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| Coordinate Plane | The plane determined by two perpendicular axes. |
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| Coordinates | Each point in the coordinate plane corresponds to an ordered pair of numbers called its coordinates. | ||
| X-Coordinate | The first coordinate in an ordered pair. This represents the independent variable. | ||
| Y-Coordinate | The second coordinate in an ordered pair. This represents the dependent variable. | ||
| Ordered Pair | A set of two numbers, with the x-coordinate (independent variable) listed first, and the y-coordinate (dependent variable) listed second. | ||
| Quadrant | The x-axis and y-axis separate the plane into four regions called quadrants. They are numbered with Roman numerals, beginning in the upper right quadrant and proceeding in a counter-clockwise rotation. | ||
Representing Data Graphically
There is an old saying that "a picture is like a thousand words." This
can also be true in mathematics. A graph (mathematical picture) can be
more meaningful than a stack of numbers. Below, in a table is given some
random results for some of the older contestants in the 1997 Boston Marathon.
It is difficult to see any patterns just by observing the table. By thinking
of each row in the table as an ordered pair,
we can plot the points representing each contestant. The graph of all of
these points is on the right.
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| While we still do not have a clear mathematical equation or exact pattern, we now can see better how all of these points look together. We have a much better chance of seeing patterns with the graph. The type of graph shown above is called a scatter plot. It is simply a plotting of all of the points given. Scatter plots are great for trying to see a relationship between two different variables. (in this case age(x) and time(y). | |||||||||||||||||||||||
| Another thing we might wish to do is to just plot a bar graph of the times of the runners in the above table. Bar graphs are great for comparing just variable. (time only) |
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| Another way of representing only one variable is to use a circle graph or pie graph. |
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Given equation. | ||||||||||||
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Make a two column chart, so that we can find some ordered pairs. Pick any values for x, keeping in mind that it is desirable to pick zero, at least some positives and at least some negatives. This gives you better distribution on a graph grid. If the points are more widely distributed, then the graph is easier to draw and interpret. | ||||||||||||
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For each value of x picked, substitute this into the equation, and
find the value of y.
What we now have is five ordered pairs, which we can place on a graph. |
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Plot the ordered pairs from the table in the last step. |
The Distance Formula
We need a formula for finding the distance between any two points on
the coordinate plane. Below we develop this formula. You need to understand
this development, and you need to memorize this formula.
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We begin by picking two points in the plane. These points are picked at random. The grid is used so that you can see the development more clearly. We will call the lower point, point 1 and the higher point, point 2. |
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We first connect the two points we wish to find the distance between, we also construct a right triangle with the distance we wish to find as the hypotenuse of that right triangle. |
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We name the coordinates of the two given points using subscripts. What are the coordinates of the new point created when we made a right triangle? If you look closely at the left, all the needed information is there for you to answer this question. |
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Because this point is directly below point 2, its x-coordinate
must be  .
Because this point is directly across from point 1, its y-coordinate
must be  . |
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As we try to find a formula between the original two points, we will need to find a way to express the distances of the two legs (perpendicular sides of the triangle). What is the distance between the two bottom points? |
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The distance between these points would have to be the difference of
the two x-coordinates or  . |
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What is the distance between the two points which form the vertical side of the triangle? |
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The distance between these points would have to be the difference of
the two y-coordinates or  . |
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We finish our diagram by labeling the side which we want to find the length of, namely d. |
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Now, we use one of the most famous theorems in mathematics which applies
to right triangles. The Pythagorean
Theorem: The sum of the squares of the legs is equal to the square
of the hypotenuse. or  .
Our
"a" is
and our "b" is .
Our "c" is the d that we are looking for. |
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We then take the square root of both sides of the equation. This is using the Equal Rights Amendment for Algebra (ERAA) |
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We now have a formula for finding the distance between any two points in the coordinate plane. Memorize this formula. |
