10.6 Parametrics

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10.6 Parametrics
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Objective

• To evaluate sets of parametric equations for
  given values of the parameter.
• To sketch curves that are represented by sets of
  parametric equations
• To rewrite sets of parametric equations as single
  rectangular equations by eliminating the
  parameter.

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• Suppose you were running around an elliptical
  shaped track.
• You might be following the elliptical path
  modeled by the equation

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• This equation only shows you where you are, it
  doesnt show you when you are at a given point
  (x, y) on the track. To determine this time, we
  introduce a third variable t, called a parameter.
  We can write both x and y as functions of t to
  obtain parametric equations.

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Definition of a Plane Curve

• If f and g are continuous functions of t on an
  interval I, the set of ordered pairs
• (f(t), g(t)) is a plane curve C. The equations
• x f(t) and y g(t)
• are parameter equations for C, and t is the
  parameter.

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• A parameterization of a curve consists of the
  parametric equation and the interval of t-values.
• Time is often the parameter in a problem
  situation, which is why we normally use t for the
  parameter

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• Sometimes parametric equations are used by
  companies in their design plans. It is easier
  for the company to make larger and smaller
  objects efficiently by simply changing the
  parameter t.

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Sketching a Plane Curve

• When sketching a curve represented by a pair of
  parametric equations, you still plot points in
  the xy-plane.
• Each set of coordinates (x, y) is determined from
  a value chosen for the parameter t.

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Example Sketching a Plane Curve
Example 1 Sketch the curve given by x t 2
and y t2, 3 ? t ? 3.
t 3 2 1 0 1 2 3
x 1 0 1 2 3 4 5
y 9 4 1 0 1 4 9
orientation of the curve
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Graphing Utility Sketching a Curve Plane
Graphing Utility Sketch the curve given by x
t 2 and y t2, 3 ? t ? 3.
Mode Menu
Set to parametric mode.
Window
Graph
Table
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Definition Eliminating the Parameter
Eliminating the parameter is a process for
finding the rectangular equation (in x and y) of
a curve represented by parametric equations.
x t 2 y t2
Parametric equations
t x 2
Solve for t in one equation.
y (x 2)2
Substitute into the second equation.
y (x 2)2
Equation of a parabola with the vertex at (2, 0)
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Example Eliminating the Parameter
Solve for t in one equation.
Substitute into the second equation.
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Example Finding Parametric Equations
Example 3
Find a set of parametric equations to represent
the graph of y 4x 3. Use the parameter t x.
x t
Parametric equation for x.
y 4t 3
Substitute into the original rectangular equation.
y 4t 3
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Example

• Use the parameter t 2 x in the previous
  example.

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Parametric Conics

• The use of two of the three Pythagorean
  Trigonometric Identities allow for easy
  parametric representation on ellipses,
  hyperbolas, and circles.

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Pythagorean Identities
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Circles

• Compare the standard form of a circle with the
  1st Pythagorean Identity
• Standard form

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Change the equation so that it equals one
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Pythagorean Identity
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Using simple substitutions
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Solving for x and y
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Example 4

• Graph

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• Set calculator Mode Parametric

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Window
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Examples 5

• Graph

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Example 7

• Sketch the curve represented by
• Eliminating the parameter.

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Example 8

• The motion of a projectile at time t (in seconds)
  is given by the parametric equations
• Where x(t) gives the horizontal position of the
  projectile in feet and y(t) gives the vertical
  position of the projectile in feet.

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a. Find the vertical and horizontal position of
the projectile when t 2

• x 50, y 6

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b. At what time will the projectile hit the
ground?
The ball will hit the ground between t 2.16 and
t 2.18. Notice y goes from positive to
negative.
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Example

• The parametric equations below represent the hawk
  and dove populations at time t, where t is
  measured in years.

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a. Use your calculator in function mode to graph
the hawk and dove populations over time.
Dove
Hawk
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b. Find the maximum and minimum values for each
population.

• Hawk minimum 10 maximum 30
• Dove minimum 50 maximum 250

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c. Now using Parametric mode on your calculator,
graph the hawk population versus the dove
As the hawk population increases, the dove
populations decreases, followed by a decrease in
hawk population and a decrease in the dove
population.
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d. Using the parametric graph, find the
population of hawks and doves after one year.

• Dove population is 250, hawk population is 20

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e. When will the population of hawks reach its
maximum value and what is that value?

• Hawk population will be 30 at year 2.

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Example 9The complete graph of the parametric
equations x 2cos t and y 2 sin t is the
circle of radius 2 centered at the origin. Find
an interval of values for t so that the graph is
the given portion of the circle.

• A) the portion in the first quadrant. (0, p/2)
• B) the portion above the x-axis. (0, p)
• C) the portion to the left of the y-axis
• (p/2, 3p/2)

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Example 10 Ron is on a Ferris wheel of radius 35
ft that turns councterclockwise at the rate of
one revolution every 12 seconds. The lowest
point of the Ferris sheel is 15 feet above ground
level at the point, (0, 15) on a rectangular
coordinate system. Find parametric equations for
the position of Ron as a function of time t in
seconds if the Ferris wheel starts with Ron at
the point (35, 50)
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Example 11 Al and Betty are on a Ferris wheel.
The wheel has a radius of 15 feet and its center
is 20 feet above the ground. How high are Al and
Betty ath the 3 oclock position? At the 12
oclock position? At the 9 oclock position?
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Example 12A dart is thrown upward with an
initial velocity of 58 ft/sec at an angle of
elevation of 41. Find the parametric equations
that model the problem situation. Whne will the
dart hit the ground? Find the maximum height of
the dart. When will this occur?
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The dart will hit the ground at about 2.51
seconds. The maximum height of the dart is 26.6
feet. This will occur at 1.22 seconds.