From Equations to Graph

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Parametric equations allow us to define complicated curves in an elegant, and often simpler, way than we can do with ordinary equations Let s look at a couple of examples, and plot their graphs in rectangular and polar coordinates

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Rectangular-coordinate graph: example 1 Suppose that x varies directly with the square of t, and y varies directly with the cube of t In this situation, we have the parametric equations

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x = t2 and y = t3

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From Equations to Graph

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Let s construct a graph of this equation by inputting several values of t to the system and then plotting the points We can break the situation down as follows: When t = 4, we have x = ( 4)2 = 16 and y = ( 4)3 = 64 When t = 3, we have x = ( 3)2 = 9 and y = ( 3)3 = 27 When t = 2, we have x = ( 2)2 = 4 and y = ( 2)3 = 8 When t = 1, we have x = ( 1)2 = 1 and y = ( 1)3 = 1 When t = 0, we have x = 02 = 0 and y = 03 = 0 When t = 1, we have x = 12 = 1 and y = 13 = 1 When t = 2, we have x = 22 = 4 and y = 23 = 8 When t = 3, we have x = 32 = 9 and y = 33 = 27 When t = 4, we have x = 42 = 16 and y = 43 = 64

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We can plot the points for these nine xy-plane coordinates and then connect them by curve fitting to get the graph of Fig 16-4 To keep the picture clean, the points aren t labeled In this illustration, we have a rectangular-coordinate graph, but not a true Cartesian graph That s because the divisions on the y axis represent different increments than those on the x axis The result is a curve that s vertically squashed compared to the way it would look if plotted on a true Cartesian coordinate grid, but we can fit more of the curve into the available space

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Polar-coordinate graph: example 1 Figure 16-5 illustrates what happens when we substitute q for x and r for y in the above example, and then graph the result in polar coordinates For simplicity, let s restrict the graph

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Figure 16-4

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Rectangularcoordinate graph of the parametric equations x = t2 and y = t3

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x 16 8 30 8 16

Parametric Equations in Two-Space

Each radial division

p /2

3p /2

is 3 units

Figure 16-5

Polar-coordinate graph of the parametric equations q = t2 and r = t3 Each radial division represents 3 units

to values of t such that 0 t (2p)1/2 To keep the picture clean, we won t label any of the points The situation breaks down as follows: When t = 0, we have q = 02 = 0 and r = 03 = 0 When t = 1, we have q = 12 = 1 and r = 13 = 1 When t = p 1/2, we have q = (p 1/2)2 = p and r = (p 1/2)3 = p 3/2 557 When t = 2, we have q = 22 = 4 and r = 23 = 8 When t = 51/2, we have q = (51/2)2 = 5 and r = (51/2)3 = 53/2 1118 When t = (2p )1/2, we have q = [(2p )1/2]2 = 2p and r = [(2p )1/2]3 = (2p )3/2 1575

Rectangular-coordinate graph: example 2 Suppose that x varies inversely with t, and y varies directly with ln t The parametric equations are

x = t 1 and y = ln t

From Equations to Graph

2 1 x 8 4 1 2 4 8

Figure 16-6