Parametric Equations 10.6 Adapted by JMerrill, 2011

Parametric Equations10.6

Adapted by JMerrill, 2011

Plane Curves

• Up to now, we have been representing graphs by a single equation in 2 variables. The y = equations tell us where an object (ball being thrown) has been.

• Now we will introduce a 3rd variable, t (time) which is the parameter. It tells us when an object was at a given point on a path.

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A pair of parametric equations are equations with both x and y written as functions of time, t.

216 24 2y t t Parametric equation for x

Parametric equation for y

t is the parameter.

Rectangular equation2

.72xy x

The path of an object thrown into the air at a 45° angle at 48 feet per second can be represented by

horizontal distance (x)vertical distance (y)

24 2x tNow the distances depend on the

time, t.


216 24 2y t t 2

72xy x

y

x

9

18

9 18 27 36 45 54 63 72(0, 0)t = 0

(36, 18)

3 24

t

3 22

t (72, 0)

two variables (x and y) for positionone variable (t) for time

Curvilinear motion:

24 2x tExample:Parametric equations


Example: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 y

x-4 4

4

8

The (x,y) ordered pairs will graph exactly the same as they always have graphed.


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


Eliminating the parameter is a process for finding the rectangular equation (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)


Solve for t in one equation.

Substitute into the second equation.

Example:2y t Identify the curve represented by x = 2t and

by eliminating the parameter.

2xt

22y x

y

x-4 4

4

8

22xy

The absolute value bars can be found in the Math

menu--Num

Eliminating an Angle Parameter

• Sketch and identify the curve represented by x = 3cosθ, y = 4sinθ

• Solve for cosθ & sinθ:

• Use the identity cos2θ + sin2θ = 1

• We have a vertical ellipse with a = 4 and b = 3


3 4cos sin

yx

22

22

13 4

19 16

yx

yx

You Try

• Eliminate the parameter in the equations


3 2

2 3

x t

y t

2 3

3

2 2

t x

xt

32 3

2 23 9

22 2

3 11

2 2

xy

y x

y x


Let x = t

Substitute into the original rectangular equation.

Writing Parametric Equations from Rectangular Equations Find a set of parametric equations to represent the graph of y = 4x – 3.

x = t

y = 4t – 3

x

y

-4 4

4

-4

8y = 4t – 3

You Try

• Find a set of parametric equations given y = x2

• x = t

• y = t2



Application:The center-field fence in a ballpark is 10 feet high and 400 feet from home plate. A baseball is hit at a point 3 feet above the ground and leaves the bat at a speed of 150 feet per second at an angle of 15. The parametric equations for its path are x = 145t and y = 3 + 39t – 16t2.

Graph the path of the baseball. Is the hit a home run?

Home Run

(364, 0)

y

5

10

0

15

20

25

x50 100 150 200 250 300 350 400

The ball only traveled 364 feet and was not a home run.

(0, 3)

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Parametric Equations 10.6 Adapted by JMerrill, 2011