Graphing in Two Dimensions

ByDr. Julia Arnold

“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

A little background about thecreator of the coordinate system.

We begin with two number lines intersecting.

0 1 2 3 4 5 6 -6 -5 -4 -3 -2 -1

1

2

3

4

-1-2

-3

-4

0 1 2 3 4 5 6 -6 -5 -4 -3 -2 -1

1

2

3

4

-1-2

-3

-4

The horizontalLine is called theX-axis

x

The vertical line is calledthe y-axis. Y

0 1 2 3 4 5 6 -6 -5 -4 -3 -2 -1

1

2

3

4

-1-2

-3

-4

x

Y

Where the two lines cross isCalled the origin.

As you can see, there are fourQuadrants.

This is quadrant I.This is quadrant II.

This is quadrant III.

This is quadrant IV.

They are numbered counter-clockwise, beginning with the upperright corner. This numbering stays the same for whatever mathcourse you take.

0 1 2 3 4 5 6 -6 -5 -4 -3 -2 -1

1

2

3

4

-1-2

-3

-4

x

Y

To graph or plot a point you need two numbers, one to tell you how farright or left to go, and one to tell you how high or low to go.

We write the point as (x,y)And we call the x, the x-coordinate, and we call y, the y-coordinate.

The point (x,y) is called anordered pair of numbers,because the order matters.

-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0

-3.0

-2.0

-1.0

1.0

2.0

3.0

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

To find the point (2,3), beginat the origin, and, since 2 is inthe x-coordinate position,go to 2 on the x axis.

At 2, go straight up to 3 andDraw the dot.

(2,3)

-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0

-3.0

-2.0

-1.0

1.0

2.0

3.0 (2,3)

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

(3,2)

As you can see, they aredifferent points.

-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0

-3.0

-2.0

-1.0

1.0

2.0

3.0

As you click your mouse, points will appear on the screen.Write the ordered pair of numbers for that point beforeClicking again.

(-3,1)

(3,0)

(-4,-3)

(0,-2)

(2,-3)

The rise is the vertical change as you move from one point to another or below as we go from A to B.

A

B

This is theRise.

To go from A to B we move up which is positive.

The rise is the vertical change as you move from one point to another or below as we go from A to B.

A

B

This is theRise.

To go from A to B we move down which is negative.

-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0

-3.0

-2.0

-1.0

1.0

2.0

3.0

What is the rise going from A to B?

Point A

Point B

(-4,3)

(1,0)

Start withThe y-coordinate of Band subtract the y-coordinate of A0-3=-3

Going downis negative.

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.

A

B

This is therun.

Going to the right is positive.

The run is the horizontal change as you move from one point to another or below as we go from A to B.

A

B

This is therun.

Going left is negative.

-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0

-3.0

-2.0

-1.0

1.0

2.0

3.0

What is the run going from A to B?

Point A

Point B

(-4,3)

(1,0)

Start withThe x-coordinate of Band subtract the x-coordinate of A1-(-4)= 5

The run is 5

Going right is positive.

The distance between two numbers on theNumber 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) = 5If you subtract backwards --- -4 – 1 = -5 you get a negative numberbut 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, orthe vertical number line, the distance between themcan be found by subtracting and taking the absolutevalue.

As a formula , we would write for the followingPicture: b - a

a bOr for the following: x2 – x1

x1 x2

-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0

-3.0

-2.0

-1.0

1.0

2.0

3.0What is the distanceBetween the two points?

Since they are on theSame vertical line, Subtract.

3 – (-3) = 6

-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0

-3.0

-2.0

-1.0

1.0

2.0

3.0

We also want to beable to find the distance betweenany two points, such as..

To do this we turn to a famous theoremdiscovered by a man named Pythagoras.The theorem is called the Pythagorean Theorem

Born: about 569 BC in Samos, IoniaDied: 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.

In a right triangle,the legs are per-pendicular. Thus,a is perpendicularto b.

a

bca2 + b2 = c2

a

b or a

Only c a2 + b2 = c2

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

or b

RightAngle90o

-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0

-3.0

-2.0

-1.0

1.0

2.0

3.0

Now back to findingThe distance betweenThe two points.

The rise.

Then the run

See how the rise and run create a right triangle!

2-(-2)= 4

3 – (-3) = 6

up

right

-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0

-3.0

-2.0

-1.0

1.0

2.0

3.0

Since the rise and run are the legs of the right triangleWe can convert the Pythagorean Theorem to(rise)2 + (run)2 = (distance)2

6

42 + 62 = (distance)2

4

42 + 62 = (distance)2

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

16 + 36 = d2

52 = d2

But, how do we find d?

By taking the square root of both sides.

13213413452 132 is what we call an exact answer

d =

132 an exact answer

We may need to give an approximate answer. To doThat we will need to use our calculator. ScientificCalculators, or the TI 83 has a square root button. IfYou know how to use it, you can come up with an approximate value for

132

You can also use the calculator found on your computer By going to Start/Programs/Accessories/Calculator

Square Root button

Put in 52 then hitSqrt button. The approximate answer is shown on calculator.

Rounded to nearest tenth, the approximate answer is7.2

-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0

-3.0

-2.0

-1.0

1.0

2.0

3.0

Let’s find the distance between the points pictured

A (-2,2)

B (1,-3)

The rise is-3 – 2 = -5(down is negative)

The run is 1 – (-2) = 3 Right is positive

-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0

-3.0

-2.0

-1.0

1.0

2.0

3.0

A (-2,2)

B (1,-3)

-5

3

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

-5

(-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 therise, run, or distance between them as long aswe have their coordinates.

Let’s create a formula for each of theseLet 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 +93025D2 = 271109

68.520

271109

D

D

Now it’s time for youto show what you know.