Precalculus Parametric Equations graphs. Parametric Equations Graph parametric equations. ...

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Precalculus Parametric Equations graphs

Transcript of Precalculus Parametric Equations graphs. Parametric Equations Graph parametric equations. ...

Page 1: Precalculus Parametric Equations graphs. Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric.

Precalculus

Parametric Equations

graphs

Page 2: Precalculus Parametric Equations graphs. Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric.

Parametric Equations

Graph parametric equations. Determine an equivalent rectangular

equation for parametric equations. Determine the location of a moving

object at a specific time.

Page 3: Precalculus Parametric Equations graphs. Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric.

Graphing Parametric Equations

We have graphed plane curves that are composed of sets of ordered pairs (x, y) in the rectangular coordinate plane. Now we discuss a way to represent plane curves in which x and y are functions of a third variable t.One method will be to construct a table in which we choose values of t and then determine the values of x and y.

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Page 4: Precalculus Parametric Equations graphs. Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric.

Example 1 Graph ParametricGraph the curve represented by the equations

x 1

2t, y t 2 3; 3 t 3.

Solve for x when t=-3 and t=3

Solve for y when t=-3 and t=3

These are the restrictions on x and y – now make a table for all values of t… 822

Page 5: Precalculus Parametric Equations graphs. Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric.

Example 1 Graph ParametricGraph the curve represented by the equations

x 1

2t, y t 2 3; 3 t 3.

Find the rectangular equation by eliminating t: First, solve for t in the first equation 822

Page 6: Precalculus Parametric Equations graphs. Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric.

Example 1 Graph ParametricGraph the curve represented by the equations

x 1

2t, y t 2 3; 3 t 3.

t = 2x – substitute 2x for t in the second equation…

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Page 7: Precalculus Parametric Equations graphs. Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric.

Example 1 Graph ParametricGraph the curve represented by the equations

x 1

2t, y t 2 3; 3 t 3.

The rectangular equation is:y 4x2 3,

3

2x

3

2.

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Page 8: Precalculus Parametric Equations graphs. Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric.

Parametric Equations

If f and g are continuous functions of t on

an interval I, then the set of ordered pairs

(x, y) such that x = f(t) and y = g(t) is a plane curve.

The equations x = f(t) and y = g(t) are parametric equations for the curve.

The variable t is the parameter.

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Page 9: Precalculus Parametric Equations graphs. Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric.

Determining a Direct Relationship for Given Parametric Equations

Solve either equation for t.

Then substitute that value of t into the other equation.

Calculate the restrictions on the variables x and y based on the restrictions on t.

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Page 10: Precalculus Parametric Equations graphs. Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric.

Example 2 Find Rectangular

Find a rectangular equation equivalent to

x t 2 , y t 1; 1 t 4Calculate the restrictions:

−1≤t ≤4

x=t2; 0 ≤x≤16y=t−1; −2 ≤y≤3

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Page 11: Precalculus Parametric Equations graphs. Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric.

Example 2 Find Rectangular

Find a rectangular equation equivalent to

Solution

x t 2 , y t 1; 1 t 4

The rectangular equation is:

y t 1

t y 1

Substitute t y 1 into x t 2 .

x y 1 2

Calculate the restrictions:

−1≤t ≤4

xt2; 1≤x≤16yt−1; −2 ≤y≤3

x y1 2 ; 0 ≤x≤16.

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Page 12: Precalculus Parametric Equations graphs. Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric.

Example 3

• X = 2cos t

• Y = 2sin t

• Square both equations and add together

• (This one we will handle differently)

0 ≤t ≤2π

Page 13: Precalculus Parametric Equations graphs. Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric.

Example 3

• X = 2cos t

• Y = 2sin t

• Add the squares….

• A circle of radius 2

x2 + y2 =4 cos2 t+ 4sin2 t

x2 + y2 =4(cos2 t+sin2 t)

x2 + y2 =4

= 1

0 ≤t ≤2π

Page 14: Precalculus Parametric Equations graphs. Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric.

Determining Parametric Equations for a Given Rectangular Equation

Many sets of parametric equations can represent the same plane curve. In fact, there are infinitely many such equations.

The most simple case is to let either x (or y)

equal t and then determine y (or x).

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Page 15: Precalculus Parametric Equations graphs. Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric.

Example 4 Find Parametric EquationsFind three sets of parametric equations for the parabola y 4 x 3 2

.

SolutionIf x t, then y 4 t 3 2 t 2 6t 5.

If x t 3, then y 4 t 3 3 2 t 2 4.

If x t

3, then y 4

t

3 3

2

t

9

2

2t 5.

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Page 16: Precalculus Parametric Equations graphs. Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric.

Parametization of a line

• Find the parametric equations going through the points, p (-2,2) and q (3,6)

• Step 1: Find the vector

• Step 2: Find the vector from p to (x,y)

• Step 3: set up:

pqu vu

Page 17: Precalculus Parametric Equations graphs. Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric.

solution

pqu ru

= 3−−2,6 −2 = 5,4

pxuru

= x−−2, y−2 = x+ 2, y−2

so...

x+ 2, y−2 =t 5,4

Set x’s and y’s equal, separately:

X+2 = 5tY-2 = 4t

Solve for x and y:

X = 5t – 2Y = 4t + 2