Parametric Equations
Think of a curve being traced out over time, sometimes doubling back
on itself or crossing itself. Such a curve cannot be described by a
function y = f(x). Instead, we will describe our position along the
curve at time t by
Then x and y are related to each other through their dependence on
the parameter t.
Example
Suppose we trace out a curve according to
| t | x | y |
| 0 | 0 | 0 |
| 1 | -3 | 3 |
| 2 | -4 | 6 |
| 3 | -3 | 9 |
| 4 | 0 | 12 |
where t ³ 0.
Drag the point along the curve and notice how x and y vary with t.
An arrow is often put on the curve to indicate the direction of increasing time or orientation of the curve.
The parameter does not always represent time.
Example
Consider the parametric equation
| x = |
3 cos(t) |
| y = |
3 sin(t). |
Here, the parameter t represents
the polar angle of the position on a circle of radius 3 centered
at the origin and oriented counterclockwise.
Differentiating Parametric Equations
Let x = x(t) and y = y(t).
Suppose for the moment that we are able to re-write this as y(t) = f(x(t)).
Then dy/dt = [dy/dx]·[dx/dt] by
the Chain Rule. Solving for dy/dx and
assuming dx/dt ¹ 0,
a formula that holds in general.
Example
If x = t2-3 and y = t8, then
dx/dt = 2t and dy/dt = 8t7. So
|
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= |
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= |
d
dx
|
é
ê
ë |
dy
dx
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ù
ú
û |
= |
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= |
24t5
2t
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= 12t4. |
|
Notes
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It is often possible to re-write the parametric equations
without the parameter. In the second example,
x/3 = cos(t),
y/3 = sin(t).
Since cos2(t)+sin2(t) = 1,
(x/3)2+(y/3)2 = 1.
Then x2+y2 = 9, which is the equation of a circle as
expected. When you do eliminate the parameter, always check that you have
not introduced extraneous portions of the curve.
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Every curve has infinitely many parametrizations, amounting to
different scales for the parameter. For example,
traces out the circle from the second example twice as "quickly,"
completing a full revolution in p rather
than 2p units of q.
-
Every equation y = f(x) may be re-written in parametric form by
letting x = t, y = f(t).
Key Concept
A curve in the xy-plane may be described by a pair of parametric equations
where x and y are related through their dependence on t. This is
particularly useful when neither x nor y is a function of the other.
The derivative of y with respect to x (in terms of the parameter t)
is given by
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