1. PARAMETRIC EQUATIONS Dr. Shildneck

2. Parametric Equations Most 2-Dimensional equations and graphs that we have dealt with involve two variables that are based on the relation of one variable, y, that depends on the value of a second variable, x. In some instances you might want to introduce a third variable to help represent a curve in a plane. To see the usefulness, consider a projectile that is launched into the air at a 45 degree angle. If the initial velocity of the object is 48 feet per second, the projectile’s path can be modeled by the equation However, this equation DOES NOT tell us anything about where the projectile is at a given time. To do this, we would have to introduce a third quantity into the formula (for time). This third variable, t, is what we call the parameter.

3. Parametric Equations The set of parametric equations for the previous situation is: To develop a set of points for the curve using the variable t, you plug in appropriate values of t for each equation. The coordinates of the curve are still (x, y). When plotted, this set of points creates a plane curve for the given situation.

4. Plane Curve If f and g are continuous functions of t on an interval, the set of pairs (x, y) = (f(t), g(t)) is a plane curve. The equations x = f(t) and y = g(t) are the parametric equations for the plane curve and t is the parameter.

5. Sketching Plane Curve To sketch the plane curve for a given parameter: 1.Make a table for all three variables 2.Plug in values of t (for the interval given, in order) 3.Create points (x, y) from the table 4.Plot the points and sketch the curve on the interval (in the appropriate direction – indicated with arrows)

6. Sketching Plane Curve t-20123 x y Example: Sketch the curve given by the parametric equations on the interval [-2, 3] (-3, 0)(-2, -3)(-1, -4)(0, -3)(1, 0)(2, 5) -3-2 0 1 2 -3 -4-3 0 0 5 Plug in to get x ’ s Plug in to get y ’ s So, now your points are… Now, plot and sketch…

7. Using the TI to Sketch a Plane Curve To sketch the plane curve for in a TI-83/84: 1.Change the [MODE] to PAR 2.Enter an equation for x and for y (based on t) 3.Set the t-value for your interval (or to an appropriate interval) 4.Press [Graph]

8. Using the TI to Sketch a Plane Curve Example: Graph the following on the interval [-4, 4].

9. Changing Forms – Parametric to Rectangular To change a parametric set into a rectangular equation. 1.Analyze the equation set (determine which equation is easiest to solve for the parameter or change the form and look for a way to combine if necessary) 2.Solve for the parameter in one equation. 3.Substitute that equation into the other equation. 4.Solve for y (if possible).

10. Changing Forms – Parametric to Rectangular Example: Eliminate the parameter.

11. Changing Forms – Parametric to Rectangular Example: Eliminate the parameter.

12. Changing Forms – Parametric to Rectangular Example: Eliminate the parameter.

13. Changing Forms – Rectangular to Parametric Infinitely many pairs of parametric equations can represent the same plane curve. If the plane curve is defined by the function y=f(x), one way to find a parametric set is: 1.Choose the x-equation to be x = t. (You can start with any choice here*). 2.Substitute the x equation for x in the rectangular function. 3.Simplify. *Warning: You must make sure that the x-equation you choose covers all possible values of the domain of the rectangular function.

14. Changing Forms – Rectangular to Parametric Example: Find two sets of parametric equations for the function.

15. Assignment 1 Blitzer P. 932 #1-39 odd, 40-44, 53,55