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If you multiply both x and y by 1, P will move diagonally to the opposite quadrant It will, in effect, be reflected by both axes If P starts out in the first quadrant, it will move to the third If P starts out in the second quadrant, it will move to the fourth If P starts out in the third quadrant, it will move to the first If P starts out in the fourth quadrant, it will move to the second
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If you have trouble envisioning these point maneuvers, draw a Cartesian plane on a piece of graph paper Then plot a point or two in each quadrant Calculate how the x and y values change when you multiply either or both of them by 1, and then plot the new points
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On a straight number line, the distance of any point from the origin is equal to the absolute value of the number corresponding to the point In the Cartesian plane, the distance of a point from the origin depends on both of the numbers in the point s ordered pair
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An example Figure 1-4 shows the point (4,3) plotted in the Cartesian plane Suppose that we want to find the distance d of (4,3) from the origin (0,0) How can this be done We can calculate d using the Pythagorean theorem from geometry In case you ve forgotten that principle, here s a refresher Suppose we have a right triangle defined by points P, Q, and R Suppose the sides of the triangle have lengths b, h, and d as shown in Fig 1-5 Then
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b2 + h2 = d 2 We can rewrite this as d = (b 2 + h 2)1/2 where the 1/2 power represents the nonnegative square root Now let s make the following point assignments between the situations of Figs 1-4 and 1-5: The origin in Fig 1-4 corresponds to the point Q in Fig 1-5 The point (4,0) in Fig 1-4 corresponds to the point R in Fig 1-5 The point (4,3) in Fig 1-4 corresponds to the point P in Fig 1-5 Continuing with this analogy, we can see the following facts: The line segment connecting the origin and (4,0) has length b = 4 The line segment connecting (4,0) and (4,3) has height h = 3 The line segment connecting the origin and (4,3) has length d (unknown)
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What s the distance d
2 4 6
(4, 0)
Figure 1-4
We can use the Pythagorean theorem to find the distance d of the point (4,3) from the origin (0,0) in the Cartesian plane
The side of the right triangle having length d is the longest side, called the hypotenuse Using the Pythagorean formula, we can calculate d = (b 2 + h 2)1/2 = (42 + 32)1/2 = (16 + 9)1/2 = 251/2 = 5 We ve determined that the point (4,3) is 5 units distant from the origin in Cartesian coordinates, as measured along a straight line connecting (4,3) and the origin
Figure 1-5
The Pythagorean theorem for right triangles
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The general formula We can generalize the previous example to get a formula for the distance of any point from the origin in the Cartesian plane In fact, we can repeat the explanation of the previous example almost verbatim, only with a few substitutions Consider a point P with coordinates (xp,yp) We want to calculate the straight-line distance d of the point P from the origin (0,0), as shown in Fig 1-6 Once again, we use the Pythagorean theorem Turn back to Fig 1-5 and follow along by comparing with Fig 1-6:
The origin in Fig 1-6 corresponds to the point Q in Fig 1-5 The point (xp,0) in Fig 1-6 corresponds to the point R in Fig 1-5 The point (xp,yp) in Fig 1-6 corresponds to the point P in Fig 1-5 The following facts are also visually evident: The line segment connecting the origin and (xp,0) has length b = xp The line segment connecting (xp,0) and (xp,yp) has height h = yp The line segment connecting the origin and (xp,yp) has length d (unknown) The Pythagorean formula tells us that d = (b 2 + h 2)1/2 = (xp2 + yp2)1/2
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