red;mso-fareast-language:ZH-CN'> M= a 2<= span style=3D'position:absolute;top:92.0%;left:98.5%;width:1.12%;height:2.25%'>= 0<= span style=3D'position:absolute;top:94.25%;left:98.5%;width:1.12%;height:2.25%'= >0<= span style=3D'position:absolute;top:96.25%;left:98.5%;width:1.12%;height:2.25%'= >6<= /div>
Translations of Graphs
        = ;            &n= bsp; Stretch and Compress
   
The graph of y=3Dc*f(x) i= s a
vertical str= etch or compression
of the graph y =3D f(x).<= /span>
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y=3Dc(2x-x2= )
If c > 1, stretch the graph vertically= by a factor o= f c=
Ó=  F= r<= span style=3D'position:absolute;top:76.25%;left:98.5%;width:1.12%;height:2.25%'= >a<= span style=3D'position:absolute;top:78.5%;left:98.5%;width:1.12%;height:2.25%'>= n<= span style=3D'position:absolute;top:80.75%;left:98.5%;width:1.12%;height:2.25%'= >k<= span style=3D'position:absolute;top:83.0%;left:98.5%;width:.18%;height:2.25%'><= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:56%;color:red;mso-fareast-language:ZH-CN'> M= a 2<= span style=3D'position:absolute;top:92.0%;left:98.5%;width:1.12%;height:2.25%'>= 0<= span style=3D'position:absolute;top:94.25%;left:98.5%;width:1.12%;height:2.25%'= >0<= span style=3D'position:absolute;top:96.25%;left:98.5%;width:1.12%;height:2.25%'= >6<= /div>
Translations of Graphs
        = ;            &n= bsp; Stretch and Compress
   
The graph of y=3Dc*f(x) i= s a
vertical str= etch or compression
of the graph y =3D f(x).<= /span>
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=
y=3D0.5(2x-x2)
If 1 > c > 0, compress vertica= lly by the fac= tor of c
Translations of Graphs
y=3Dc(2x-x2= )
        = ;            &n= bsp; Stretch and Compress
   
The graph of y=3Dc*f(x) i= s a
vertical str= etch or compression
of the graph y =3D f(x).<= /span>
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Translations of Graphs
     Flip across the x-axis if c<0.
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Translations of Graphs
     Flip across the x-axis if c<0.
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y=3D – (2x-x<= span lang=3DEN-US style=3D'font-family:Tahoma;mso-ascii-font-family:Tahoma;mso-= fareast-font-family: \5B8B\4F53;mso-hansi-font-family:Tahoma;font-size:67%;color:black;position: relative;top:-.45em;mso-text-raise:30%;mso-fareast-language:ZH-CN'>2= )
     Flip across the x-axis if c<0.
Translations of Graphs
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Ó<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:56%;color:red;mso-fareast-language:ZH-CN'> F= r<= span style=3D'position:absolute;top:68.5%;left:98.5%;width:1.12%;height:2.25%'>= a<= span style=3D'position:absolute;top:70.75%;left:98.5%;width:1.12%;height:2.25%'= >n<= span style=3D'position:absolute;top:73.0%;left:98.5%;width:1.12%;height:2.25%'>= k<= span style=3D'position:absolute;top:75.25%;left:98.5%;width:.18%;height:2.25%'>=  Ma = 2<= span style=3D'position:absolute;top:84.25%;left:98.5%;width:1.12%;height:2.25%'= >0<= span style=3D'position:absolute;top:86.25%;left:98.5%;width:1.12%;height:2.25%'= >0<= span style=3D'position:absolute;top:88.5%;left:98.5%;width:1.12%;height:2.25%'>= 6<= /div>
     Flip across the x-axis if c<0.
Translations of Graphs
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        Stretch and Compress  &n= bsp;
Given t= he graph of y=3Df(x), the graph of y=3Dc*f(x) is the vertical-stretch or <= /span>compression of y=3Df(x).
Ó= =  F= r<= span style=3D'position:absolute;top:76.25%;left:98.5%;width:1.12%;height:2.25%'= >a<= span style=3D'position:absolute;top:78.5%;left:98.5%;width:1.12%;height:2.25%'>= n<= span style=3D'position:absolute;top:80.75%;left:98.5%;width:1.12%;height:2.25%'= >k<= span style=3D'position:absolute;top:83.0%;left:98.5%;width:.18%;height:2.25%'><= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:56%;color:red;mso-fareast-language:ZH-CN'> M= a 2<= span style=3D'position:absolute;top:92.0%;left:98.5%;width:1.12%;height:2.25%'>= 0<= span style=3D'position:absolute;top:94.25%;left:98.5%;width:1.12%;height:2.25%'= >0<= span style=3D'position:absolute;top:96.25%;left:98.5%;width:1.12%;height:2.25%'= >6<= /div>
Translations of Graphs
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        Stretch and Compress  &n= bsp;
Given t= he graph of y=3Df(x), the graph of y=3Dc*f(x) is the vertical-stretch or <= /span>compression of y=3Df(x).
If c>1, it is a vertical stretch.
If 0<c<1, it is a vertical compression.
Ó=  F= r<= span style=3D'position:absolute;top:76.25%;left:98.5%;width:1.12%;height:2.25%'= >a<= span style=3D'position:absolute;top:78.5%;left:98.5%;width:1.12%;height:2.25%'>= n<= span style=3D'position:absolute;top:80.75%;left:98.5%;width:1.12%;height:2.25%'= >k<= span style=3D'position:absolute;top:83.0%;left:98.5%;width:.18%;height:2.25%'><= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:56%;color:red;mso-fareast-language:ZH-CN'> M= a 2<= span style=3D'position:absolute;top:92.0%;left:98.5%;width:1.12%;height:2.25%'>= 0<= span style=3D'position:absolute;top:94.25%;left:98.5%;width:1.12%;height:2.25%'= >0<= span style=3D'position:absolute;top:96.25%;left:98.5%;width:1.12%;height:2.25%'= >6<= /div>
Translations of Graphs
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        Stretch and Compress  &n= bsp;
Given t= he graph of y=3Df(x), the graph of y=3Dc*f(x) is the vertical-stretch or <= /span>compression of y=3Df(x).
If c>1, it is a vertical stretch.
If 0<c<1, it is a vertical compression.
If c is= negative (c<0), it flips vertically
across the x-axis.=
Ó=  F= r<= span style=3D'position:absolute;top:76.25%;left:98.5%;width:1.12%;height:2.25%'= >a<= span style=3D'position:absolute;top:78.5%;left:98.5%;width:1.12%;height:2.25%'>= n<= span style=3D'position:absolute;top:80.75%;left:98.5%;width:1.12%;height:2.25%'= >k<= span style=3D'position:absolute;top:83.0%;left:98.5%;width:.18%;height:2.25%'><= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:56%;color:red;mso-fareast-language:ZH-CN'> M= a 2<= span style=3D'position:absolute;top:92.0%;left:98.5%;width:1.12%;height:2.25%'>= 0<= span style=3D'position:absolute;top:94.25%;left:98.5%;width:1.12%;height:2.25%'= >0<= span style=3D'position:absolute;top:96.25%;left:98.5%;width:1.12%;height:2.25%'= >6<= /div>
Translations of Graphs
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Translations of Graphs
Vertical Translations
The graph of y=3Df(x)+c is a vertical translation of y=3Df(x).
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y=3Dx2+5
y=3Dx2
Translations of Graphs
Vertical Translations
The graph of y=3Df(x)+c is a vertical translation of y=3Df(x).
5
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y=3Dx2+5
5
y=3Dx2
y=3Dx2–5
-5
Translations of Graphs
Vertical Translations
The graph of y=3Df(x)+c is a vertical translation of y=3Df(x).
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y=3Dx2+5
5
y=3Dx2
y=3Dx2–5
-5
Translations of Graphs
Vertical Translations
The graph of y=3Df(x)+c is a vertical translation of y=3Df(x).
If c>0, the graph moves = up.
If c<0, the graph moves = down.
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Translations of Graphs
        = ;            &n= bsp;    Vertical Translations  &= nbsp;
<= /div>
The graph of y=3Df(x)+c i= s a vertical translation of y=3Df(x).
If c>0, the graph moves = up.
If c<0, the graph moves = down.
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        = ;            Hor= izontal Translations
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        = ;            Hor= izontal Translations
<= /div> The graphs of y=3Df(x&pl= usmn;c) are the horizontal = translations of y=3Df(x) = by c units.
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        = ;            Hor= izontal Translations
<= /div> The graphs of y=3Df(x&pl= usmn;c) are the horizontal = translations of y=3Df(x) = by c units.
For y=3Df(x + c), moves y=3Df(x) to the left. (c>0)
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        = ;            Hor= izontal Translations
<= /div> The graphs of y=3Df(x&pl= usmn;c) are the horizontal = translations of y=3Df(x) = by c units.
For y=3Df(x + c), moves y=3Df(x) to the left. (c>0)
For y=3Df(x – c),= moves y=3Df(x) to the right. (c>0)
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        = ;            Hor= izontal Translations
<= /div> The graphs of y=3Df(x&pl= usmn;c) are the horizontal = translations of y=3Df(x) = by c units.
For y=3Df(x + c), moves y=3Df(x) to the left. (c>0)
For y=3Df(x – c),= moves y=3Df(x) to the right. (c>0)
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        = ;            Hor= izontal Translations
<= /div> The graphs of y=3Df(x&pl= usmn;c) are the horizontal = translations of y=3Df(x) = by c units.
For y=3Df(x + c), moves y=3Df(x) to the left. (c>0)
For y=3Df(x – c),= moves y=3Df(x) to the right. (c>0)
y=3D(x – 3)2 
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        = ;            Hor= izontal Translations
<= /div> The graphs of y=3Df(x&pl= usmn;c) are the horizontal = translations of y=3Df(x) = by c units.
For y=3Df(x + c), moves y=3Df(x) to the left. (c>0)
For y=3Df(x – c),= moves y=3Df(x) to the right. (c>0)
y=3D(x + 3)2 
y=3D(x – 3)2 
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Absolute-Value Flip
Example A:
Given the g= raph of y =3D f(x),
graph y =3D -2f(x – 1) + 3
(-3, 1)
(-1, -1)
(1, 1)
(2, 1)
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Absolute-Value Flip
Example A:
Given the g= raph of y =3D f(x),
graph y =3D -2f(x – 1) + 3
(-3, 1)
(-1, -1)
(1, 1)
(2, 1)
"-2f(x ̵= 1; 1)" corresponds moving the graph 1 unit to the right a= nd stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
 
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Absolute-Value Flip
Example A:
Given the g= raph of y =3D f(x),
graph y =3D -2f(x – 1) + 3
(-3, 1)
(-1, -1)
(1, 1)
(2, 1)
"-2f(x ̵= 1; 1)" corresponds moving the graph 1 unit to the right a= nd stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3&qu= ot; corresponds to moving the graph vertically by 3. 
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Absolute-Value Flip
Example A:
Given the g= raph of y =3D f(x),
graph y =3D -2f(x – 1) + 3
(-3, 1)
(-1, -1)
(1, 1)
(2, 1)
"-2f(x ̵= 1; 1)" corresponds moving the graph 1 unit to the right a= nd stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3&qu= ot; corresponds to moving the graph vertically by 3. 
A good way to do = this is to shift the important points on the gra= ph.
------=_NextPart_01C957D9.19A60440 Content-Location: file:///C:/4D764A33/6TransformationsofGraphsofFunctions_files/slide0075.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Translations of Graphs
Absolute-Value Flip
Example A:
Given the g= raph of y =3D f(x),
graph y =3D -2f(x – 1) + 3
(-3, 1)
(-1, -1)
(1, 1)
(2, 1)
"-2f(x ̵= 1; 1)" corresponds moving the graph 1 unit to the right a= nd stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3&qu= ot; corresponds to moving the graph vertically by 3. 
A good way to do = this is to shift the important points on the gra= ph.
(-3, 1) à (-2, -2) <= /span>
------=_NextPart_01C957D9.19A60440 Content-Location: file:///C:/4D764A33/6TransformationsofGraphsofFunctions_files/slide0076.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Translations of Graphs
Absolute-Value Flip
Example A:
Given the g= raph of y =3D f(x),
graph y =3D -2f(x – 1) + 3
(-3, 1)
(-1, -1)
(1, 1)
(2, 1)
"-2f(x ̵= 1; 1)" corresponds moving the graph 1 unit to the right a= nd stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3&qu= ot; corresponds to moving the graph vertically by 3. 
A good way to do = this is to shift the important points on the gra= ph.
(-3, 1) à (-2, -2) <= /span>à (-2, 1)
------=_NextPart_01C957D9.19A60440 Content-Location: file:///C:/4D764A33/6TransformationsofGraphsofFunctions_files/slide0077.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Translations of Graphs
Absolute-Value Flip
Example A:
Given the g= raph of y =3D f(x),
graph y =3D -2f(x – 1) + 3
(-3, 1)
(-1, -1)
(1, 1)
(2, 1)
"-2f(x ̵= 1; 1)" corresponds moving the graph 1 unit to the right a= nd stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3&qu= ot; corresponds to moving the graph vertically by 3. 
A good way to do = this is to shift the important points on the gra= ph.
(-3, 1) à (-2, -2) <= /span>à (-2, 1)
(-1, -1)
------=_NextPart_01C957D9.19A60440 Content-Location: file:///C:/4D764A33/6TransformationsofGraphsofFunctions_files/slide0078.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Translations of Graphs
Absolute-Value Flip
Example A:
Given the g= raph of y =3D f(x),
graph y =3D -2f(x – 1) + 3
(-3, 1)
(-1, -1)
(1, 1)
(2, 1)
"-2f(x ̵= 1; 1)" corresponds moving the graph 1 unit to the right a= nd stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3&qu= ot; corresponds to moving the graph vertically by 3. 
A good way to do = this is to shift the important points on the gra= ph.
(-3, 1) à (-2, -2) <= /span>à (-2, 1)
(-1, -1) à (0, 2)
------=_NextPart_01C957D9.19A60440 Content-Location: file:///C:/4D764A33/6TransformationsofGraphsofFunctions_files/slide0079.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Translations of Graphs
Absolute-Value Flip
Example A:
Given the g= raph of y =3D f(x),
graph y =3D -2f(x – 1) + 3
(-3, 1)
(-1, -1)
(1, 1)
(2, 1)
"-2f(x ̵= 1; 1)" corresponds moving the graph 1 unit to the right a= nd stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3&qu= ot; corresponds to moving the graph vertically by 3. 
A good way to do = this is to shift the important points on the gra= ph.
(-3, 1) à (-2, -2) <= /span>à (-2, 1)
(-1, -1) à (0, 2) à (0, 5)
------=_NextPart_01C957D9.19A60440 Content-Location: file:///C:/4D764A33/6TransformationsofGraphsofFunctions_files/slide0080.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Translations of Graphs
Absolute-Value Flip
Example A:
Given the g= raph of y =3D f(x),
graph y =3D -2f(x – 1) + 3
(-3, 1)
(-1, -1)
(1, 1)
(2, 1)
"-2f(x ̵= 1; 1)" corresponds moving the graph 1 unit to the right a= nd stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3&qu= ot; corresponds to moving the graph vertically by 3. 
A good way to do = this is to shift the important points on the gra= ph.
(-3, 1) à (-2, -2) <= /span>à (-2, 1)
(-1, -1) à (0, 2) à (0, 5)
(1, 1) à (2, -2) à (2, 1)
------=_NextPart_01C957D9.19A60440 Content-Location: file:///C:/4D764A33/6TransformationsofGraphsofFunctions_files/slide0081.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Translations of Graphs
Absolute-Value Flip
Example A:
Given the g= raph of y =3D f(x),
graph y =3D -2f(x – 1) + 3
(-3, 1)
(-1, -1)
(1, 1)
(2, 1)
"-2f(x ̵= 1; 1)" corresponds moving the graph 1 unit to the right a= nd stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3&qu= ot; corresponds to moving the graph vertically by 3. 
A good way to do = this is to shift the important points on the gra= ph.
(-3, 1) à (-2, -2) <= /span>à (-2, 1)
(-1, -1) à (0, 2) à (0, 5)
(1, 1) à (2, -2) à (2, 1)
(2, 1) à (3, -2) à (3, 1)
------=_NextPart_01C957D9.19A60440 Content-Location: file:///C:/4D764A33/6TransformationsofGraphsofFunctions_files/slide0082.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Translations of Graphs
Absolute-Value Flip
Example A:
Given the g= raph of y =3D f(x),
graph y =3D -2f(x – 1) + 3
(-3, 1)
(-1, -1)
(1, 1)
(2, 1)
"-2f(x ̵= 1; 1)" corresponds moving the graph 1 unit to the right a= nd stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3&qu= ot; corresponds to moving the graph vertically by 3. 
A good way to do = this is to shift the important points on the gra= ph.
(-3, 1) à (-2, -2) <= /span>à (-2, 1)
(-1, -1) à (0, 2) à (0, 5)
(1, 1) à (2, -2) à (2, 1)
(2, 1) à (3, -2) à (3, 1)
(-2, 1)
(0, 5)
(2, 1)
(3, 1)
------=_NextPart_01C957D9.19A60440 Content-Location: file:///C:/4D764A33/6TransformationsofGraphsofFunctions_files/slide0083.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Translations of Graphs
Absolute-Value Flip
Example A:
Given the g= raph of y =3D f(x),
graph y =3D -2f(x – 1) + 3
(-3, 1)
(-1, -1)
(1, 1)
(2, 1)
"-2f(x ̵= 1; 1)" corresponds moving the graph 1 unit to the right a= nd stretch the graph by a factor of 2 then reflect the entire graph across the x axis.
"+3&qu= ot; corresponds to moving the graph vertically by 3. 
A good way to do = this is to shift the important points on the gra= ph.
(-3, 1) à (-2, -2) <= /span>à (-2, 1)
(-1, -1) à (0, 2) à (0, 5)
(1, 1) à (2, -2) à (2, 1)
(2, 1) à (3, -2) à (3, 1)
(-2, 1)
(0, 5)
(2, 1)
(3, 1)
------=_NextPart_01C957D9.19A60440 Content-Location: file:///C:/4D764A33/6TransformationsofGraphsofFunctions_files/slide0088.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Translations of Graphs
Absolute-Value Flip
------=_NextPart_01C957D9.19A60440 Content-Location: file:///C:/4D764A33/6TransformationsofGraphsofFunctions_files/slide0065.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Translations of Graphs
Absolute-Value Flip
y =3D f(x) =3D x
1
0
-1
-2
y
1
0
-1
-2
x
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Absolute-Value Flip
y =3D f(x) =3D x