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| The Rectangular (Two-dimensional,
Cartesian) Coordinate System |
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Rectangular coordinate axes, x-axis
and y-axis,
origin, quadrants
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Ordered
pair (x, y), coordinates of a point (abscissa x, ordinate y) |
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Midpoint of a line segment
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Graphing
Linear Equation, Linear Function (First Degree Polynomial)
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Linear
function f(x) = mx + c |
| The graph of
the linear function |
| Slope
or gradient, y-intercept and x-intercept |
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Slope-intercept form of a line
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Properties
of the linear function
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Lines
parallel to the axes, horizontal and vertical lines
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The
point-slope form of a line
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Parallel
and perpendicular lines
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General
form of the equation of a line
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The
two point form of the equation of a line
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| Rectangular coordinate axes, x-axis
and y-axis,
origin, quadrants
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| The Cartesian coordinate system is defined by two
axes at right angles to each other, forming a plane. |
| The horizontal axis is
labeled x, and the vertical axis is labeled
y. |
| The point of intersection,
where the axes meet, is called the origin labeled O. |
| Given each axis,
choose a unit length, and mark off each unit along the axis, forming
a grid. |
| The position of each point in a plane is identified with an
ordered pair of real numbers, in the form (x,
y), |
| called the
coordinates of the point. |
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| The
x-coordinate, called the
abscissa, equal to the distance of the point from the y-axis measured parallel
to the x-axis, and the
y-coordinate, called the
ordinate, the distance of the point from the x-axis measured parallel to the
y-axis.
The
origin O
has coordinates (0, 0). |
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| The intersection of the two axes creates four quadrants
indicated by numerals I, II, III, and IV. The quadrants |
| are labeled
counterclockwise starting from that in which both coordinates are
positive. |
 |
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| Ordered pair (
x,
y
),
coordinates of a point (abscissa x, ordinate y)
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| On the Cartesian plane the points are placed as shown:
A( 3, 0 ), B( 2, 3 ), C( -
5/2, 4 ), D( 0, 2
), |
| E(
- 4,
0 ), F( -2,
-11/3
), G( 0, -1
), and H( 4, -2
). |
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| Midpoint of a line segment
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| The coordinates of the midpoint
M(xM,
yM)
of the line segment AB
where, A(x1,
y1)
and B(x2,
y2)
are |
| endpoints: |
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| Example:
Find the midpoint of the
line segment AB
where the endpoints, A(-5,
3)
and B(-1,
-1). |
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| Graphing
Linear Equation, Linear Function (First Degree Polynomial)
|
|
|
Linear
function f(x) = mx + c |
| The graph of
the linear function |
| Slope
or gradient, y-intercept and x-intercept |
|
Slope-intercept form of a line
|
|
Properties
of the linear function
|
|
Lines
parallel to the axes, horizontal and vertical lines
|
|
The
point-slope form of a line
|
|
Parallel
and perpendicular lines
|
|
General
form of the equation of a line
|
|
The
two point form of the equation of a line
|
|
| Linear function f
(x)
= mx
+ c |
| The expression |
|
y =
mx
+ c
or f(x)
= ax + b
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| we call linear function,
where x is the argument or the independent variable,
f(x) or
y is the
dependent |
| variable or the function,
numbers, a and
b or
m and
c are
constants. |
| If the coefficient
m =
0, then y does not depend of
x. |
| Linear function
f(x)
= mx
+ c,
m is
not 0, to each value of the argument
x associates a unique value
of y. |
|
| The graph of the linear function |
| A function
y =
f(x)
can be considered as the set of ordered pairs (x,
y) where each pair represents a point in |
| a Cartesian coordinate
system xOy. |
| When plotted on an
x,
y graph linear function forms a
straight line. |
| The constant
m is called the
slope or gradient
and the constant c is the
y-intercept.
This is the point of |
| intersection between
the graph of the function and the y-axis. |
| If the coefficient
m is
not 0 and c =
0 then, the linear function has the form
y =
f(x) =
mx. |
| To the
graph of the function f(x) =
mx belong the points
P(x,
mx ) of a
coordinate plane of which the |
| abscissas are values of argument x
taken arbitrarily, and the ordinates are the calculated values of the |
|
function. |
| Thus, for
x =
0
=> f(x) =
f(0)
=
m
· 0
=
0,
this means that the graph of the function f(x) =
mx is running
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| through the origin
O(0,
0) or
the x-intercept
is at the origin. |
|
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|
| As the above
right triangles OAP1
and OBP
are similar the ratios of the ordinates and the abscissas of the
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|
points
P and
P1
are equal: |
| |
 |
or |
 |
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| The value of the ratio
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|
determines the rate at which the y-coordinate of a straight line
changes with respect to the x-coordinate. |
| On a line graph, changing
m
makes the line steeper or more gentle (shallower). |
| If m
is negative, y
decreases
as x
increases. |
|
| Example:
Find the equation of the line that passes through the
origin and the point A(-3,
2). |
| By
plugging the coordinates of the point A
into the equation y =
mx we
determine the slope m. |
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| Slope-intercept form of a line
|
| By moving the graph of the linear function
f(x)
= mx
vertically by
the vector c, |
| obtained is the linear function |
f(x)
= mx
+ c, |
|
|
| every
point of which has the y value that differs from the source graph
f(x)
= mx by
the same constant c. |
| This form of the linear function is called the
slope-intercept form. |
|
 |
|
| The graph of the function
f(x)
= mx
+ c
intersects the y-axis at the
point P0(0,
c). |
| Every point on the
y-axis has abscissa
0 that is, |
x
= 0 |
denotes the
y-axis. |
|
|
| Thus, we find the
y-intercept by calculating
f(0)
= m
· 0
+ c,
f(0)
= c. |
| Every point on the
x-axis has ordinate
0 that is, |
y
= 0 |
denotes
the
x-axis. |
|
|
| Therefore, the intersection point of the
graph and the x-axis we calculate from the equation
f(x)
= 0. |
| Thus
solving for x, |
0
= mx
+ c, |
we get |
 |
the root or
the zero point (or the x-intercept) |
|
| of the
linear function f(x)
= mx
+ c. |
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| Example:
Find the equation of the line that passes through the points A(-4,
-1)
and B(6,
4). |
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| Properties
of the linear function
|
| We examine the behavior of a function
y =
f(x)
by moving from left to right in the direction of x-axis by |
| inspecting its graph. |
| The linear
function f(x)
= mx
+ c,
m
> 0 is increasing, the graph rises from left to right, that is, |
| f(x1)
<
f(x2)
for all x1
< x2 |
 |
|
| If
m
< 0 linear function decreasing, f(x1)
>
f(x2) for all
x1
< x2 i.e., the graph falls from left to right. |
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 |
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| Lines
parallel to the axes, horizontal and vertical lines
|
| If
m =
0, the function does not depend of
x. To every number
x
associated is the same constant value y =
c.
|
| A
line parallel to the x-axis
is called a horizontal line (or constant). |
| If
the x
value never changes a line is parallel to the y-axis.
A line parallel to the y-axis
is called a vertical line. |
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| The linear function changes the sign at the root or zero point. |
| Thus,
if |
m > 0, |
then |
f(x)
<
0 |
for all |
 |
while
|
f(x) > 0 |
for all |
 |
|
|
| That
is, f(x)
= mx
+ c,
m
> 0 is negative for all x
less than the root,
positive for all x
greater than the root, |
| and at the root
f(x)
= 0. |
| If |
m < 0, |
then |
f(x)
>
0 |
for all |
|
while
|
f(x)
< 0 |
for all |
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| The
point-slope form of a line
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| The equation of the line
y
= mx
+ c
is defined by the slope m
and by one of its points P1(x1,
y1). |
| As
coordinates of the point P1must
satisfy
the equation of the line, we calculate the y-intercept,
by |
|
substituting them into the equation of the line, |
|
P1(x1,
y1)
=>
y
= mx
+ c |
|
y1
= mx1
+ c
or
c
= y1 -
mx1
|
| Therefore, the line that passes through the given point
P1(x1,
y1)
and given is its slope m, has the equation |
| |
y
= mx
+ y1 -
mx1 |
or |
y -
y1
= m(x
- x1) |
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|
| Example:
Find the equation of the line that is parallel with the line y
= - x
- 2 and passes through the point |
|
P1( 2,
1) . |
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| Parallel
and perpendicular lines
|
| Two lines having slopes
m1
and m2
are parallel if |
| |
m1
= m2 |
|
that is, if
they have the same slope. |
|
|
| To acquire the criteria when two lines,
y
= m1x
and y
= m2x |
| are
perpendicular or orthogonal we can use the principle of similar triangles,
OA'A
and OB'B in the picture. |
| Therefore, |
m1
: 1
= -1
: m2 |
=> |
 |
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 |
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| This relation will stay
unchanged if we translate the perpendicular lines, that is, when
lines |
|
y
= m1x
+ c1
and y
= m2x
+ c2
are written in the slope-intercept form. |
| Two lines are perpendicular if the slope of one line is the negative
reciprocal of the other. |
|
| Example:
Find the equation of the line that is perpendicular to the line |
 |
and passes through |
|
|
the point A(-2,
5). |
 |
|
| General
form of the equation of a line
|
| The linear equation
Ax +
By +
C
= 0 in two unknowns, x
and y, of
which at least one of the coefficients, A |
|
or B, are different then zero,
is called the general form for the equation of a line. |
| Dividing the
equation by B, where
B
is not 0, gives |
| |
 |
where |
 |
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|
| the slope-intercept form
y
= mx
+ c
of a line. |
| By putting
C
= 0 into the general form obtained is |
 |
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|
|
the equation of the
line that passes through the origin. |
| Setting
A = 0, gives |
 |
the
line parallel to the x-axis. |
| And
setting
B = 0 gives |
 |
the
line parallel to the y-axis. |
|
| The
two point form of the equation of a line
|
| Two points
P1(x1,
y1)
and P2(x2,
y2)
determine a unique line on the
Cartesian plane. |
| Therefore, their coordinates satisfy the equation
y
= mx
+ c
of the line. |
| The line which passes through the point
P1(x1,
y1) |
|
is determined by the equation y -
y1
= m(x
- x1). |
| As the
point P2(x2,
y2)
lies on the same line, |
| its coordinates should satisfy
the same equation
y2 -
y1
= m(x2
- x1). |
| That way determined is
the slope |
 |
where
x2
- x1
is not 0. |
|
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|
| Thus, obtained is |
 |
the equation of the line passing through the two points. |
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|
| Example:
Find the equation of the line which passes through points P(-2, 3) and
Q(6,
-1). |
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| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
