Parametric Equations 10.6 Adapted by JMerrill, 2011

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 3rd 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.

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

Example: Parametric Equation Parametric equations y x 9 18 27 36 45 54 63 72 (36, 18) (72, 0) (0, 0) 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. Example: Parametric Equation

Example: Sketching a Plane Curve Sketch the curve given by x = t + 2 and y = t2, – 3  t  3. t – 3 – 2 – 1 1 2 3 x 4 5 y 9 y x -4 4 8 The (x,y) ordered pairs will graph exactly the same as they always have graphed. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Sketching a Plane Curve

Graphing Utility: Sketching a Curve Plane 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. Graphing Utility: Sketching a Curve Plane

Definition: Eliminating the Parameter Eliminating the parameter is a process for finding the rectangular equation (y =) of a curve represented by parametric equations. x = t + 2 y = t2 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) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Eliminating the Parameter

Example: Eliminating the Parameter Identify the curve represented by x = 2t and by eliminating the parameter. Solve for t in one equation. Substitute into the second equation. y x -4 4 8 The absolute value bars can be found in the Math menu--Num Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Eliminating the Parameter

Eliminating an Angle Parameter Sketch and identify the curve represented by x = 3cosθ, y = 4sinθ Solve for cosθ & sinθ: Use the identity cos2θ + sin2θ = 1 We have a vertical ellipse with a = 4 and b = 3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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

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

You Try Find a set of parametric equations given y = x2 x = t y = t2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Application: Parametric Equations 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? y 5 10 15 20 25 x 50 100 150 200 250 300 350 400 The ball only traveled 364 feet and was not a home run. (364, 0) (0, 3) Home Run Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Application: Parametric Equations