GRAPHS

2D and 3D coordinate

systems

• The Cartesian coordinate system is the most commonly used coordinate system. In two dimensions, this system consists of a pair of lines on a flat surface or plane, that intersect at right angles.

• The lines are called axes and the point at which they intersect is called the origin. The axes are usually drawn horizontally and vertically and are referred to as the x- and y-axes, respectively.

• The Cartesian coordinate system is the most commonly used coordinate system. In two dimensions, this system consists of a pair of lines on a flat surface or plane, that intersect at right angles.

• The lines are called axes and the point at which they intersect is called the origin. The axes are usually drawn horizontally and vertically and are referred to as the x- and y-axes, respectively.

• A point in the plane with coordinates (a, b) is a units to the right of the y axis and b units up from the x axis if a and b are positive numbers.

• If a and b are both negative numbers, the point is a units to the left of the y axis and b units down from the x axis. In the figure above point P1 has coordinates (3, 4), and point P2 has coordinates (-1, -3).

3D COORDINATE SYSTEM

• In a 3D Cartesian coordinate system, a point P is referred to by three real numbers (coordinates), indicating the positions of the perpendicular projections from the point to three fixed, perpendicular, graduated lines, called the axes which intersect at the origin.

• Often the x-axis is imagined to be horizontal and pointing roughly toward the viewer (out of the page), the y-axis is also horizontal and pointing to the right, and the z-axis is vertical, pointing up.

• The system is called right-handed if it can be rotated so that the three axes are in the position as shown in the figure above. The x-coordinate of of the point P in the figure is a, the y-coordinate is b, and the z-coordinate is c.

Coordinate Systems: Right Hand Rule

Place your fingers in the direction of the positive x-axis and rotate them in the direction of the y-axis. Your thumb will point in the direction of the positive z-axis.

Left or Right-Handed? The systems are right-handed (positive).

Z X

Y

Y Z X

These systems are left-handed (negative). Z X

Y

Y X Z

Coordinates in 2 Dimensions

x

y

3

2

(3,2)

The 3rd Dimension

x

y

3

2

z

(3,2,0)

4(3,2,4)

BC

D

E F

G

Consider the Point A(5,4,2)

A(2,4,0)

B(0,4,3)

C(2,4,3)

D(2,0,3)

B(3,0,0)

C(3,4,0)

Q(3,0,2)

R(3,4,2)

(0,2,0) (0,4,1) (1½,4,0)

(1½,2,0) (1½,2,2) (1½,4,1)

Centre of Box?

(1½,2,1)

Graphical MethodWhen two quantities are so related that a change in one produces a corresponding change in other, the relation between them can well be shown by means of a graphical method. The two quantities are said to be variables.

Variables…

• Ex: y=3x-5, y=2x2-6x+10• When x is given a value y will

have a definite corresponding value. X and y are called the variables.

Axes of reference

• In a suitably chosen position two lines are drawn; one horizontally OX, and one vertically OY, meeting at the point O. the position of the point O is determined by the values of the variables. How to establish this position will be shown later.

• These two lines OX,OY, at right angles are called the “Axes of reference” or usually “The axes.”

• The point O is called the “origin of axes” or “the origin.”

• The horizontal axis OX is the axis along which x values are plotted and is called the axes of abscissa.

• The vertical axis OY is the axis along which y values are plotted and is called the axis of ordinates.

• Along OX the axis from O is divided into equal parts, each part being equal to the same number of x units. Similarly the axis OY is divided into y units.

COORDINATE PLANE

Parts of a plane1. X-axis2. Y-axis3. Origin4. Quadrants I-IV

X-axis

Y-axis

Origin ( 0 , 0 )

1ST QUADRANT2nd QUADRANT

3rd QUADRANT 4th QUADRANT

PLOTTING POINTSRemember when plotting points you always start at the origin. Next you go left (if x-coordinate is negative) or right (if x-coordinate is positive. Then you go up (if y-coordinate is positive) or down (if y-coordinate is negative)

Plot these 4 points A (3, -4), B(5, 6), C (-4, 5) and D (-7, -5)

A

BC

D

Example 1.

Plot the points A (3, -4), B(5, 6), C (-4, 5) and D (-7, -5) on the Cartesian plane.

SLOPESlope is the ratio of the vertical rise to the horizontal run between any two points on a line. Usually referred to as the rise over run. Slope triangle between

two points. Notice that the slope triangle can be drawn two different ways.

Rise is -10 because we went down

Run is -6 because we went to the left

3

5

6

10

iscasethisinslopeThe

Rise is 10 because we went up

Run is 6 because we went to the right

3

5

6

10iscasethisinslopeThe

Another way to find slope

FORMULA FOR FINDING SLOPE

21

21

12

12

YY

XX

YY

XX

RUN

RISESLOPE

The formula is used when you know two points of a line.

),(),( 2211 YXBandYXAlikelookThey

EXAMPLE

Find the slope of the line between the two points (-4, 8) and (10, -4)

If it helps label the points. 1X 1Y2X 2Y

Then use the formula

12

12

YY

XX

)8()4(

)4()10(

FORMULAINTOSUBSTITUTE

6

7

12

14

)8(4

410

)8()4(

)4()10(

SimplifyThen

X AND Y INTERCEPTSThe x-intercept is the x-coordinate of a point where the graph crosses the x-axis.

The y-intercept is the y-coordinate of a point where the graph crosses the y-axis.

The x-intercept would be 4 and is located at the point (4, 0).

The y-intercept is 3 and is located at the point (0, 3).

SLOPE-INTERCEPT FORM OF A LINEThe slope intercept form of a line is y = mx + b, where “m” represents the slope of the line and “b” represents the y-intercept.

When an equation is in slope-intercept form the “y” is always on one side by itself. It can not be more than one y either.

If a line is not in slope-intercept form, then we must solve for “y” to get it there.

Examples

IN SLOPE-INTERCEPT NOT IN SLOPE-INTERCEPT

y = 3x – 5 y – x = 10

y = -2x + 10 2y – 8 = 6x

y = -.5x – 2 y + 4 = 2x

Put y – x = 10 into slope-intercept form

Add x to both sides and would get y = x + 10

Put 2y – 8 = 6x into slope-intercept form.

Add 8 to both sides then divide by 2 and would get y = 3x + 4

Put y + 4 = 2x into slope-intercept form.

Subtract 4 from both sides and would get y = 2x – 4.

GRAPHING LINESBY MAKING A TABLE OR USING THE

SLOPE-INTERCEPT FORM

I could refer to the table method by input-output table or x-y table. For now I want you to include three values in your table. A negative number, zero, and a positive number.

Graph y = 3x + 2 INPUT (X) OUTPUT (Y)

-2 -4

0 2

1 5

By making a table it gives me three points, in this case (-2, -4) (0, 2) and (1, 5) to plot and draw the line.

See the graph.

Plot (-2, -4), (0, 2) and (1, 5)

Then draw the line. Make sure your line covers the graph and has arrows on both ends. Be sure to use a ruler.

Slope-intercept graphing

Slope-intercept graphingSteps1. Make sure the equation is in slope-intercept form.2. Identify the slope and y-intercept.3. Plot the y-intercept.4. From the y-intercept use the slope to get another point to draw the line.

1. y = 3x + 22. Slope = 3 (note that this means the

fraction or rise over run could be (3/1) or (-3/-1). The y-intercept is 2.

3. Plot (0, 2)4. From the y-intercept, we are going rise

3 and run 1 since the slope was 3/1.

FIND EQUATION OF A LINE GIVEN 2 POINTS

1. Find the slope between the two points.

2. Plug in the slope in the slope-intercept form.

3. Pick one of the given points and plug in numbers for x and y.

4. Solve and find b.5. Rewrite final form.

Find the equation of the line between (2, 5) and (-2, -3).

1. Slope is 2.2. y = 2x + b3. Picked (2, 5) so

(5) = 2(2) + b4. b = 15. y = 2x + 1

Two other ways

Steps if given the slope and a point on the line.1. Substitute the slope into

the slope-intercept form.

2. Use the point to plug in for x and y.

3. Find b.4. Rewrite equation.

If given a graph there are three ways.

One way is to find two points on the line and use the first method we talked about.

Another would be to find the slope and pick a point and use the second method.

The third method would be to find the slope and y-intercept and plug it directly into y = mx + b.

Exercise: Plot the following points1.(5,6);(4,2)2.(-1,2);(3,0)3.(-3,-4);(-3,-1)4.(8,-3);(3,-8)5.(0,2);(3,-4)6.(1,-6);(-5,2)7.(5,7);(-3,-8)8.(-4,-5);(-4,6)9.(-1,-6);(3,3)10.(3,-2);(-2,-3)

DRAWING A CURVEGiven a series of values of x and the corresponding values of y, the relation between x and y can be shown by plotting the given points and then drawing their curve. The curve is found by joining up the points, using the “smoothest” curve which will pass through all the points.

When, however, the points are gained from experimental data, say, the curve obtained by joining all the points would consist of irregular angles and sharp bends. In this case the rule is to draw the smoothest curve which is the best approximation to that which would pass through all the points.

• Some points will be on the curve, some above it and some below it. The curve can then be used to find the error in those values lying off it.

Example:

• The following table gives values of x corresponding the values of y.

x -3 -2 -1 0 1 2 3

y 9 4.2 1 0 0.7 3.9 9

-4 -3 -2 -1 0 1 2 3 40

1

2

3

4

5

6

7

8

9

10

Series1; 9

4.2

1

0

0.7

3.9

9

X

Y

• Here the values of x are evenly distributed on either side of zero. Hence the