Parametric Equations and Motion

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Parametric Equations and Motion

• 9.7

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Parametric Equations

• Let x f(t) and y g(t), where f and g are two
  functions whose common domain is some interval I.
  The collection of points defined by (x,y)
  (f(t),g(t)) is called a plane curve. The
  equations x f(t) and y g(t) where t is in I,
  are called parametric equations of the curve.
  The variable t is called the parameter.

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Parametric Equations

• Useful in defining movement along a curve. Also
  used to have two variables change when another
  changes. For example, as time changes, the x
  direction and y direction change.

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Graph Parametric Equations

• Successive values of t give rise to a directed
  movement along a curve. The curve is traced out
  in a certain direction.
• Choose values of t within the given interval and
  find the corresponding (x,y) and plot each point.
  Connect with arrows to find the orientation of
  the curve.

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Graphing Parametric Equations on the Calculator

• Press the mode button
• Arrow down and over to Par and hit enter.
• Press y and place each parametric equation in
  for x1t and y1t.
• Press the window button and set the Tmin and Tmax
  as desired or use zoom standard, etc.

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tstep

• The calculator will start plotting a parametric
  equation by taking tmin and finding the
  corresponding (x,y) point for that t value and
  plotting it. The it will look at tstep and use
  it to decide what the next t value will be. Then
  the calculator will find the (x,y) for that
  t-value, until it reaches tmax.

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Eliminating the Parameter or Finding the
Rectangular Equation

• Solve one of the parametric equations for t.
  Unless sine and cosine are present.
• Substitute this in for t in the other equation.
• Solve for y if linear or quadratic, otherwise put
  in the form for the equation of a circle,
  ellipse, etc.

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Parametric Equation for a Circle

• If x a cos t and y a sin t, then square both
  sides and add together.
• x2 a2 cos2 t
• y2 a2 sin2 t
• x2y2 a2((cos2 t)(sin2 t))
• x2y2 a2

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Parametric Equation for a Ellipse

• If x a cos t and y b sin t, then solve for
  cos t and sin t, then square both sides and add
  together.
• x/a cos t
• y/b sin t
• (y/b)2 sin2 t
• (x/a)2 cos2 t
• (x/a)2 (y/b)2 cos2 t sin2 t
• (x/a)2 (y/b)2 1

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Parametric Equation for a Circle

• If x a cos t h and y a sin t k, then solve
  for a cos t and a sin t. Then square both sides
  and add together.
• x - h a cos t
• y k a sin t
• (x-h)2 a2 cos2 t
• (y-k)2 a2 sin2 t
• (x-h)2 (y-k)2 a2 cos2 t a2 sin2 t
• (x-h)2 (y-k)2 a2

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Parametric Equation for a Circle

• (x-2)2 (y-5)2 16
• X 4 cos t 2
• Y 4 sin t 5

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Finding Parametric Equations from Rectangular

• Set xt and then put ts in for the xs to get y in
  terms of t. Solve for y.
• Answer is xt and yequation with t
• Set yt and then put ts in for the ys to get x in
  terms of t. Solve for x.
• Answer is yt and xequation with t

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Finding the Parametric Equation of a Line

• Let point A (-3,-3) and point C (5,1).
  Assume t starts at 0 and then becomes 3.
• Pick one of the points to start withlets take A.
• X is equal to -3 when t is zero and 5 when t is
  3, so let x -3 (5-(-3))/3 t
• Y is equal to -3 and 1, so let y -3
  (1-(-3))/3 t

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Finding the Parametric Equation of a Line

• Let point A (a,b) and point C (c,d). Assume
  t starts at 0 and ends at z.
• Pick one of the points to start withlets take A.
• X is equal to a when t is zero and c when t is z,
  so let x a (c-a)/z t
• Y is equal to b and d, so let y b (d-b)/z t