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First Degree Functions
------=_NextPart_01C97BD1.58F592C0 Content-Location: file:///C:/ECF94D13/2GraphsofLinesandSlopes_files/slide0272.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
The graphs of the equa= tions Ax + By =3D C are straight
lines. 
First Degree Functions
------=_NextPart_01C97BD1.58F592C0 Content-Location: file:///C:/ECF94D13/2GraphsofLinesandSlopes_files/slide0273.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
The graphs of the equa= tions Ax + By =3D C are straight
lines.  Its easy to graph lines by graphi= ng the x and y
intercepts,
First Degree Functions
------=_NextPart_01C97BD1.58F592C0 Content-Location: file:///C:/ECF94D13/2GraphsofLinesandSlopes_files/slide0274.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
The graphs of the equa= tions Ax + By =3D C are straight
lines.  Its easy to graph lines by graphi= ng the x and y
intercepts, i.e. set x= =3D 0 to get y-intercept, and set y =3D 0 for the x-intercept.
First Degree Functions
------=_NextPart_01C97BD1.58F592C0 Content-Location: file:///C:/ECF94D13/2GraphsofLinesandSlopes_files/slide0275.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
a.2x – 3y =3D 12        = ;   
The graphs of the equa= tions Ax + By =3D C are straight
lines.  Its easy to graph lines by graphi= ng the x and y
intercepts, i.e. set x= =3D 0 to get y-intercept, and set y =3D 0 for the x-intercept. <= /span>
First Degree Functions
------=_NextPart_01C97BD1.58F592C0 Content-Location: file:///C:/ECF94D13/2GraphsofLinesandSlopes_files/slide0276.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
a.2x – 3y =3D 12        = ;   
The graphs of the equa= tions Ax + By =3D C are straight
lines.  Its easy to graph lines by graphi= ng the x and y
intercepts, i.e. set x= =3D 0 to get y-intercept, and set y =3D 0 for the x-intercept. 
(0,-4) =
First Degree Functions
------=_NextPart_01C97BD1.58F592C0 Content-Location: file:///C:/ECF94D13/2GraphsofLinesandSlopes_files/slide0277.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
a.2x – 3y =3D 12        = ;   
The graphs of the equa= tions Ax + By =3D C are straight
lines.  Its easy to graph lines by graphi= ng the x and y
intercepts, i.e. set x= =3D 0 to get y-intercept, and set y =3D 0 for the x-intercept. <= /span>
(6,0) <= /b>
(0,-4) =
First Degree Functions
------=_NextPart_01C97BD1.58F592C0 Content-Location: file:///C:/ECF94D13/2GraphsofLinesandSlopes_files/slide0278.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
a.2x – 3y =3D 12        = ;   
The graphs of the equa= tions Ax + By =3D C are straight
lines.  Its easy to graph lines by graphi= ng the x and y
intercepts, i.e. set x= =3D 0 to get y-intercept, and set y =3D 0 for the x-intercept. <= /span>
(6,0) <= /b>
(0,-4) =
First Degree Functions
------=_NextPart_01C97BD1.58F592C0 Content-Location: file:///C:/ECF94D13/2GraphsofLinesandSlopes_files/slide0279.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
a.2x – 3y =3D 12        = ;   
The graphs of the equa= tions Ax + By =3D C are straight
lines.  Its easy to graph lines by graphi= ng the x and y
intercepts, i.e. set x= =3D 0 to get y-intercept, and set y =3D 0 for the x-intercept. If there is only one variable in the equation, we get a ver= tical or a horizontal line. 
(6,0) <= /b>
(0,-4) =
First Degree Functions
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a.2x – 3y =3D 12        = ;    b. -3y =3D 12      
The graphs of the equa= tions Ax + By =3D C are straight
lines.  Its easy to graph lines by graphi= ng the x and y
intercepts, i.e. set x= =3D 0 to get y-intercept, and set y =3D 0 for the x-intercept. If there is only one variable in the equation, we get a ver= tical or a horizontal line. 
(6,0) <= /b>
(0,-4) =
First Degree Functions
------=_NextPart_01C97BD1.58F592C0 Content-Location: file:///C:/ECF94D13/2GraphsofLinesandSlopes_files/slide0281.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
a.2x – 3y =3D 12        = ;    b. -3y =3D 12        = ;  
The graphs of the equa= tions Ax + By =3D C are straight
lines.  Its easy to graph lines by graphi= ng the x and y
intercepts, i.e. set x= =3D 0 to get y-intercept, and set y =3D 0 for the x-intercept. If there is only one variable in the equation, we get a ver= tical or a horizontal line. 
(6,0) <= /b>
(0,-4) =
y =3D –4
First Degree Functions
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a.2x – 3y =3D 12        = ;    b. -3y =3D 12        = ;            c. 2x =3D 12
The graphs of the equa= tions Ax + By =3D C are straight
lines.  Its easy to graph lines by graphi= ng the x and y
intercepts, i.e. set x= =3D 0 to get y-intercept, and set y =3D 0 for the x-intercept. If there is only one variable in the equation, we get a ver= tical or a horizontal line. 
(6,0) <= /b>
(0,-4) =
y =3D –4
First Degree Functions
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a.2x – 3y =3D 12        = ;    b. -3y =3D 12        = ;            c. 2x =3D 12
The graphs of the equa= tions Ax + By =3D C are straight
lines.  Its easy to graph lines by graphi= ng the x and y
intercepts, i.e. set x= =3D 0 to get y-intercept, and set y =3D 0 for the x-intercept. If there is only one variable in the equation, we get a ver= tical or a horizontal line. 
(6,0) <= /b>
(0,-4) =
y =3D –4
x =3D 6
First Degree Functions
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a.2x – 3y =3D 12        = ;    b. -3y =3D 12        = ;            c. 2x =3D 12
If both x and y are &#= 13;
present, we get a =
tilted line.
The graphs of the equa= tions Ax + By =3D C are straight
lines.  Its easy to graph lines by graphi= ng the x and y
intercepts, i.e. set x= =3D 0 to get y-intercept, and set y =3D 0 for the x-intercept. If there is only one variable in the equation, we get a ver= tical or a horizontal line. 
(6,0) <= /b>
(0,-4) =
y =3D –4
x =3D 6
First Degree Functions
------=_NextPart_01C97BD1.58F592C0 Content-Location: file:///C:/ECF94D13/2GraphsofLinesandSlopes_files/slide0285.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
a.2x – 3y =3D 12        = ;    b. -3y =3D 12        = ;            c. 2x =3D 12
If both x and y are &#= 13;
present, we get a =
tilted line.
If the equation is = 3;
y =3D c, we get a horizonta= l line.
The graphs of the equa= tions Ax + By =3D C are straight
lines.  Its easy to graph lines by graphi= ng the x and y
intercepts, i.e. set x= =3D 0 to get y-intercept, and set y =3D 0 for the x-intercept. If there is only one variable in the equation, we get a ver= tical or a horizontal line. 
(6,0) <= /b>
(0,-4) =
y =3D –4
x =3D 6
First Degree Functions
------=_NextPart_01C97BD1.58F592C0 Content-Location: file:///C:/ECF94D13/2GraphsofLinesandSlopes_files/slide0286.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
a.2x – 3y =3D 12        = ;    b. -3y =3D 12        = ;            c. 2x =3D 12
If both x and y are &#= 13;
present, we get a =
tilted line.
If the equation is = 3;
y =3D c, we get a horizonta= l line.
The graphs of the equa= tions Ax + By =3D C are straight
lines.  Its easy to graph lines by graphi= ng the x and y
intercepts, i.e. set x= =3D 0 to get y-intercept, and set y =3D 0 for the x-intercept. If there is only one variable in the equation, we get a ver= tical or a horizontal line. 
(6,0) <= /b>
(0,-4) =
y =3D –4
x =3D 6
If the equation is = 3;
x =3D c, we get a vertical = line.
First Degree Functions
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First Degree Functions
Given Ax = + By =3D C with B =3D 0,
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First Degree Functions
Given Ax + By =3D C wi= th B =3D 0, treating x as the input
and y as the output, w= e may solve for y
------=_NextPart_01C97BD1.58F592C0 Content-Location: file:///C:/ECF94D13/2GraphsofLinesandSlopes_files/slide0288.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
First Degree Functions
Given Ax + By =3D C wi= th B =3D 0, treating x as the input
and y as the output, w= e may solve for y and put the
equation in a function= form: y =3D f(x) =3D mx + b, 
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First Degree Functions
Given Ax + By =3D C wi= th B =3D 0, treating x as the input
and y as the output, w= e may solve for y and put the
equation in a function= form: y =3D f(x) =3D mx + b, 
m is call= ed the slope <= /span>and b is = the y intercept,
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First Degree Functions
Given Ax + By =3D C wi= th B =3D 0, treating x as the input
and y as the output, w= e may solve for y and put the
equation in a function= form: y =3D f(x) =3D mx + b, 
m is called the slope <= /span>and b is = the y intercept, and the
form is called the slope-intercept form.
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First Degree Functions
Given Ax + By =3D C wi= th B =3D 0, treating x as the input
and y as the output, w= e may solve for y and put the
equation in a function= form: y =3D f(x) =3D mx + b, 
m is called the slope <= /span>and b is = the y intercept, and the
form is called the slope-intercept form.
From the = examples above,
a. 2x = 211; 3y =3D 12 à 
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First Degree Functions
Given Ax + By =3D C wi= th B =3D 0, treating x as the input
and y as the output, w= e may solve for y and put the
equation in a function= form: y =3D f(x) =3D mx + b, 
m is called the slope <= /span>and b is = the y intercept, and the
form is called the slope-intercept form.
2
3 
2
3 
From the = examples above,
a. 2x = 211; 3y =3D 12 à y =3D      x –= 4, so the slope =3D
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First Degree Functions
Given Ax + By =3D C wi= th B =3D 0, treating x as the input
and y as the output, w= e may solve for y and put the
equation in a function= form: y =3D f(x) =3D mx + b, 
m is called the slope <= /span>and b is = the y intercept, and the
form is called the slope-intercept form.
2
3 
2
3 
From the = examples above,
a. 2x = 211; 3y =3D 12 à y =3D      x –= 4, so the slope =3D
b. -3y = =3D 12
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First Degree Functions
Given Ax + By =3D C wi= th B =3D 0, treating x as the input
and y as the output, w= e may solve for y and put the
equation in a function= form: y =3D f(x) =3D mx + b, 
m is called the slope <= /span>and b is = the y intercept, and the
form is called the slope-intercept form.
2
3 
2
3 
From the = examples above,
a. 2x = 211; 3y =3D 12 à y =3D      x –= 4, so the slope =3D
b. -3y = =3D 12 à y =3D 0x –= 4, so the slope =3D 0
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First Degree Functions
Given Ax + By =3D C wi= th B =3D 0, treating x as the input
and y as the output, w= e may solve for y and put the
equation in a function= form: y =3D f(x) =3D mx + b, 
m is called the slope <= /span>and b is = the y intercept, and the
form is called the slope-intercept form.
2
3 
2
3 
From the = examples above,
a. 2x = 211; 3y =3D 12 à y =3D      x –= 4, so the slope =3D
b. -3y = =3D 12 à y =3D 0x –= 4, so the slope =3D 0
c. 2x =3D= 12,
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First Degree Functions
Given Ax + By =3D C wi= th B =3D 0, treating x as the input
and y as the output, w= e may solve for y and put the
equation in a function= form: y =3D f(x) =3D mx + b, 
m is called the slope <= /span>and b is = the y intercept, and the
form is called the slope-intercept form.
2
3 
2
3 
From the = examples above,
a. 2x = 211; 3y =3D 12 à y =3D      x –= 4, so the slope =3D
b. -3y = =3D 12 à y =3D 0x –= 4, so the slope =3D 0
c. 2x =3D= 12, the slope is undefined since we can't
solve for= y.
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First Degree Functions
Given Ax + By =3D C wi= th B =3D 0, treating x as the input
and y as the output, w= e may solve for y and put the
equation in a function= form: y =3D f(x) =3D mx + b, 
m is called the slope <= /span>and b is = the y intercept, and the
form is called the slope-intercept form.
The slope m is also the ratio of the change in the
output vs. the change = in the input.
2
3 
2
3 
From the = examples above,
a. 2x = 211; 3y =3D 12 à y =3D      x –= 4, so the slope =3D
b. -3y = =3D 12 à y =3D 0x –= 4, so the slope =3D 0
c. 2x =3D= 12, the slope is undefined since we can't
solve for= y.
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First Degree Functions
Given Ax + By =3D C wi= th B =3D 0, treating x as the input
and y as the output, w= e may solve for y and put the
equation in a function= form: y =3D f(x) =3D mx + b, 
m is called the slope <= /span>and b is = the y intercept, and the
form is called the slope-intercept form.
The slope m is also the ratio of the change in the
output vs. the change = in the input. If two points on the
line are given, the sl= ope may be calculated via the
following formula. 
2
3 
2
3 
From the = examples above,
a. 2x = 211; 3y =3D 12 à y =3D      x –= 4, so the slope =3D
b. -3y = =3D 12 à y =3D 0x –= 4, so the slope =3D 0
c. 2x =3D= 12, the slope is undefined since we can't
solve for= y.
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Slope Formula: Let (x1, y1) and (x2, y) be two = points = on a line= , then the slope
Δy
Δx
m =3D 
First Degree Functions
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Slope Formula: Let (x1, y1) and (x2, y) be two = points = on a line= , then the slope
Δy
Δx
m =3D 
First Degree Functions
(The Greek letter Δ means "the difference&quo= t;.)
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Slope Formula: Let (x1, y1) and (x2, y) be two = points = on a line= , then the slope
Δy
Δx
y2 – y1
x2 – x1 
m =3D 
=3D
First Degree Functions
(The Greek letter Δ means "the difference&quo= t;.)
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Slope Formula: Let (x1, y1) and (x2, y) be two = points = on a line= , then the slope
Δy
Δx
y2 – y1
x2 – x1 
m =3D 
rise
run
=3D
=3D
First Degree Functions
(The Greek letter Δ means "the difference&quo= t;.)
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Slope Formula: Let (x1, y1) and (x2, y) be two = points = on a line= , then the slope
Δy
Δx
y2 – y1
x2 – x1 
m =3D 
rise
run
=3D
=3D
(x1, y= 1)
(x2, y= 2)
First Degree Functions
(The Greek letter Δ means "the difference&quo= t;.)
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Slope Formula: Let (x1, y1) and (x2, y) be two = points = on a line= , then the slope
Δy
Δx
y2 – y1
x2 – x1 
m =3D 
rise
run
=3D
=3D
(x1, y= 1)
(x2, y= 2)
Δy=3Dy2–y1=3Drise
First Degree Functions
(The Greek letter Δ means "the difference&quo= t;.)
------=_NextPart_01C97BD1.58F592C0 Content-Location: file:///C:/ECF94D13/2GraphsofLinesandSlopes_files/slide0304.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Slope Formula: Let (x1, y1) and (x2, y) be two = points = on a line= , then the slope
Δy
Δx
y2 – y1
x2 – x1 
m =3D 
rise
run
=3D
=3D
(x1, y= 1)
(x2, y= 2)
Δy=3Dy2–y1=3Drise
Δx=3Dx2–x1=3Drun
First Degree Functions
(The Greek letter Δ means "the difference&quo= t;.)
------=_NextPart_01C97BD1.58F592C0 Content-Location: file:///C:/ECF94D13/2GraphsofLinesandSlopes_files/slide0305.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Slope Formula: Let (x1, y1) and (x2, y) be two = points = on a line= , then the slope
Δy
Δx
y2 – y1
x2 – x1 
m =3D 
rise
run
=3D
=3D
(x1, y= 1)
(x2, y= 2)
Δy=3Dy2–y1=3Drise
Δx=3Dx2–x1=3Drun
The Point Slope For= mula:  Let y =3D f(x) be a first = = degree eq= uation with slope m, and (x1, y1) is a point
on the line, then  
y =3D f(x= ) =3D m(x – x1) = + y1   
First Degree Functions
(The Greek letter Δ means "the difference&quo= t;.)
------=_NextPart_01C97BD1.58F592C0 Content-Location: file:///C:/ECF94D13/2GraphsofLinesandSlopes_files/slide0306.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Slope Formula: Let (x1, y1) and (x2, y) be two = points = on a line= , then the slope
Δy
Δx
y2 – y1
x2 – x1 
m =3D 
rise
run
=3D
=3D
(x1, y= 1)
(x2, y= 2)
Δy=3Dy2–y1=3Drise
Δx=3Dx2–x1=3Drun
The Point Slope For= mula:  Let y =3D f(x) be a first = = degree eq= uation with slope m, and (x1, y1) is a point
on the line, then  
y =3D f(x= ) =3D m(x – x1) = + y1   
First degree functions= are also called linear functions
because their graphs a= re straight lines.
First Degree Functions
(The Greek letter Δ means "the difference&quo= t;.)
------=_NextPart_01C97BD1.58F592C0 Content-Location: file:///C:/ECF94D13/2GraphsofLinesandSlopes_files/slide0307.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Notation and Algebra of Functions
Slope Formula: Let (x1, y1) and (x2, y) be two = points = on a line= , then the slope
Δy
Δx
y2 – y1
x2 – x1