Parametric Equations

Greg Kelly, Hanford High School, Richland, Washington

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 ).

x f t y g t These are calledparametric equations.

“t” is the parameter. (It is also the independent variable)Think of “t” in terms of time (except here, “time” can benegative).

Example 1: 0x t y t t

Hit zoom square to see the correct, undistorted curve.

We can confirm this algebraically:

x t y t

x y

2x y 0x

2y x 0x

parabolic function

Exploration 1x = 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.

Exploration 14. 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 = 3•cos(t) y = 3•sin(t)

3cos 4sinx t y t

cos sin3 4

x yt t

2 22 2cos sin

3 4

x yt t

2 2

13 4

x y

This is the equation of an ellipse.

General Parametric Equations

• x = a•cos(t) y = a•sin(t) circle

• x = a•cos(t) y = b•sin(t) ellipse

• x = sec(t) y = tan(t) hyperbola

• x = sin3(t) y = cos3(t) sinusoid