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

We begin with two number lines intersecting

The horizontal Line is called the X-axis x The vertical line is called the y-axis. Y

x Y Where the two lines cross is Called the origin. As you can see, there are four Quadrants. This is quadrant I. This is quadrant II. This is quadrant III. 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.

x Y 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. 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 an ordered pair of numbers, because the order matters.

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

To emphasize that order matters, let’s now locate the point (3,2) (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) (-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 the Rise. 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 the Rise. To go from A to B we move down which is negative.

What is the rise going from A to B? Point A Point B (-4,3) (1,0) Start with The y- coordina te of B and subtract the y- coordinate of A 0-3=-3 Going down is 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 the run. 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 the run. Going left is negative.

What is the run going from A to B? Point A Point B (-4,3) (1,0) Start with The x- coordina te of B and subtract the x- coordinate of A 1-(-4)= 5 The run is 5 Going right is positive.

The distance between two numbers on the Number line is easy to compute 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 – 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:  x 2 – x 1  x1x1 x2x2

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 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)

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. a b c a 2 + b 2 = c 2

a b or a Only c a 2 + b 2 = c 2 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 Right Angle 90 o

Now back to finding The distance between The 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

Since the rise and run are the legs of the right triangle We can convert the Pythagorean Theorem to (rise) 2 + (run) 2 = (distance) = (distance) 2 4

(rise) 2 + (run) 2 = (distance) = d 2 52 = d 2 But, how do we find d? By taking the square root of both sides. is what we call an exact answer d =

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

Square Root button Put in 52 then hit Sqrt button. The approximate answer is shown on calculator. Rounded to nearest tenth, the approximate answer is 7.2

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

A (-2,2) B (1,-3) -5 3 (-5) 2 + (3) 2 = d 2 -5

(-5) 2 + (3) 2 = d = d 2 34 = d 2 = 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 = (x 1,y 1 ) and B = (x 2,y 2 ) The rise from A to B is y 2 - y 1 The run from A to B is x 2 - x 1 The distance between any two points is (distance) 2 = (rise) 2 + (run) 2 or D 2 = (y 2 - y 1 ) 2 + (x 2 - x 1 ) 2

Find the rise, run, and distance between the points A(-256, 340) and B(49, -82) The rise from A to B is y 2 - y 1 or –82 – 340 = -422 The run from A to B is x 2 - x 1 or 49 – (-256)=305 D 2 = (y 2 - y 1 ) 2 + (x 2 - x 1 ) 2 D 2 = (-422) 2 + (305) 2 = D 2 =

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