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…

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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