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From Equations to Graph
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Plot a rectangular-coordinate graph of the pair of parametric equations where x varies directly with et and y varies directly with t2 Then plot a polar-coordinate graph of the pair of parametric equations where q varies directly with et and r varies directly with t2 For simplicity, restrict the polar graph to values of t such that 0 q 2p
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The parametric equations for plotting the system in the rectangular xy plane are x = et and y = t2 Let s tabulate the x and y values for several points, based on various values of t: When t = 2, we have x = e 2 014 and y = ( 2)2 = 4 When t = 1, we have x = e 1 037 and y = ( 1)2 = 1 When t = 0, we have x = e0 = 1 and y = 02 = 0 When t = 1, we have x = e1 = e 272 and y = 12 = 1 When t = 3/2, we have x = e3/2 448 and y = (3/2)2 = 9/4 = 225 When t = 2, we have x = e2 739 and y = 22 = 4
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Figure 16-8 shows the graph we obtain by plotting the points in the xy plane This is a true Cartesian graph; the divisions on the x and y axes are the same size
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Figure 16-8
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Cartesian-coordinate graph of the parametric equations x = et and y = t2
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y 6 4 2 x 4 2 2 4 6 2 4 6 8
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Now let s tabulate some values for a polar graph We substitute q for x and r for y, input values of t to get a good sampling of polar angles in the output, and restrict t to keep q within the closed interval [0,2p] The situation breaks down into the following cases: When t = 2, we have q = e 2 014 and r = ( 2)2 = 4 When t = 1, we have q = e 1 037 and r = ( 1)2 = 1 When t = 0, we have q = e0 = 1 and r = 02 = 0 When t = 1, we have q = e1 = e 272 and r = 12 = 1 When t = 3/2, we have q = e3/2 448 and r = (3/2)2 = 9/4 = 225 When t = ln 2p, we have q = e(ln 2p ) = 2p and r = (ln 2p )2 338
Figure 16-9 shows the resulting curve in the polar plane If the above tabulation doesn t generate enough points to satisfy you, feel free to work out a few more As you gain experience in plotting graphs like this, you ll learn to get a sense of where the curves go without having to calculate very many discrete values
Each radial division
p /2
3p /2
is 1 unit
Figure 16-9
Polar-coordinate graph of the parametric equations q = et and r = t2 Each radial division represents 1 unit
From Graph to Equations
We ve seen how we can go from parametric equations to graphs Now we ll do an exercise going from a graph to a pair of parametric equations
From Graph to Equations
Cartesian-coordinate graph to equations Consider a circle of radius a, centered at the origin in the Cartesian xy plane as shown in Fig 16-10 From trigonometry, we remember that
x = a cos f and y = a sin f where f is the angle going counterclockwise from the positive x axis Both x and y depend on the value of f Let s rename f and call it t, so our equations become x = a cos t and y = a sin t This is a pair of parametric equations representing a circle of radius a, centered at the origin in the Cartesian xy plane For any particular circle, a is a constant (not a variable), so the parameter t is the only variable on the right-hand side of either equation
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