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produces no output, that s okay) The Cartesian plane gives us an excellent way to illustrate relations and functions
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The axes In a Cartesian plane, both axes are linear, and both axes are graduated in increments of the same size On either axis, the change in value is always directly proportional to the physical displacement For example, if we travel 5 millimeters along an axis and the value changes by 1 unit, then that fact is true everywhere along that axis, and it s also true everywhere along the other axis The quadrants Any pair of intersecting lines divides a plane into four parts In the Cartesian system, these parts are called quadrants, as shown in Fig 1-3:
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In the first quadrant, both variables are positive In the second quadrant, the independent variable is negative and the dependent variable is positive In the third quadrant, both variables are negative In the fourth quadrant, the independent variable is positive and the dependent variable is negative
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Second quadrant
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First quadrant x
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Figure 1-3
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The Cartesian plane is divided into quadrants The first, second, third, and fourth quadrants are sometimes labeled I, II, III, and IV, respectively
How It s Assembled
The quadrants are sometimes labeled with Roman numerals, so that Quadrant I is at the upper right Quadrant II is at the upper left Quadrant III is at the lower left Quadrant IV is at the lower right
If a point lies on one of the axes or at the origin, then it is not in any quadrant
Are you confused
Why do we insist that the increments be the same size on both axes in a Cartesian two-space graph The answer is simple: That s how the Cartesian plane is defined! But there are other types of coordinate systems in which this exactness is not required In a more generalized system called rectangular coordinates or the rectangular coordinate plane, the two axes can be graduated in divisions of different size For example, the value on one axis might change by 1 unit for every 5 millimeters, while the value on the other axis changes by 1 unit for every 10 millimeters
Here s a challenge!
Imagine an ordered pair (x,y), where both variables are nonzero real numbers Suppose that you ve plotted a point (call it P) on the Cartesian plane Because x 0 and y 0, the point P does not lie on either axis What will happen to the location of P if you multiply x by 1 and leave y the same If you multiply y by 1 and leave x the same If you multiply both x and y by 1
Solution
If you multiply x by 1 and do not change the value of y, P will move to the opposite side of the y axis, but will stay the same distance away from that axis The point will, in effect, be reflected by the y axis, moving to the left if x is positive to begin with, and to the right if x is negative to begin with If P starts out in the first quadrant, it will move to the second If P starts out in the second quadrant, it will move to the first If P starts out in the third quadrant, it will move to the fourth If P starts out in the fourth quadrant, it will move to the third
If you multiply y by 1 and leave x unchanged, P will move to the opposite side of the x axis, but will stay the same distance away from that axis In a sense, P will be reflected by the x axis, moving straight downward if y is initially positive and straight upward if y is initially negative If P starts out in the first quadrant, it will move to the fourth If P starts out in the second quadrant, it will move to the third If P starts out in the third quadrant, it will move to the second If P starts out in the fourth quadrant, it will move to the first
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