Digital Lesson Parametric Equations The path of an object thrown into the air at a 45° angle at 48 feet per second can be represented by 2 y = −+x x. Rectangular equation 72 horizontal distance (x) vertical distance (y) A pair of parametric equations are equations with both x and y written as functions of a third variable such as time, t. Now the distances x = 24 2t Parametric equation for x depend on the 2 time, t. yt=−16 + 24 2 tParametric equation for y t is the parameter. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Example: 2 x = 24 2t y =−x +x Parametric 72 equations yt=−162 + 24 2 t t = 32 y 4 (36, 18) 18 t = 32 2 9 (72, 0) x (0, 0) 9 18 27 36 45 54 63 72 t = 0 two variables (x and y) for position Curvilinear motion: one variable (t) for time Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 t = 32 4 y (36, 18) t = 32 18 2 9 (72, 0) (0, 0) x t = 0 9 18 27 36 45 54 63 72 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) parameter parametric equations for C Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Example: Sketch the curve given by x = t + 2 and y = t2, – 3 ≤ t ≤ 3. t –3 –2 –1 0 1 2 3 x –1 0 1 2 3 4 5 y 9 4 1 0 1 4 9 y 8 orientation of the curve 4 x -4 4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Graphing Utility: Sketch the curve given by x = t + 2 and y = t2, – 3 ≤ t ≤ 3. Mode Menu: Set to parametric mode. Window Graph Table Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Eliminating the parameter is a process for finding the rectangular equation (in x and y) of a curve represented by parametric equations. x = t + 2 Parametric equations y = t2 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) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Example: Identify the curve represented by x = 2t and yt = − 2 by eliminating the parameter. t = x 2 Solve for t in one equation. y =−x 2 Substitute into the second equation. 2 y 8 4 y = x − 2 2 x -4 4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Example: Find a set of parametric equations to represent the graph of y = 4x – 3. Use the parameter t = x. x = t Parametric equation for x. y = 4t –3 Substitute into the original rectangular equation. y 8 y = 4t –3 4 x -4 4 -4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 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 –16t2. Graph the path of the baseball. Is the hit a home run? 25 y The ball only traveled 20 364 feet and was not a 15 home run. 10 (364, 0) 5 (0, 3) x 0 50 100 150 200 250 300 350 400 Home Run Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10.