Parametric Equations

1
Parametric Equations

• Lesson 6.7

2
Movement of an Object

• Consider the position of an object as a function
  of time
• The x coordinate is a function of timex f(t)
• The y coordinate is a function of timey g(t)

time
0
3
Table of Values

• We have t as an independent variable
• Both x and y are dependent variables
• Given
• x 3t
• y t2 4
• Complete the table

t -4 -3 -2 -1 0 1 2 3 4
x
y
4
Plotting the Points

• Use the Data Matrix on the TI calculator
• Choose APPS, 6, and Current
• Data matrix appears
• Use F1, 8 to clear previous values

5
Plotting the Points

• Enter the values for t in Column C1
• Place cursor on the C2
• Enter formula for x f(t) 3C1
• Place cursor on the C3
• Enter formula for y g(t) C12 4

6
Plotting the Points

• Choose F2 Plot Setup
• Then F1, Define
• Now specify that thex values come fromcolumn 2,
  the y's from column 3
• Press Enter to proceed

7
Plotting the Points

• Go to the Y screen
• Clear out (or toggle off) any other functions
• Choose F2, Zoom Data

8
Plotting the Points

• Graph appears
• Note that each x value is a function of t
• Each y value is a function of t

y g(t)
x f(t)
9
Parametric Plotting on the TI

• Press the Mode button
• For Graph, choose Parametric
• Now the Y screen will have two functions for
  each graph

10
Parametric Plotting on the TI

• Remember that both xand y are functions of t
• Note the results whenviewing the Table, ?Y
• Compare to theresults in the data matrix

11
Parametric Plotting on the TI

• Set the graphing window parameters as shown here
• Note the additional specificationof values for
  t, our new independent variable
• Now graph the parametric functions
• Note how results coincidewith our previous points

12
Try These Examples

• See if you can also determine what the equivalent
  would be in y f(x) form.
• x 2ty 4t 1
• x t 5
• y 3t 2
• x 2 cos ty 6 cos t
• x sin 4ty cos 2t
• x 3 sin 3 ty cos t

Which one is it?
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Assignment A

• Lesson 6.7A
• Page 440
• Exercises 1 9 odd 27, 29

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Parametric EquationsThe Sequal

• Lesson 6.7B

15
Eliminating the Parameter

• Possible to represent the parametric curve with a
  single (x, y) equation
• Example
• Given x 1 t2 and y 2 t2
• Solution
• Solve 1st equation for t2 in terms of y
• Substitute into 2ndequation
• Result
• y 2 (x 1)

Verify by graphing
16
Try It

• Given x 3 2 cos t and y - 1 3
  sin t
• Hint manipulate the equations by using
  Pythagorean identity sin2 t cos2 t 1

17
Path of a Projectile

• Consider a projectile (such as a pumpkin)
  launched at a specified angle and initial
  velocity
• Based on vector components and effects of gravity
  the actual path can be represented by

Experiment with Pumpkin Launch Spreadsheet
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Launch Another!

• Graph the path of a pumpkin launched at an angle
  of 35 with an initial velocity of 195 ft/sec
• How far did it go?
• How long was it in the air?

19
Assignment B

• Lesson 6.7B
• Page 440
• Exercises 11 17 odd 31, 33