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The Coordinate System

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Axis Of The Real Numbers

P In one dimension, each point on a straight line can be represented

by a real number.

P The range of real numbers runs from negative infinity to positiveinfinity.

P

A straight line marked with an origin corresponding to thenumber 0 and with all other real numbers in order is called theaxis of the real numbers.

P In the Cartesian Plane, this would be known as thex axis.

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Cartesian Coordinate System

The two dimensionalCartesian coordinate systemis formed by drawing thetwo axes of the real numbers(x-axis and y-axis)perpendicular to each other

with the intersection pointas the origin. The plane isthen divided into fourquadrants

According to Ren Descartes, a French mathematician of the seventeenthcentury, each point on a plane can be represented by a pair of real numbers

in a like manner.

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Quadrant Properties

Reading Coordinates on the Cartesian Plane

P x coordinates are read first,followed by the ycoordinates in the form (x,y)

P Quadrant 1 will havecoordinates (+,+)

P Quadrant 2 will havecoordinates (-,+)

P Quadrant 3 will havecoordinates (-,-)

P Quadrant 4 will havecoordinates (+,-)

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Origin, Ordered Pairs, Abscissa, and Ordinate

PThe origin is the point wherethe x and y axes intersect.

P The coordinates of the point Ais the pair of real numbers(XA,YA)

P In the figure, point A inquadrant I has coordinates(4, 3). We call the pair ofnumbers an ordered pair.

PThe first component of thecoordinates is called theabscissa and the secondcomponent is called theordinate in a Cartesian

coordinate system

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There exists a one-to-one correspondence between the set of

ordered pairs of real numbers in a specific pair of axes and theset of points in a plane.

For any two points A,B on x-axis,

mAB = XB - XA = |difference of abscissas|.

For any two points C,D on y-axis,

mCD = YD - YC = |difference of ordinates|.

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Slope

P A number that represents thesteepness of a line on a grid.

P The number always indicateshow far we move vertically aswe move 1 unit horizontally.

P The slope is an indication ofhow steep the line is.

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Slope Properties

PThe slope of any horizontal segment is zero.

PThe slope of any vertical segment is undefined.

PThe slope of a line segment rising to the right is positive.

PThe slope of a line segment falling to the right is negative.

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Calculation of Slope (1)

slope = rise / run

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Calculation of Slope 2

Examples 1 and 2 illustrate two ways to determine the slope of a linesegment.

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Determine the Slope ofLine Segment AB

Solution

P Determine the slope bycounting squares.

P Moving from A to B, the riseis -8 and the run is 3.

P Slope AB = rise / run

= -8 / 3= -2.667

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Calculation of Slope (3)

GIVE IT A TRY!

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Calculation of Slope (4)Sample Calculations (a) slope = rise / run

= -1 / 4= -0.25

(b) slope = rise / run= 4 / 6= 2 / 3= 0.667

(c) slope = rise / run= 2 / 6= 1 / 3= 0. 333

(d) slope = rise / run

= -5 / 2= - 2.5

(e) slope = rise / run= 0 / 3

= 0

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Undefined Slopes

Slopes With a Value of Zero