Parametric Equations Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 A pair of parametric equations are equations.

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1 Parametric Equations Digital Lesson

2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 A pair of parametric equations are equations with both x and y written as functions of a third variable such as 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.

3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Example: Parametric Equation y x 9 18 9 273645546372 (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

4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Definition: Plane Curve y x 9 18 9 273645546372 (0, 0) t = 0 (36, 18) (72, 0) If f and g are continuous functions of t, the set of ordered pairs (f(t), g(t)) is the plane curve, C. x = f(t) and y = g(t) parametric equations for C parameter

5 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– 10123 x 012345 y9410149 y x -4 4 4 8 orientation of the curve

6 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

7 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Eliminating the parameter is a process for finding the rectangular equation (in x and 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)

8 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 -4 4 4 8

9 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Example: Finding Parametric Equations Parametric equation for x. Substitute into the original rectangular equation. Example: Find a set of parametric equations to represent the graph of y = 4x – 3. Use the parameter t = x. x = t y = 4t – 3 x y -4 4 4 8 y = 4t – 3

10 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 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 = 3 + 39t – 16t 2. Graph the path of the baseball. Is the hit a home run? Home Run (364, 0) y 5 10 0 15 20 25 x 50100150200250300350400 The ball only traveled 364 feet and was not a home run. (0, 3)