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Variations
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We say the variable y varies (directly)= to an expression f if = 3;
        = ;            &n= bsp;            = ;         y =3D k·f
where k is a constant.
Variations
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We say the variable y varies (directly)= to an expression f if = 3;
        = ;            &n= bsp;            = ;         y =3D k·f
where k is a constant. The formula y =3D k* f is called the general (direct) variation equation.
Variations
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Example A.  Translate the following phrases i= nto equations.
a. y varies directly to x.
We say the variable y varies (directly)= to an expression f if = 3;
        = ;            &n= bsp;            = ;         y =3D k·f
where k is a constant. The formula y =3D k* f is called the general (direct) variation equation.
Variations
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Example A.  Translate the following phrases i= nto equations.
a. y varies directly to x.
    y =3D kx for some k.<= /span>
We say the variable y varies (directly)= to an expression f if = 3;
        = ;            &n= bsp;            = ;         y =3D k·f
where k is a constant. The formula y =3D k* f is called the general (direct) variation equation.
Variations
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Example A.  Translate the following phrases i= nto equations.
a. y varies directly to x.
    y =3D kx for some k.<= /span>
We say the variable y varies (directly)= to an expression f if = 3;
        = ;            &n= bsp;            = ;         y =3D k·f
where k is a constant. The formula y =3D k* f is called the general (direct) variation equation.
Variations
b. y varies to xz=
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Example A.  Translate the following phrases i= nto equations.
a. y varies directly to x.
    y =3D kx for some k.<= /span>
We say the variable y varies (directly)= to an expression f if = 3;
        = ;            &n= bsp;            = ;         y =3D k·f
where k is a constant. The formula y =3D k* f is called the general (direct) variation equation.
Variations
b. y varies to xz
    y =3D kxz for some k.=
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Example A.  Translate the following phrases i= nto equations.
a. y varies directly to x.
    y =3D kx for some k.<= /span>
 
We say the variable y varies (directly)= to an expression f if = 3;
        = ;            &n= bsp;            = ;         y =3D k·f
where k is a constant. The formula y =3D k* f is called the general (direct) variation equation.
Variations
b. y varies to xz
    y =3D kxz for some k.=
c. y varies to x<= /span>2z2
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Example A.  Translate the following phrases i= nto equations.
a. y varies directly to x.
    y =3D kx for some k.<= /span>
 
We say the variable y varies (directly)= to an expression f if = 3;
        = ;            &n= bsp;            = ;         y =3D k·f
where k is a constant. The formula y =3D k* f is called the general (direct) variation equation.
Variations
b. y varies to xz
    y =3D kxz for some k.=
c. y varies to x<= /span>2z2 <= /span>
      y =3D kx2z2 for some k.
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Example A.  Translate the following phrases i= nto equations.
a. y varies directly to x.
    y =3D kx for some k.<= /span>
 
We say the variable y varies (directly)= to an expression f if = 3;
        = ;            &n= bsp;            = ;         y =3D k·f
where k is a constant. The formula y =3D k* f is called the general (direct) variation equation.
Variations
b. y varies to xz
    y =3D kxz for some k.=
c. y varies to x<= /span>2z2 <= /span>
      y =3D kx2z2 for some k.
d. The cost C var= ies directly with the square of the length L.
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Example A.  Translate the following phrases i= nto equations.
a. y varies directly to x.
    y =3D kx for some k.<= /span>
 
We say the variable y varies (directly)= to an expression f if = 3;
        = ;            &n= bsp;            = ;         y =3D k·f
where k is a constant. The formula y =3D k* f is called the general (direct) variation equation.
Variations
b. y varies to xz
    y =3D kxz for some k.=
c. y varies to x<= /span>2z2 <= /span>
      y =3D kx2z2 for some k.
d. The cost C var= ies directly with the square of the length L.
    The square of the len= gth L is L2.
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Example A.  Translate the following phrases i= nto equations.
a. y varies directly to x.
    y =3D kx for some k.<= /span>
 
We say the variable y varies (directly)= to an expression f if = 3;
        = ;            &n= bsp;            = ;         y =3D k·f
where k is a constant. The formula y =3D k* f is called the general (direct) variation equation.
Variations
b. y varies to xz
    y =3D kxz for some k.=
c. y varies to x<= /span>2z2 <= /span>
      y =3D kx2z2 for some k.
d. The cost C var= ies directly with the square of the length L.
    The square of the len= gth L is L2.
    Hence the general equ= ation is C =3D kL2 for some k.
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We say the variable y varies inver= sely to an expression f if = 3;
<= /div>
        = ;            &n= bsp;            = ;         y =3D
where k is a constant.
Variations
k
f
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We say the variable y varies inver= sely to an expression f if = 3;
<= /div>
        = ;            &n= bsp;            = ;         y =3D
where k is a constant. The formula y =3D      is called= the general inverse variation equation.
Variations
k
f
k
f
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Example B.<= span style=3D'mso-spacerun:yes'>  Translate the following phrases i= nto equations.
<= span style=3D'font-size:133%;color:#003300'>a.y varies inversel= y to x.
We say the variable y varies inver= sely to an expression f if = 3;
<= /div>
        = ;            &n= bsp;            = ;         y =3D
where k is a constant. The formula y =3D      is called= the general inverse variation equation.
Variations
k
f
k
f
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Example B.<= span style=3D'mso-spacerun:yes'>  Translate the following phrases i= nto equations.
<= span style=3D'font-size:133%;color:#003300'>a.y varies inversel= y to x.
k
x
We say the variable y varies inver= sely to an expression f if = 3;
<= /div>
        = ;            &n= bsp;            = ;         y =3D
where k is a constant. The formula y =3D      is called= the general inverse variation equation.
Variations
k
f
k
f
y =3D
where k is a constant
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Example B.<= span style=3D'mso-spacerun:yes'>  Translate the following phrases i= nto equations.
<= span style=3D'font-size:133%;color:#003300'>a.y varies inversel= y to x.
 
k
x
We say the variable y varies inver= sely to an expression f if = 3;
<= /div>
        = ;            &n= bsp;            = ;         y =3D
where k is a constant. The formula y =3D      is called= the general inverse variation equation.
Variations
k
f
k
f
b. y varies inversely to x2z
y =3D
where k is a constant
------=_NextPart_01C9BC25.49555230 Content-Location: file:///C:/09764A33/0Variations_files/slide0048_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAAAR AAIAgAAAAAAAAAIFjI+pqwUAOw== ------=_NextPart_01C9BC25.49555230 Content-Location: file:///C:/09764A33/0Variations_files/slide0048_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFQADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQAR AAEAgAAAAAAAAAIEjI9pBQA7 ------=_NextPart_01C9BC25.49555230 Content-Location: file:///C:/09764A33/0Variations_files/slide0048_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFQADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C9BC25.49555230 Content-Location: file:///C:/09764A33/0Variations_files/slide0049.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Variations
Example B.<= span style=3D'mso-spacerun:yes'>  Translate the following phrases i= nto equations.
<= span style=3D'font-size:133%;color:#003300'>a.y varies inversel= y to x.
 
k
x
k
x2z
We say the variable y varies inver= sely to an expression f if = 3;
<= /div>
        = ;            &n= bsp;            = ;         y =3D
where k is a constant. The formula y =3D      is called= the general inverse variation equation.
Variations
k
f
k
f
b. y varies inversely to x2z
y =3D
where k is a constant
y =3D
where k is a constant
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Example B.<= span style=3D'mso-spacerun:yes'>  Translate the following phrases i= nto equations.
<= span style=3D'font-size:133%;color:#003300'>a.y varies inversel= y to x.
 
k
x
k
x2z
We say the variable y varies inver= sely to an expression f if = 3;
<= /div>
        = ;            &n= bsp;            = ;         y =3D
where k is a constant. The formula y =3D      is called= the general inverse variation equation.
Variations
k
f
k
f
b. y varies inversely to x2z
y =3D
where k is a constant
y =3D
where k is a constant
c. The intensity I of light varies inversely to the square of distance D<= /span>
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Example B.<= span style=3D'mso-spacerun:yes'>  Translate the following phrases i= nto equations.
<= span style=3D'font-size:133%;color:#003300'>a.y varies inversel= y to x.
 
k
x
k
x2z
We say the variable y varies inver= sely to an expression f if = 3;
<= /div>
        = ;            &n= bsp;            = ;         y =3D
where k is a constant. The formula y =3D      is called= the general inverse variation equation.
Variations
k
f
k
f
b. y varies inversely to x2z
y =3D
where k is a constant
y =3D
where k is a constant
c. The intensity I of light varies inversely to the square of distance D<= /span>
The square = of distance D is D2.
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Example B.<= span style=3D'mso-spacerun:yes'>  Translate the following phrases i= nto equations.
<= span style=3D'font-size:133%;color:#003300'>a.y varies inversel= y to x.
 
k
x
k
x2z
We say the variable y varies inver= sely to an expression f if = 3;
<= /div>
        = ;            &n= bsp;            = ;         y =3D
where k is a constant. The formula y =3D      is called= the general inverse variation equation.
Variations
k
f
k
f
b. y varies inversely to x2z
y =3D
where k is a constant
y =3D
where k is a constant
c. The intensity I of light varies inversely to the square of distance D<= /span>
The square = of distance D is D2.
Hence I =3D=
k
D2
where k is a constant.
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In general, a variation problem gives the type of the variation and the v= alues of the variables.
Variations
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In general, a variation problem gives the type of the variation and the v= alues of the variables. From these, we solve for the  constant k and find the specific (exact) variation equation.
Variations
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Example C. <= /span>
a. Given that y varies directly to x and y =3D –4&n= bsp; when x =3D –6.  
    Find the constant k a= nd the specific variation equation.
In general, a variation problem gives the type of the variation and the v= alues of the variables. From these, we solve for the  constant k and find the specific (exact) variation equation.
Variations
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Example C. <= /span>
a. Given that y varies directly to x and y =3D –4&n= bsp; when x =3D –6.  
    Find the constant k a= nd the specific variation equation.
In general, a variation problem gives the type of the variation and the v= alues of the variables. From these, we solve for the  constant k and find the specific (exact) variation equation.
Variations
Since y varies directly to x, the general equation is y =3D kx.
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Example C. <= /span>
a. Given that y varies directly to x and y =3D –4&n= bsp; when x =3D –6.  
    Find the constant k a= nd the specific variation equation.
In general, a variation problem gives the type of the variation and the v= alues of the variables. From these, we solve for the  constant k and find the specific (exact) variation equation.
Variations
Since y var= ies directly to x, the general equation is y =3D kx.  Set y =3D –= 4  and x =3D –6 in the general equation.
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Example C. <= /span>
a. Given that y varies directly to x and y =3D –4&n= bsp; when x =3D –6.  
    Find the constant k a= nd the specific variation equation.
In general, a variation problem gives the type of the variation and the v= alues of the variables. From these, we solve for the  constant k and find the specific (exact) variation equation.
Variations
Since y var= ies directly to x, the general equation is y =3D kx.  Set y =3D –= 4  and x =3D –6 in the general equation.
–4 =3D k(–6)
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Example C. <= /span>
a. Given that y varies directly to x and y =3D –4&n= bsp; when x =3D –6.  
    Find the constant k a= nd the specific variation equation.
k =3D      = =3D
–4
–6
2
3
In general, a variation problem gives the type of the variation and the v= alues of the variables. From these, we solve for the  constant k and find the specific (exact) variation equation.
Variations
Since y var= ies directly to x, the general equation is y =3D kx.  Set y =3D –= 4  and x =3D –6 in the general equation.
–4 =3D k(–6)  so
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Example C. <= /span>
a. Given that y varies directly to x and y =3D –4&n= bsp; when x =3D –6.  
    Find the constant k a= nd the specific variation equation.
Put this into the general equation, we’ve the specific equation:
In general, a variation problem gives the type of the variation and the v= alues of the variables. From these, we solve for the  constant k and find the specific (exact) variation equation.
Variations
k =3D      = =3D
–4
–6
2
3
Since y var= ies directly to x, the general equation is y =3D kx.  Set y =3D –= 4  and x =3D –6 in the general equation.
–4 =3D k(–6)  so
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Example C. <= /span>
a. Given that y varies directly to x and y =3D –4&n= bsp; when x =3D –6.  
    Find the constant k a= nd the specific variation equation.
Put this into the general equation, we’ve the specific equation:
In general, a variation problem gives the type of the variation and the v= alues of the variables. From these, we solve for the  constant k and find the specific (exact) variation equation.
Variations
y =3D     x
3
2
k =3D      = =3D
–4
–6
2
3
Since y var= ies directly to x, the general equation is y =3D kx.  Set y =3D –= 4  and x =3D –6 in the general equation.
–4 =3D k(–6)  so
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b. Find the value= of x if y =3D 20.
Variations
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b. Find the value= of x if y =3D 20.
Variations
Substitute y =3D = 20 into the specific equation
y =3D     x
3
2
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b. Find the value= of x if y =3D 20.
Variations
20 =3D     x
3
2
Substitute y =3D = 20 into the specific equation
y =3D     x
3
2
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b. Find the value= of x if y =3D 20.
Variations
20 =3D     x
3
2
Substitute y =3D = 20 into the specific equation
20 *      =3D x
3
2
y =3D     x
3
2
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b. Find the value= of x if y =3D 20.
Variations
20 =3D     x
3
2
Substitute y =3D = 20 into the specific equation
20 *      =3D x
3
2
So     x =3D 30
y =3D     x
3
2
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b. Find the value= of x if y =3D 20.
Variations
20 =3D     x
3
2
Substitute y =3D = 20 into the specific equation
20 *      =3D x
3
2
So     x =3D 30
In application, the language of variation sums up the basic  relation of two or more types of variables.
y =3D     x
3
2
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b. Find the value= of x if y =3D 20.
Variations
20 =3D     x
3
2
Substitute y =3D = 20 into the specific equation
20 *      =3D x
3
2
So     x =3D 30
In application, the language of variation sums up the basic  relation of two or more types of variables.
Example = D. The weight W of an object varies inversely to the square of the distance D from the center of the earth. =
A person weighs 160 pounds on the surface = which is 4000 miles to the center of the eart= h. What will the person weigh if he’s 6000 mil= es above the surface of the earth?
y =3D     x
3
2
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b. Find the value= of x if y =3D 20.