1. Parametric Equations Greg Kelly, Hanford High School, Richland, Washington

2. There are times when we need to describe motion or path of a particle that may or may not be a function. We can do this by writing equations for the x and y coordinates in terms of a third variable (usually t or ). These are called parametric equations. “ t ” is the parameter. (It is also the independent variable) Think of “t” in terms of time (except here, “time” can be negative).

3. Example 1:

4. Hit zoom square to see the correct, undistorted curve. We can confirm this algebraically: parabolic function

5. Exploration 1 x = a∙cos(t)y = a∙sin(t) 1. Let a = 1. What does this graph look like by hand? 2. Let a = 2 and 3. Using your calculator, graph in a square viewing window. How does changing a affect the graph? 3. Let a = 2 and use the following parametric intervals: [0, π/2], [0, π], and [0, 4 π] Describe the role of the parameter interval.

6. Exploration 1 4. Let a = 3. Graph using the following intervals: [π/2, 3 π/2], [π, 2 π], [π, 5 π]. What are the initial and terminal points in each case? 5. Graph x = 2∙cos(–t) and y = 2∙sin(–t) using the parameter intervals [0, 2 π] and [π, 3 π]. Describe how the graphs are traced. What is the Cartesian equation for a curve that is represented parametrically by: x = 3cos(t)y = 3sin(t)

7. This is the equation of an ellipse.

8. General Parametric Equations x = acos(t)y = asin(t)  circle x = acos(t)y = bsin(t)  ellipse x = sec(t)y = tan(t)  hyperbola x = sin 3 (t)y = cos 3 (t)  sinusoid