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
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