Transcript Document

Graphing in Two Dimensions
By
Dr. Julia Arnold
A little background about the
creator of the coordinate system.
“Descartes was a "jack of all trades", making major
contributions to the areas of anatomy, cognitive
science, optics, mathematics and philosophy.
Underlying his methodology is the belief that all
science is based on mathematics. This is manifested
in his unification of ancient geometry and his new
alegbra based on the Cartesian coodinate system.
“(1)
(1) Copied from http://www.trincoll.edu/depts/phil/philo/phils/descartes.html
4
3
2
1
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
-1
-2
-3
-4
We begin with two number lines intersecting.
The vertical line is called
the y-axis. Y
4
3
2
1
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
-1
-2
-3
-4
The horizontal
Line is called the
X-axis
x
As you can see, there are four
Quadrants.
Y
This is quadrant II.
4
This is quadrant I.
3
2
1
x
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
-1
Where the two lines cross is
-2
Called the origin.
This is quadrant III.
-3
-4
This is quadrant IV.
They are numbered counter-clockwise, beginning with the upper
right corner. This numbering stays the same for whatever math
course you take.
To graph or plot a point you need two numbers, one to tell you how far
right or left to go, and one to tell you how high or low to go.
Y
We write the point as (x,y)
And we call the x, the
x-coordinate, and we call y,
4
the y-coordinate.
3
2
1
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
-1
-2
The point (x,y) is called an
ordered pair of numbers,
because the order matters.
-3
-4
x
This is how a coordinate system or graph would look with a grid.
(2,3)
3.0
To find the point (2,3), begin
at the origin, and, since 2 is in
the x-coordinate position,
go to 2 on the x axis.
2.0
1.0
-4.0 -3.0 -2.0 -1.0
1.0 2.0
3.0 4.0
-1.0
-2.0
-3.0
At 2, go straight up to 3 and
Draw the dot.
5.0
To emphasize that order matters, let’s now locate the point (3,2)
(2,3)
3.0
(3,2)
2.0
1.0
-4.0 -3.0 -2.0 -1.0
1.0 2.0
3.0 4.0
-1.0
-2.0
-3.0
As you can see, they are
different points.
5.0
As you click your mouse, points will appear on the screen.
Write the ordered pair of numbers for that point before
Clicking again.
3.0
2.0
(-3,1)
1.0
(3,0)
-4.0 -3.0 -2.0 -1.0
1.0 2.0
3.0 4.0
-1.0
-2.0
(-4,-3)
-3.0
(0,-2)
(2,-3)
5.0
The rise is the vertical change as you move
from one point to another or below as we go
from A to B.
To go from A to B we move up which is positive.
This is the
Rise.
B
A
The rise is the vertical change as you move
from one point to another or below as we go
from A to B.
To go from A to B we move down which is
negative.
A
This is the
Rise.
B
What is the rise going from A to B?
Point A (-4,3)
3.0
Start
Going down
2.0
with
is negative.
The y1.0
coordina
te of B
and
subtract -4.0 -3.0 -2.0 -1.0
the y-1.0
1.0 2.0
3.0 4.0
Point B (1,0)
coordinate
of A
0-3=-3
-2.0
-3.0
The rise is -3
5.0
The run is the horizontal change as you move
from one point to another or below as we go
from A to B.
Going to the right is positive.
B
This is the
run.
A
The run is the horizontal change as you move
from one point to another or below as we go
from A to B.
Going left is negative.
B
A
This is the
run.
What is the run going from A to B?
Point A (-4,3)
3.0
Start
Going right is positive.
2.0
with
The x1.0
coordina
te of B
and
1.0 2.0 3.0 4.0 5.0
subtract -4.0 -3.0 -2.0 -1.0
(1,0)
Point
B
the x-1.0
coordinate
of A
1-(-4)= 5
-2.0
-3.0
The run is 5
The distance between two numbers on the
Number line is easy to compute.
-6 -5 -4
-3
-2 -1
0
1
2
3
4
5
How far apart are the two points pictured?
Don’t click till you have an answer.
5 units The formula is to subtract 1 – (-4) = 5
If you subtract backwards --- -4 – 1 = -5 you get a negative number
but distance can’t be negative, so to make sure the answer is positive
no matter which way you subtract we take the absolute value of the
number.
If two points are on the horizontal number line, or
the vertical number line, the distance between them
can be found by subtracting and taking the absolute
value.
As a formula , we would write for the following
Picture: b - a
a
Or for the following: x2 – x1
x1
b
x2
What is the distance
Between the two points? 3.0
Since they are on the
Same vertical line,
Subtract.
2.0
1.0
-4.0 -3.0 -2.0 -1.0
3 – (-3) = 6
-1.0
-2.0
-3.0
1.0 2.0
3.0 4.0
5.0
We also want to be
able to find the
distance between
any two points, such as..
3.0
2.0
1.0
-4.0 -3.0 -2.0 -1.0
-1.0
-2.0
-3.0
1.0 2.0
3.0 4.0
5.0
To do this we turn to a famous theorem
discovered by a man named Pythagoras.
The theorem is called the Pythagorean Theorem
Born: about 569 BC in Samos, Ionia
Died: about 475 BC
Pythagoras of Samos is often described as the first pure mathematician. He
is an extremely important figure in the development of mathematics yet we
know relatively little about his mathematical achievements. Unlike many later
Greek mathematicians, where at least we have some of the books which they
wrote, we have nothing of Pythagoras's writings. The society which he led,
half religious and half scientific, followed a code of secrecy which certainly
means that today Pythagoras is a mysterious figure. (2)
(2) http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html
the Pythagorean Theorem
His theorem says that the sum of the squares of
the legs of a right triangle equals the square of
the hypotenuse.
a2 + b2 = c2
c
b
a
In a right triangle,
the legs are perpendicular. Thus,
a is perpendicular
to b.
It is important for you to know that when you
label a right triangle, or when a, b, and c, are given
in a problem that c is ALWAYS the hypotenuse,
which is the side opposite the right angle.
Ahh, there’s
C
Only
c
a or b
b or a
a2 + b2 = c2
Right
Angle
90o
Now back to finding
The distance between
The two points.
right
The rise.
3.0
2.0
Then the run
3 – (-3) = 6
1.0
up
-4.0 -3.0 -2.0 -1.0
2-(-2)= 4
1.0 2.0
3.0 4.0
-1.0
-2.0
-3.0
See how the rise and run create a right triangle!
5.0
Since the rise and run are the legs of the right triangle
We can convert the Pythagorean
Theorem to
3.0
(rise)2 + (run)2 = (distance)2
6
2.0
1.0
4
-4.0 -3.0 -2.0 -1.0
-1.0
-2.0
42 +
62
-3.0
2
= (distance)
1.0 2.0
3.0 4.0
5.0
(rise)2 + (run)2 = (distance)2
42 + 62 = (distance)2
16 + 36 = d2
52 = d2
But, how do we find d?
By taking the square root of both sides.
d=
52  4 13  4  13  2 13
2 13 is what we call an exact answer
2 13
an exact answer
We may need to give an approximate answer. To do
That we will need to use our calculator. Scientific
Calculators, or the TI 83 has a square root button. If
You know how to use it, you can come up with an
approximate value for
2 13
You can also use the calculator found on your computer
By going to Start/Programs/Accessories/Calculator
Put in 52 then hit
Sqrt button. The
approximate answer is
shown on calculator.
Square Root button
Rounded to nearest tenth, the approximate answer is
7.2
Let’s find the distance between the points pictured
3.0
A (-2,2)
The rise is
-3 – 2 = -5
(down is
negative)
2.0
The run is 1 – (-2) = 3
Right is positive
1.0
-4.0 -3.0 -2.0 -1.0
1.0 2.0
-1.0
-2.0
-3.0
B (1,-3)
3.0 4.0
5.0
3.0
A (-2,2)
2.0
-5
3
1.0
-5
-4.0 -3.0 -2.0 -1.0
1.0 2.0
-1.0
(-5)2
+
(3)2
=
d2
-2.0
-3.0
B (1,-3)
3.0 4.0
5.0
(-5)2 + (3)2 = d2
25 + 9 = d2
34 = d2
34 = d
This is the exact value.
The approximate value rounded to the
nearest hundredth is 5.83
What you have learned:
How to plot or graph points on the Cartesian coordinate
system
How to find the rise
How to find the run
How to find the distance between any two points in the
Cartesian coordinate system.
We don’t need to view the points to find the
rise, run, or distance between them as long as
we have their coordinates.
Let’s create a formula for each of these
Let A = (x1,y1) and B = (x2,y2)
The rise from A to B is y2 - y1
The run from A to B is x2 - x1
The distance between any two points is
(distance)2 = (rise)2 + (run)2 or
D2 = (y2 - y1)2 + (x2 - x1)2
Find the rise, run, and distance between the points
A(-256, 340) and B(49, -82)
The rise from A to B is y2 - y1 or –82 – 340 = -422
The run from A to B is x2 - x1 or 49 – (-256)=305
D2 = (y2 - y1)2 + (x2 - x1)2
D2 = (-422)2 + (305)2
= 178084 +93025
D2 = 271109
D  271109
D  520.68
Now it’s time for you
to show what you
know.