11.10.notebook 1 April 20, 2016 Parametric Equations Section 11.10
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Parametric Equations
Section 11.10
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Introduction
• There are interesting curves which, when plotted in the xyplane
• neither represent y as a function of x • nor x as a function of y. • In this section, we present a new concept which allows
us to use functions to study these kinds of curves.
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Motivation • Imagine a bug crawling across a table top starting at
the point O and tracing out a curve C in the plane• The bug can be in only one place P(x,y) at any given
time t• We define • the xcoordinate of P as a function of t• The ycoordinate of P as a (usually) different function of
t.
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Parametrization • The independent variable t in this case is called a
parameter• The system of equations
is called a system of parametric equations or a parametrization of the curve C
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Orientation • The curve itself is a set of points and as such is devoid of
any orientation. • The parametrization determines the orientation• Different parametrizations can determine different
orientations. • The system of equations
• parametrizes the Unit Circle, giving it a counterclockwise orientation.
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Example • Sketch the curve described by
• for t ≥ 2
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Parameter Elimination • The curve sketched out in Example certainly looks like a
parabola,• We can use the technique of substitution as to eliminate
the parameter t and get an equation involving just x and y.
• If the parametrization involves the trigonometric functions, it is often best to solve for the trigonometric functions and relate them using an identity.
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Example • Sketch the curves described by the following
parametric equations.
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Parametrizations of Common Curves
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Example • Find a parametrization for each of the following curves
and check your answers.
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Adjusting Parametric Equations
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Find a parametrization for the following curvesThe curve which starts at (2,4) and follows the parabola y = x2 to end at (1,1). Shift the parameter so that the path starts at t=0.
The two part path which starts at (0,0), travels along a line to (3,4), then travels along a line to (5,0).
The Unit Circle, oriented clockwise, with t=0 corresponding to (0,1).
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Cycloid
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Example • Find the parametric equations of a cycloid which results
from a circle of radius 3 rolling down the positive xaxis as described above.
• Graph your answer using a calculator.
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