Algebra/The Coordinate (Cartesian) Plane
A Quick Review...[edit]
What is the Cartesian Plane?
Named for "the father of analytical geometry," 17th century French mathematician René Descartes (Cartesius), uniform regular grid (Cartesian) coordinates is one system used for graphing. Many algebraic expressions lend themselves to graphical analysis. The location of a point's Cartesian plot is found by indexing numerical values (coordinates) along numbered grid lines. The trivial single R number line comprises a one dimensional, single ordinate, system with all locations existing only on that line. This study begins with graphing in two dimensions (see the two diagrams below). Plots and points are located and labeled with the offsets of their 'projections' from two number lines (R^{2} axes) anywhere on the page and plane.
We choose the values or ordinates for points on the Cartesian plane using two perpendicular numbered axes. By convention the point where the two axes cross is labeled as 0 on each axis, making the ordinate for their intersection a special point called the origin and labeled (0,0). Generally, the horizontal axis is labeled x and the vertical axis y. An ordered pair (x,y) specifies the location of a point P on the plane. If we don't want to talk about ordered pairs as x and y we can refer to the first variable in the ordered pair as the abscissa and the second as the ordinate. The axes cross where the abscissa is 0 and the ordinate is 0. We can generalize about the signs of the abscissa and ordinate in ordered pairs because the two axes form four quadrants when they cross. Moving counter clockwise, starting in the upper right quadrant, the quadrants are labeled I, II, III, and IV. In quadrant I, all x- and y-values are positive, in II, x is negative and y positive, in III, all values are negative, and only x is positive in IV. The ordered pair (0,0) represents the origin where the axes intersect.
We use the axes to define relationships between two sets of numbers for which we substitute variables. A relationship implies that changing the value of one variable determines the value of the other. We call the variable that is changing the independent variable and the variable whose value changes the dependent variable. Generally we let x represent the independent variable, y represent the dependent variable so that we can model the relationship between x and y with mathematical symbols. The set of numbers represented by the independent variable is called the domain. The set of numbers represented by the dependent variable is called the range. A special kind of relationship is called a function. A function is a relationship where any value in the domain maps onto one and only one value in the range.
When we graph the points of a relationship on a Cartesian plane then we can determine if the relationship is a function--all vertical lines of the plane cross our graph once and only once. Functions are useful for modeling cause and effect relationships - where the cause is the independent variable and the effect is the dependent variable.
Quadrant | x-axis (abscissa) | y-axis (ordinate) |
---|---|---|
I | Positive | Positive |
II | Negative | Positive |
III | Negative | Negative |
IV | Positive | Negative |
Note that the only thing unique about the Cartesian plane is that it contains a point that we call "the origin" and have labeled (0,0) If we draw another axis perpendicular to the two dimensional graph through the origin we can graph 3 dimensional objects (R^{3}). Imagine a thin extensible numbered wire through the origin of the two dimensional graph. By using a 3rd coordinate this line (X, Y, Z) we can locate points that are above or below the page in space. This idea can be expanded to higher dimensions (including time) and is the basis for a field of theoretical physics called [[w:String_theory| string theory
Determining Points[edit]
Continuous sets of points can be represented by lines or curves on a graph showing how continuous paired quantities relate to each other. Such numbers or quantities are called variables. A function, relation, or equation can define how, as an input quantity varies, a related output quantity can vary. A graph can then illustrate how these variables are related to each other. The independent variable is one which is sent to the function because a person can control or vary it at will. The dependent variable is the one that comes out of the function. The definition of the function determines the value of the dependent variable depending on the value of the independent variable.
Any equation which has two variables can effectively define a relation between the two variables and the relation can be plotted on a two-dimensional Cartesian coordinate graph. In any particular equation, relation, or function definition, numbers which stay the same, i. e. which are not variables, are often called constants. Even if someone does not know much fancy algebra, a person may start getting an idea of what a function or a two-variable equation or relation looks like in a graph by choosing various numbers for one of the variables from the domain and calculating corresponding numbers of the other variable (in the range) to determine as many ordered pairs as practical and plotting those points on a graph. After enough points are plotted, one may be able to estimate what a continuous relation looks like by connecting the calculated points by a straight or curved line, depending on the situation.
Exercise: For the function y = x^{2} - 1, set up a table so that you pick values for x (the independent value) and then calculate the value for y (the dependent variable)
x | y |
---|---|
0 | ? |
1 | ? |
-1 | ? |
2 | ? |
-2 | ? |
3 | ? |
-3 | ? |
? | ? |
? | ? |
Determine as many points as you feel comfortable to make a plot of this equation on a graph. Then on graph paper draw x and y axes on a grid of squares. Then plot the (x,y) ordered pairs as points on the graph. Finally, when you think you have plotted enough points to get a feel for the shape of the function, connect the points to see what the function graph looks like. If you are not sure in a part of the graph, you can always calculate more points to fill in that place.
Plotting Points on a Graph[edit]
A formula, equation or inequality that is solved for just a single instance of one variable is an implicit relation of, and dependent in, that variable in terms of the remaining and independent variable(s). An explicit relation is one noting which variable(s) is(are) independent. A point solution is a group of values (one for each independent variable) together with a corresponding calculated dependent value. Consensus-notation comma-separates these with the dependent variable's value last in an ordered group (parenthesis are optional iff there's no ambiguity). The independent variable's(s') valid-values solution set(s) constitutes the relation's domain and the dependent's its range. A function is a relation restricted to independent value(s) which generate unique dependent values.
A complete plot depicts all possible point solutions to a relation. 'One' independent variable plus 'the' dependent variable presents a 'two' dimensional map (graph) with 'two' perpendicular (orthogonal) number lines (axes) which cross (intercept) at the origin (O). By consensus the first coordinate relates to offset along the horizontal axis and the second to the vertical (an ordered pair).
This Function Graphing (section) uses a two-dimensional rectangular Cartesian coordinate system, the first coordinate is abscissa and commonly references an x-axis and the second is ordinate for a y-axis. The axes values are numbers and are elements of and closed to R and together constitute R^{2}. The origin point is the number pair 0,0. Positive numbers increment to the right and up and negative numbers decrement to the left and down. The notation x,y is the common form/formula of an ordered-pair point and is plotted on/in the following graph.
To remember the order of coordinates think of a roller coaster--"If you do it RIGHT, you might throw UP"; or if the graph were a street with tall buildings—you have to go along the street (RIGHT/LEFT) before you can enter and go UP the stairs; or you crawl (RIGHT/LEFT) before you walk (Stand UP).
The axes and plane field extend out as close to infinity (positive and negative) as is necessary for any analysis. In practice graphs are abbreviated to areas of examination. In our example the generic point x,y is assigned to number values (x = 3, y =4) by plotting it at point 3,4. A line and a dotted line show that this position is located by starting at (0,0), moving 3 units in the positive x direction (right), and moving 4 units in the positive y direction (upwards).