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Module 3.1 Graphing in Two Dimensions. By Dr. Julia Arnold. A little background about the creator of the coordinate system.

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slide1

Module 3.1

Graphing in Two Dimensions

By

Dr. Julia Arnold

slide2

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

slide3

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.

slide4

The vertical line is called

the y-axis.

Y

4

3

The horizontal

Line is called the

X-axis

2

1

x

-6 -5 -4 -3 -2 -1

0 1 2 3 4 5 6

-1

-2

-3

-4

slide5

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

Called the origin.

-2

This is quadrant III.

-3

This is quadrant IV.

-4

They are numbered counter-clockwise, beginning with the upper

right corner. This numbering stays the same for whatever math

course you take.

slide6

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,

the y-coordinate.

4

3

2

1

x

-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

slide7

This is how a coordinate system or graph would look with a grid.

(2,3)

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.

At 2, go straight up to 3 and

Draw the dot.

slide8

To emphasize that order matters, let’s now locate the point (3,2)

(2,3)

(3,2)

As you can see, they are

different points.

slide9

As you click your mouse, points will appear on the screen.

Write the ordered pair of numbers for that point before

Clicking again.

(-3,1)

(3,0)

(0,-2)

(2,-3)

(-4,-3)

slide10

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

slide11

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

slide12

What is the rise going from A to B?

(-4,3)

Point A

Start with

The y-coordinate of B

and subtract the y-

coordinate of A

0-3=-3

Going down

is negative.

(1,0)

Point B

The rise is -3

slide13

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

slide14

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

This is the

run.

A

slide15

What is the run going from A to B?

(-4,3)

Point A

Start with

The x-coordinate of B

and subtract the x-

coordinate of A

1-(-4)= 5

Going right is positive.

(1,0)

Point B

The run is 5

slide16

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.

slide17

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

b

Or for the following: x2 – x1

x1

x2

slide18

What is the distance

Between the two points?

Since they are on the

Same vertical line,

Subtract.

3 – (-3) = 6

slide19

We also want to be

able to find the

distance between

any two points, such as..

slide20

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

slide21

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.

In a right triangle,

the legs are per-

pendicular. Thus,

a is perpendicular

to b.

a2 + b2 = c2

c

b

a

slide22

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

a2 + b2 = c2

b or a

Right

Angle

90o

a

or b

slide23

Now back to finding

The distance between

The two points.

Then the run

3 – (-3) = 6

right

The rise.

up

2-(-2)= 4

See how the rise and run create a right triangle!

slide24

Since the rise and run are the legs of the right triangle

We can convert the Pythagorean Theorem to

(rise)2 + (run)2 = (distance)2

6

4

42 + 62 = (distance)2

slide25

(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 =

is what we call an exact answer

slide26

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

You can also use the calculator found on your computer

By going to Start/Programs/Accessories/Calculator

slide27

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

slide28

Let’s find the distance between the points pictured

A (-2,2)

The run is 1 – (-2) = 3

Right is positive

The rise is

-3 – 2 = -5

(down is

negative)

B (1,-3)

slide29

A (-2,2)

3

-5

-5

(-5)2 + (3)2 = d2

B (1,-3)

slide30

(-5)2 + (3)2 = d2

25 + 9 = d2

34 = d2

= d

This is the exact value.

The approximate value rounded to the

nearest hundredth is 5.83

slide31

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.

slide32

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

slide33

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

slide34

Now it’s time for you

To show what you

Know. Go to the HW

Problems for this lesson

In your PAN.