Transcript Slide 1

3-Ext
3-Ext Parametric
ParametricEquations
Equations
Lesson Presentation
Holt
Algebra
Holt
Algebra
22
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Parametric Equations
Objectives
Graph parametric equations, and use
them to model real-world applications.
Write the function represented by a
pair of parametric equations.
Holt Algebra 2
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Parametric Equations
Vocabulary
parameter
Parametric equations
Holt Algebra 2
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Parametric Equations
As an airplane ascends after
takeoff, its altitude increases at a
rate of 45 ft/s while its distance on
the ground from the airport
increases at 210 ft/s.
Both of these rates can be
expressed in terms of time. When
two variables, such as x and y, are
expressed in terms of a third
variable, such as t, the third
variable is called a parameter.
The equations that define this relationship are
parametric equations.
Holt Algebra 2
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Parametric Equations
Example 1A: Writing and Graphing Parametric
Equations
As a cargo plane ascends after takeoff, its
altitude increases at a rate of 40 ft/s. while its
horizontal distance from the airport increases
at a rate of 240 ft/s.
Write parametric equations to model the
location of the cargo plane described above.
Then graph the equations on a coordinate grid.
Holt Algebra 2
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Parametric Equations
Example 1A Continued
Using the horizontal and vertical speeds given
above, write equations for the ground distance x
and altitude y in terms of t.
x = 240t
Use the distance formula d = rt.
y = 40t
Make a table of values to help you draw the graph.
Use different t-values to find x- and y-values. The x
and y rows give the points to plot.
Holt Algebra 2
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Parametric Equations
Example 1A Continued
t
0
1
2
3
4
x
0
240 480 720 960
y
0
40
80 120 160
Plot and connect
(0, 0), (240, 40),
(480, 80), (720, 120),
and (960, 160).
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Parametric Equations
Example 1B: Writing and Graphing Parametric
Equations
Find the location of the cargo plane 20
seconds after takeoff.
x = 240t = 240(20) = 4800
Substitute t = 20.
y = 40t = 40(20) = 800
At t = 20, the airplane has a ground distance of
4800 feet from the airport and an altitude of 800 feet.
Holt Algebra 2
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Parametric Equations
Check It Out! Example 1a
A helicopter takes off with a horizontal speed
of 5 ft/s and a vertical speed of 20 ft/s.
Write equations for and draw a graph of the
motion of the helicopter.
Using the horizontal and vertical speeds given
above, write equations for the ground distance x
and altitude y in terms of t.
x = 5t
y = 20t
Holt Algebra 2
Use the distance formula d = rt.
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Parametric Equations
Check It Out! Example 1a Continued
Make a table of values to help you draw the graph.
Use different t-values to find x- and y-values. The x
and y rows give the points to plot.
t
0
2
4
6
8
x
0
10
20
30
40
y
0
40
80
120
160
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Holt Algebra 2
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Parametric Equations
Check It Out! Example 1b
Describe the location of the helicopter at t =
10 seconds.
x = 5t =5(10) = 50
Substitute t = 10.
y = 20t =20(10) = 200
At t = 10, the helicopter has a ground distance of
50 feet from its takeoff point and an altitude of
200 feet.
Holt Algebra 2
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Parametric Equations
You can use parametric equations to write a
function that relates the two variables by using
the substitution method.
Holt Algebra 2
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Parametric Equations
Example 2: Writing Functions Based on Parametric
Equations
Use the data from Example 1 to write an
equation from the cargo plane’s altitude y in
terms of its horizontal distance x.
Solve one of the two parametric equations for t.
Then substitute to get one equation whose
variables are x and y.
Holt Algebra 2
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Parametric Equations
Example 2 Continued
Solve for t in the first equation.
y = 40t
Second equation
Substitute and simply.
The equation for the airplane’s altitude in terms of
ground distance is
Holt Algebra 2
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Parametric Equations
Check It Out! Example 2
Recall that the helicopter in Check It Out
Problem 1 takes off with a horizontal speed of
5 ft/s and a vertical speed of 20 ft/s.
Write an equation for the helicopter's motion
in terms of only x and y.
x = 5t, so
y = 20t
y = 20
Solve for t in the first equation.
Second equation
= 4x
Substitute and simply.
y = 4x
The equation for the airplane’s altitude in terms of
ground distance is y = 4x.
Holt Algebra 2