1. 3.6 – Parametric Equations Objectives Graph a pair of parametric equations, and use them to model real-world applications. Write the function represented by a pair of parametric equations. Standard: 2.5.11.A. Use appropriate mathematical techniques to solve non-routine problems.

2. What are Parametric equations? A pair of parametric equations is a pair of continuous functions that define the x- and y-coordinates of a point in a coordinate plane in terms of a third variable, such as t, called the parameter. To graph a pair of parametric equations, you can make a table of values or use a graphics calculator.

3. Example 1

4. Using the Graphing Calculator

5. Example 2 * Graph the pair of parametric equations for -3 ≤ x ≤3.

6. Example 3

7. Method 2

8. Example 4 * Write the pair of parametric equations as a single equation in x and y.

9. Example 5 An outfielder throws a baseball to the catcher 120 feet away to prevent a runner from scoring. The ball is released 6 feet above the ground with a horizontal speed of 70 feet per second and a vertical speed of 25 feet per second. The catcher holds his mitt 2 feet off the ground. The following parametric equations describe the path of the ball. a. When does the ball reach its greatest altitude? b. Can the catcher catch the ball?

10. Calculator Keystrokes: Mode: Go to y = Plug in the parametric equation Change the window to see the graph Finally, graph Set to Par for parametric

11. Now to solve the problem… When does the ball reach its greatest altitude? Can the catcher catch the ball? After.8 seconds, the ball is 56 ft away from the outfielder and is approximately 16 feet in the air. When the ball is 119 feet away from the outfielder, the ball is 2.26 feet above ground, so therefore, the catcher could catch this ball.

12. Example 6 An outfielder throws a baseball to the 3 rd baseman 290 feet away to prevent a runner from scoring. The ball is released 6 feet above the ground with a horizontal speed of 75 feet per second and a vertical speed of 44 feet per second. The 3 rd baseman holds his mitt 3 feet off the ground. The following parametric equations describe the path of the ball. a. When does the ball reach its greatest altitude? b. Can the 3 rd basemen catch the ball?

13. To solve: Plug in the equations the same way you used in example 5. Graph: When does the ball reach its greatest altitude? Can the 3 rd basemen catch the ball? After 1.4 seconds, the ball is 105 feet away from the outfielder, and approximately 36 feet off the ground. After 2.8 seconds, the ball is only 210 feet away from the outfielder and approximately 4 feet off the ground. Therefore, the 3 rd basemen will not be able to catch this ball from his current location.