The document discusses coordinate systems and graphs. It describes: 1) The Cartesian coordinate system uses perpendicular x and y axes to locate points on a plane using coordinates like (x,y). 2) 3D coordinate systems extend this to locate points in space using x, y, and z coordinates like (x,y,z). 3) Graphs can show the relationship between variables by plotting their coordinates and drawing a line or curve through the points.

1. GRAPHS

2. 2D and 3D
coordinate
systems

4. • 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.

5. • 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.

7. • 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.

8. • 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.

9. • 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.

10. • 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).

11. 3D COORDINATE SYSTEM

13. • 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.

14. • 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.

15. • 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.

16. 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.

17. Left or Right-Handed?
The systems are right-handed (positive).
X Z
Y
Y Z X
These systems are left-handed (negative).
Z X
Y
Y X Z

18. Coordinates in 2 Dimensions
y (3,2)
2
3 x

19. The 3rd Dimension
z (3,2,4)
4 y
(3,2,0)
x

20. Consider the Point A(5,4,2)
C B
D G
E F

21. A(2,4,0)
B(0,4,3)
C(2,4,3)
D(2,0,3)

22. 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)

23. Centre of Box?
(1½,2,1)

24. Graphical Method
When 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.

25. 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.

26. 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.”

27. • 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.

28. • 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.

29. COORDINATE PLANE
Y-axis
Parts of a plane
1. X-axis
2. Y-axis
2nd QUADRANT 1ST QUADRANT 3. Origin
4. Quadrants I-IV
Origin ( 0 , 0 )
X-axis
3rd QUADRANT 4th QUADRANT

30. PLOTTING POINTS
Remember when plotting points
you always start at the origin.
B Next you go left (if x-coordinate
C 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)
A
D Plot these 4 points A (3, -4),
B(5, 6), C (-4, 5) and D (-7, -5)

31. Example 1.
Plot the points A (3, -4), B(5, 6),
C (-4, 5) and D (-7, -5) on the
Cartesian plane.

32. SLOPE
Slope 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. Run is 6 Slope triangle between
because we two points. Notice that the
went to the slope triangle can be
Rise is 10 right
drawn two different ways.
because we Rise is -10
went up because we
went down
10 5
The slope in this case is
Run is -6 6 3
because we
went to the
left 10 5
The slope in this case is
6 3
Another way to
find slope

33. FORMULA FOR FINDING SLOPE
The formula is used when you know two
points of a line.
They look like A( X 1 , Y1 ) and B( X 2 , Y2 )
RISE X 2 X1 X1 X 2
SLOPE
RUN Y2 Y1 Y1 Y2
EXAMPLE

34. Find the slope of the line between the two points (-4, 8) and (10, -4)
If it helps label the points. X 1 Y1 X2 Y2
Then use the
formula X 2 X1 (10 ) ( 4)
Y2 Y1 SUBSTITUTE INTO FORMULA ( 4) (8)
(10 ) ( 4) 10 4 14 7
Then Simplify
( 4) (8) 4 ( 8) 12 6

35. X AND Y INTERCEPTS
The 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).

36. SLOPE-INTERCEPT FORM OF A LINE
The 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

37. 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.

38. GRAPHING LINES
BY 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.

39. 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

40. Slope-intercept graphing
Steps
1. 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 + 2
2. 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.

41. FIND EQUATION OF A LINE GIVEN 2
POINTS
Find the equation of the line between (2, 5) and (-2, -3).
1. Find the slope between the two
points.
1. Slope is 2.
2. Plug in the slope in the slope-
2. y = 2x + b
intercept form.
3. Picked (2, 5) so
3. Pick one of the given points and plug
(5) = 2(2) + b
in numbers for x and y.
4. b = 1
4. Solve and find b.
5. y = 2x + 1
5. Rewrite final form.
Two other ways

42. Steps if given the slope and If given a graph there are three
ways.
a point on the line.
1. Substitute the slope into One way is to find two points on
the slope-intercept the line and use the first method
we talked about.
form.
2. Use the point to plug in Another would be to find the
for x and y. slope and pick a point and use the
second method.
3. Find b.
4. Rewrite equation. The third method would be to find
the slope and y-intercept and plug
it directly into y = mx + b.

43. Exercise: Plot the following points
1.(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)

44. DRAWING A CURVE
Given 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.

45. 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.

46. • 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.

47. 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

48. 10
-3, 9 9 3, 9
8
7
6
5
Y
-2, 4.2
4 2, 3.9
3
2
-1, 1 1
1, 0.7
0 0, 0
-4 -3 -2 -1 0 1 2 3 4
X

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