Cartesian Coordinates Help (page 2)
Introduction to Cartesian Coordinates—Number Lines
Trigonometry involves angles and their relationships to distances. All of these relationships arise from the characteristics of a circle, and can be defined on the basis of the graph of a circle in the Cartesian plane.
The Cartesian plane, also called the rectangular coordinate plane or rectangular coordinates, consists of two number lines that intersect at a right angle. This makes it possible to graph equations that relate one variable to another. Most such graphs look like lines or curves.
Two Perpendicular Number Lines
Figure 1-1 illustrates the simplest possible set of rectangular coordinates. Both number lines have uniform increments. That is, the points on the axes that represent consecutive integers are always the same distance apart. The two number lines intersect at their zero points. The horizontal (or east/west) axis is called the x axis; the vertical (or north/south) axis is called the y axis.
Ordered Pairs, Abscissa, and Ordinate
Figure 1-2 shows two points plotted in rectangular coordinates. Points are denoted as ordered pairs in the form ( x , y ) in which the first number represents the value on the x axis and the second number represents the value on the y axis. The word “ordered” means that the order in which the numbers are listed is important. For example, the ordered pair (3.5,5.0) is not the same as the ordered pair (5.0,3.5), even though both pairs contain the same two numbers.
In ordered-pair notation, there is no space after the comma, as there is in the notation of a set or sequence. When denoting an ordered pair, it is customary to place the two numbers or variables together right up against the comma.
In Fig. 1-2, two points are shown, one with an abscissa of 3.5 and an ordinate of 5.0, and the other with an abscissa of –5.2 and an ordinate of –4.7.
Abscissa And Ordinate
In most sets of coordinates where the axes are labeled x and y , the variable y is called the dependent variable (because its value “depends” on the value of x ), and the variable x is called the independent variable . The independent-variable coordinate (usually x ) of a point on the Cartesian plane is called the abscissa , and the dependent-variable coordinate (usually y) is called the ordinate . The point (0,0) is called the origin .
Relations and Functions
Mathematical relationships, technically called relations , between two variables x and y can be written in such a way that y is expressed in terms of x . The following are some examples of relations denoted in this form:
y = 5
y = x + 1
y = 2 x
y = x 2
Some Simple Graphs
Figure 1-3 shows how the graphs of the above equations look on the Cartesian plane. Mathematicians and scientists call such graphs curves , even if they are straight lines.
The graph of y = 5 (curve A) is a horizontal line passing through the point (0,5) on the y axis. The graph of y = x + 1 (curve B) is a straight line that ramps upward at a 45° angle (from left to right) and passes through the point (0,1) on the y axis. The graph of y = 2x (curve C) is a straight line that ramps upward more steeply, and that passes through the origin. The graph of y = x 2 (curve D) is known as a parabola. In this case the parabola rests on the origin, opens upward, and is symmetrical with respect to the y axis.
Relations Vs. Functions
All of the relations graphed in Fig. 1-3 have something in common. For every abscissa, each relation contains at most one ordinate. Never does a curve portray two or more ordinates for a single abscissa, although one of them (the parabola, curve D) has two abscissas for all positive ordinates.
A mathematical relation in which every abscissa corresponds to at most one ordinate is called a function . All of the curves shown in Fig. 1-3 are graphs of functions of y in terms of x . In addition, curves A, B, and C show functions of x in terms of y (if we want to “go non-standard” and consider y as the independent variable and x as the dependent variable).
Curve D does not represent a function of x in terms of y. If x is considered the dependent variable, then there are some values of y (that is, some abscissas) for which there exist two values of x (ordinates).
The Cartesian Plane Practice Problems
Suppose a certain relation has a graph that looks like a circle. Is this a function of y in terms of x ? Is it a function of x in terms of y?
The answer is no in both cases. Figure 1-4 shows why. A simple visual “test” to determine whether or not a given relation is a function is to imagine an infinitely long, straight line parallel to the dependent-variable axis, and that can be moved back and forth. If the curve ever intersects the line at more than one point, then the curve is not a function.
A “vertical line” (parallel to the y axis) test can be used to determine whether or not the circle is a function of the form y = f ( x ), meaning “y is a function of x .” Obviously, the answer is no, because there are some positions of the line for which the line intersects the circle at two points.
A “horizontal line” (parallel to the x axis) test can be used to determine if the circle is a function of the form x = f ( y ), meaning “ x is a function of y.” Again the answer is no; there are some positions of the line for which the line intersects the circle twice.
How could the circle as shown in Fig. 1-4 be modified to become a function of y in terms of x ?
Part of the circle must be removed, such that the resulting curve passes the “vertical line” test. For example, either the upper or the lower semicircle can be taken away, and the resulting graph will denote y as a function of x . But these are not the only ways to modify the circle to get a graph of a function. There are infinitely many ways in which the circle can be partially removed or broken up in order to get a graph of a function. Use your imagination!
Practice Problems for these concepts can be found at: The Circle Model Practice Test
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