1. Start Up Day 51

2. PARAMETRIC EQUATIONS
OBJECTIVES: SWBAT demonstrate an understanding of how to graph a curve in the plane to represent parametric equations and indicating the direction of motion.
SWBAT to convert from a parametric to a rectangular equation and vice versa. .
ESSENTIAL QUESTIONS: What are parametric equations? How do we graph them? What does it mean to “eliminate” the parameter? What is meant by “the orientation” of the curve?
HOME LEARNING: “Spring Training” Calculator Lab + p. 482 #5, 6, 10, 13, 15, 22, 23, 27, 37 & 43

3. Plane Curves
In parametric equations, x and y are continuous functions of t and the resulting path is a plane curve. (Recall that a continuous function is one whose graph can be traced without lifting the pencil from the paper.)

4. How do we sketch a Plane Curve?
When sketching a curve represented by a pair of parametric equations, you still plot points in the xy-plane.
Each set of coordinates (x, y) is determined from a value chosen for the parameter t.
Plotting the resulting points in the order of increasing values of t traces the curve in a specific direction. This is called the ORIENTATION of the curve.

5. Example 1 – Sketching a Curve
Sketch the curve given by the parametric equations
x = t 2 – 4 and y = , –2  t  3.
Solution:
Using values of t in the specified interval, the parametric equations yield the points (x, y) shown in the table.

6. Example 1 – Solution
cont’d
By plotting these points in the order of increasing t, you obtain the curve C shown below.
Note that the arrows on the curve indicate its orientation as t increases from –2 to 3.
So, if a particle were moving on this curve, it would start at (0, –1) and then move along the curve to the point
Figure 10.53

7. Sketching a Plane Curve
Note that the graph does not define y as a function of x. This points out one benefit of parametric equations—they can be used to represent graphs that are more general than graphs of functions.
It often happens that two different sets of parametric equations have the same graph. For example, the set of parametric equations
x = 4t 2 – 4 and y = t, –1  t 
has the same graph as the set given in Example 1.

8. Sketching a Plane Curve
However, by comparing the values of t, you can see that this second graph is traced out more rapidly (considering t as time) than the first graph.
Figure 10.53

9. Eliminating the Parameter –Ex. 2
Example 1 uses simple point plotting to sketch the curve. This tedious process can sometimes be simplified by finding a rectangular equation (in x and y) that has the same graph. This process is called eliminating the parameter.
x = t 2 – t = 2y x = (2y) 2 – x = 4y 2 – 4
y =

10. Ex. 3--Let’s try it! Eliminating the Parameter
Sketch the curve represented by the equations
and
by eliminating the parameter and adjusting the domain of the resulting rectangular equation.
Solving for t in the equation for x produces
which implies that

11. Solution
cont’d

12. Parametric MODE on your TI
Set the mode to PARametric (and Radian, PLEASE)
2) Choose Y=, and enter your equations for x(t) and y(t)
3) Select the viewing window. ZOOM (5) Square is a good one for this graph. But if you need to set a specific window, In addition to setting Xmin , Xmax, Xscl, etc. set minimum and maximum values for the parameter t and an increment setting for t (Tstep).
Set Drawing cursor to Animated: -0 {the moving cursor shows the orientation of the curve!
5) Graph x=2cost, y = sint
(Any ideas as to what type of curve it will be?)
(The smaller the t step, the smoother the curve!)

13. Example 4: Eliminating the Parameter— A Conic Connection
Eliminate the parameter: x= 2 cost, y = sint
(sometimes it is not possible to isolate the “t” or “theta”)
Think of our basic Pythagorean identity:
Instead of solving for “t”, just solve for “cos t”
Substitute our specific values in for cos t and sin t:
Finally, rewrite your ELLIPSE in standard form:

14. Example 4: Eliminating the Parameter & identify the graph of the parametric curve

15. This vector equation is equivalent to the parametric equations:
This vector equation is equivalent to the parametric equations:
x = t and y = 3 + 3t.

16. Sketching a Plane Curve—Example 9
Different parametric representations can be used to represent various speeds at which objects travel along a given path.
The basic parametric equations to model projectile motion are as follows:
Where “h” is the starting height or distance from the ground, g is the acceleration due to gravity (32 ft/sec2 or 9.8 m/sec2), and v0 is the initial velocity.

17. Start Up Day 52
Sketch the plane curve, indicate the orientation & then eliminate the parameter: