1. Section 10.7 Parametric Equations
Copyright ©2013, 2009, 2006, 2005 Pearson Education, Inc.

2. Objectives Graph parametric equations.
Determine an equivalent rectangular equation for parametric equations.
Determine parametric equations for a rectangular equation.
Solve applied problems involving projectile motion.

3. Parametric Equations
If f and g are continuous functions of t on an interval I, then the set of ordered pairs (x, y) such that x = f (t) and y = g(t) is a plane curve. The equations x = f (t) and y = g(t) are parametric equations for the curve. The variable t is the parameter.

4. Example Graph the curve represented by the equations
Solution: Choose values of t between –3 and 3 and find the corresponding values of x and y.
When t = –3, we have
Let’s make a table of values.

5. Example (continued)

6. Example (continued)
The curve appears to be part of a parabola. Find the equivalent rectangular equation.
We must include the restrictions;

7. Example
Graph the plane curve represented by x = cos t and y = sin t, with t in [0, 2π]. Then determine an equivalent rectangular equation.
Solution: Using a squared window and a Tstep of π/48, we obtain the graph:
This appears to be a unit circle.

8. Example (continued)
The equivalent rectangular equation can be obtained by squaring both sides of each parametric equation.
Add the two equations together.
Now use the trigonometric identity sin2  + cos2  = 1.
As expected, this is an equation of the unit circle.

9. Example Find three sets of parametric equations for the parabola
Solution:
If x = t, then
If x = t – 3, then

10. Projectile Motion
The motion of an object that is propelled upward can be described with parametric equations. Such motion is called projectile motion. It can be shown using more advanced mathematics that, neglecting air resistance, the following equations describe the path of a projectile propelled upward at an angle  with the horizontal from a height h, in feet, at an initial speed v0, in feet per second:
We can use these equations to determine the location of the object at time t, in seconds.

11. Example
A baseball is thrown from a height of 6 ft with an initial speed of 100 ft/sec at an angle of 45º with the horizontal. a) Find parametric equations that give the position of the ball at time t, in seconds. b) Graph the plane curve represented by the equations found in part (a). c) Find the height of the ball after 1 sec, 2 sec, and 3 sec. d) Determine how long the ball is the air. e) Determine the horizontal distance that the ball travels. f) Find the maximum height of the ball.

12. Example (cont)
a. Substitute 6 for h, 100 for v0, and 45º for .

13. Example (cont) b)

14. Example (cont)
c) The height of the ball at time t is represented by y.
When t = 1,
When t = 2,
When t = 3,

15. Example (cont) d) The ball hits the ground when y = 0.
The negative value of t has no meaning. Thus the ball is in the air for about 4.5 sec.

16. Example (cont)
e) Since the ball is in the air for about 4.5 sec, the horizontal distance that it travels is given by
f) Find the maximum value of y, which occurs at the vertex of the parabola represented by y.

17. Cycloid
The path of a fixed point on the circumference of a circle as it rolls along a line is called a cycloid. For example, a point on the rim of a bicycle wheel traces a cycloid curve.

18. Cycloid The parametric equations of a cycloid are
where a is the radius of the circle that traces the curve and t is in radian measure.

19. Example Graph the cycloid described by the parametric equations
Solution: