1. Parametric Equations 10.6
Adapted by JMerrill, 2011

2. 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.

3. 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

4. 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

5. Example: Sketching a Plane Curve
Sketch the curve given by
x = t 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

6. Graphing Utility: Sketching a Curve Plane
Graphing Utility: Sketch the curve given by
x = t 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

7. 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

8. 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

9. 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.

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

11. 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

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

13. 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 = t – 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