Parametric Equations
Many of the equations we have dealt with are plane curves, y = f(x), but this definition is restrictive since it
excludes many useful graphs. A more useful definition of a plane curve:
A
defined on an interval.
is a set of ordered pairs
Here t is a parameter that allows us to view how the
are used to construct the graph of the plane curve.
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, where f and g are functions
and the
Parametric Equations
You decide to drive your remote controlled motorcyle, which travels at 3 ft/sec, off a 176 foot high building
with a 12 foot ramp. By using a parameter, t, we can describe the motion of the RC motorcycle falling from
the cliff.
The curve, C, consisting of all ordered pairs
on an interval I.
The equations, x =
and y =
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where f and g are defined
for t  I, are called
Eliminating a parameter
Eliminating the parameter allows us to view the curve in the traditional way.
For example: Let
and
the curve in rectangular coordinates
, remove the parameter to describe
Example: Remove the parameter to describe the curve given by,
and
Example: Describe the graph of a curve, C, that has the parametrization given by
and
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Path of a projectile
The path of a projectile at time t can be modeled using the parametric equations:
) t and
where, at t = 0,
• s is the speed of the projectile in (ft/sec or m/sec)
•
is the angle the path makes with the horizontal
• h is the height (in feet or meters)
• The acceleration gravity is
or we use
sec per sec)
(meters per
Example: A projectile is fired at a speed of 1200 ft/sec at an angle of
from the
horizontal from a height of 2200 feet.
a) What are the parametric equations for this projectile?
b) Find the range, r, of the projectile (the horizontal distance it travels before it hits the
ground)
c) find an equation for y in terms of x for the projectile
d) Find the point and time at which the projectile reaches its maximum altitude
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Parametric Equations
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