1. PARAMETRIC Q U A T I 0 N S Section 1.5 Day 2

2. Parametric Equations Example: The “parameter’’ is t. It does not appear in the graph of the curve!

3. Why? The x coordinates of points on the curve are given by a function. The y coordinates of points on the curve are given by a function.

4. Two Functions, One Curve? Yes. then in the xy-plane the curve looks like this, for values of t from 0 to 10... If

6. Why use parametric equations? Use them to describe curves in the plane when one function won’t do. Use them to describe paths.

7. Paths? A path is a curve, together with a journey traced along the curve.

8. Huh? When we write we might think of x as the x-coordinate of the position on the path at time t and y as the y-coordinate of the position on the path at time t.

9. From that point of view... The path described by is a particular route along the curve.

11. As t increases from 0, x first decreases, Path moves left! then increases. Path moves right!

12. The variable t (the parameter) often represents time. We can picture this like a particle moving along and we know its x position over time and its y position over time and we figure out each of these and plot them together to see the movement of the particle.

13. Graph the plane curve represented by the parametric equations We'll make a chart and choose some t values and find the corresponding x and y values. The t values we pick must be the values that are given to you We see the "path" of the particle. The orientation is the direction it would be moving over time (shown by the arrows) x = t + 1 y = t 2 + 2t Where t = -3, -2, -1, 0, 1, 2, 3 1 3 t -3 -2 0 2 -2 0 1 2 3 4 3 0 0 3 8 15 (-2, 3) (-1, 0) (0, -1) (1, 0) (2, 3) (3, 8) (4, 15)

14. Eliminate the Parameter Find an algebraic relationship between x and y. This is called “eliminating the parameter” x = t + 1 y = t 2 + 2t Graph your result

15. Compare the Graphs

16. Graph the plane curve represented by the parametric equations We'll make a chart and choose some t values and find the corresponding x and y values. t 0 The t values we pick must be greater than or equal to 0. Let's start with 0. 1 2 3 We see the "path" of the particle. The orientation is the direction it would be moving over time (shown by the arrows)

17. We could take these parametric equations and find an equivalent rectangular equation with substitution. This is called "eliminating the parameter." Solve for the parameter t in one of equations (whichever one is easier). Substitute for t in the other equation. 2 2 We recognize this as a parabola opening up. Since our domain for t started at 0, it is only the right half.

18. USE A GRAPHING CALCULATOR IN PARAMETRIC MODE 1.Put calculator in parametric mode 2.x = t 2 + 2t y = t + 1 3.Use the table to find the points determined by t = -3, -2, -1 0, 1, 2, 3 4.Set window to follow the t values t step:1 t min: -3 t max: 3

19. 5.Is y a function of x? 6.Find an algebraic relationship between x and y. Graph the rectangular equation and compare the graphs.

20. Practice Page 135 1 - 5