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Parametric Equations 10.6 Adapted by JMerrill, 2011

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Plane Curves Up to now, we have been representing graphs by a single equation in 2 variables. The y = equations tell us where an object (ball being thrown) has been. Now we will introduce a 3 rd variable, t (time) which is the parameter. It tells us when an object was at a given point on a path. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

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3 A pair of parametric equations are equations with both x and y written as functions of time, t. Definition: Parametric Equation Parametric equation for x Parametric equation for y t is the parameter. Rectangular equation The path of an object thrown into the air at a 45° angle at 48 feet per second can be represented by horizontal distance (x) vertical distance (y) Now the distances depend on the time, t.

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Example: Parametric Equation y x (0, 0) t = 0 (36, 18) (72, 0) two variables (x and y) for position one variable (t) for time Curvilinear motion: Example: Parametric equations

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Example: Sketching a Plane Curve Example: Sketch the curve given by x = t + 2 and y = t 2, – 3 t 3. t– 3– 2– x y y x The (x,y) ordered pairs will graph exactly the same as they always have graphed.

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Graphing Utility: Sketching a Curve Plane Graphing Utility: Sketch the curve given by x = t + 2 and y = t 2, – 3 t 3. Mode Menu: Set to parametric mode. WindowGraphTable

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Eliminating the parameter is a process for finding the rectangular equation (y =) of a curve represented by parametric equations. Definition: Eliminating the Parameter x = t + 2 y = t 2 Parametric equations t = x – 2 Solve for t in one equation. y = (x –2) 2 Substitute into the second equation. y = (x –2) 2 Equation of a parabola with the vertex at (2, 0)

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Example: Eliminating the Parameter Solve for t in one equation. Substitute into the second equation. Example: Identify the curve represented by x = 2t and by eliminating the parameter. y x The absolute value bars can be found in the Math menu--Num

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Eliminating an Angle Parameter Sketch and identify the curve represented by x = 3cosθ, y = 4sinθ Solve for cosθ & sinθ: Use the identity cos 2 θ + sin 2 θ = 1 We have a vertical ellipse with a = 4 and b = 3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9

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You Try Eliminate the parameter in the equations Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Example: Finding Parametric Equations Let x = t Substitute into the original rectangular equation. Writing Parametric Equations from Rectangular Equations Find a set of parametric equations to represent the graph of y = 4x – 3. x = t y = 4t – 3 x y y = 4t – 3

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You Try Find a set of parametric equations given y = x 2 x = t y = t 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Application: Parametric Equations Application: The center-field fence in a ballpark is 10 feet high and 400 feet from home plate. A baseball is hit at a point 3 feet above the ground and leaves the bat at a speed of 150 feet per second at an angle of 15 . The parametric equations for its path are x = 145t and y = t – 16t 2. Graph the path of the baseball. Is the hit a home run? Home Run (364, 0) y x The ball only traveled 364 feet and was not a home run. (0, 3)

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