Module 3.1 Graphing in Two Dimensions

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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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Module 3.1

Graphing in Two Dimensions

By

Dr. Julia Arnold

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

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

As you can see, there are four

Y

4

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

-3

-4

They are numbered counter-clockwise, beginning with the upper

right corner. This numbering stays the same for whatever math

course you take.

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

(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.

(2,3)

(3,2)

As you can see, they are

different points.

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)

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?

(-4,3)

Point A

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

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

This is the

run.

A

What is the run going from A to B?

(-4,3)

Point A

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

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

b

Or for the following: x2 – x1

x1

x2

What is the distance

Between the two points?

Since they are on the

Same vertical line,

Subtract.

3 – (-3) = 6

We also want to be

able to find the

distance between

any two points, such as..

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

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.

In a right triangle,

the legs are per-

pendicular. Thus,

a is perpendicular

to b.

a2 + b2 = c2

c

b

a

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

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!

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

(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

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

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

A (-2,2)

The run is 1 – (-2) = 3

Right is positive

The rise is

-3 – 2 = -5

(down is

negative)

B (1,-3)

A (-2,2)

3

-5

-5

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

B (1,-3)

(-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

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

Now it’s time for you

To show what you

Know. Go to the HW

Problems for this lesson