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

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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.

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

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

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

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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

4 y = x − 2 2 x -4 4

x = t Parametric equation for x. y = 4t –3 Substitute into the original rectangular equation. y

8 y = 4t –3

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