Precalculus Parametric Equations graphs. Parametric Equations Graph parametric equations. ...
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Transcript of Precalculus Parametric Equations graphs. Parametric Equations Graph parametric equations. ...
Precalculus
Parametric Equations
graphs
Parametric Equations
Graph parametric equations. Determine an equivalent rectangular
equation for parametric equations. Determine the location of a moving
object at a specific time.
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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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
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
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…
822
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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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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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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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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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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Example 3
• X = 2cos t
• Y = 2sin t
• Square both equations and add together
• (This one we will handle differently)
0 ≤t ≤2π
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
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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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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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
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