Digital Lesson Parametric Equations The path of an

The path of an object thrown into the air at a 45° angle at

Example: Parametric equations y 18 (36, 18) 9 (72, 0) x 9 18 27

y (36, 18) 18 (72, 0) 9 (0, 0) t=0 9 18 27 36

Example: Sketch the curve given by x = t + 2 and y =

Graphing Utility: Sketch the curve given by x = t + 2 and y

Eliminating the parameter is a process for finding the rectangular equation (in x and

Example: Identify the curve represented by x = 2 t and by eliminating the

Example: Find a set of parametric equations to represent the graph of y =

Application: The center-field fence in a ballpark is 10 feet high and 400 feet

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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 Rectangular equation 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 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. 2

Example: Parametric equations y 18 (36, 18) 9 (72, 0) x 9 18 27 36 45 54 63 72 (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. 3

y (36, 18) 18 (72, 0) 9 (0, 0) t=0 9 18 27 36 45 54 63 72 x 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 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. parameter 4

Example: Sketch the curve given by x = t + 2 and y = t 2, – 3 t 3. t – 3 – 2 – 1 x – 1 0 1 y 9 4 1 0 2 0 1 3 1 2 4 4 3 5 9 y 8 orientation of the curve 4 x -4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 5

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 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) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7

Example: Identify the curve represented by x = 2 t and by eliminating the parameter. Solve for t in one equation. Substitute into the second equation. y 8 4 -4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 x 8

Example: Find a set of parametric equations to represent the graph of y = 4 x – 3. Use the parameter t = x. x=t Parametric equation for x. y = 4 t – 3 Substitute into the original rectangular equation. y 8 y = 4 t – 3 4 4 -4 x -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 = 145 t and y = 3 + 39 t – 16 t 2. Graph the path of the baseball. Is the hit a home run? y 25 The ball only traveled 364 feet and was not a home run. 20 15 10 (0, 3) (364, 0) 5 0 50 100 150 200 250 300 350 400 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. x Home Run 10