1. Parametric Equations
3-Ext
Lesson Presentation
Holt Algebra 2

2. Objectives
Graph parametric equations, and use them to model real-world applications.
Write the function represented by a pair of parametric equations.

3. Vocabulary
parameter
Parametric equations

4. 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.

5. 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.

6. 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
y = 40t
Use the distance formula d = rt.
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.

7. Example 1A Continued t 1 2 3 4 x 240 480 720 960 y 40 80 120 1601 2 3 4 x
240
480
720
960 y 40 80
120
160
Plot and connect (0, 0), (240, 40), (480, 80), (720, 120), and (960, 160).    

8. 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 feet from the airport and an altitude of 800 feet.

9. 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
Use the distance formula d = rt.

10. 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 2 4 6 8 x 10 20 30 40 y 80
120
160    

11. 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.

12. You can use parametric equations to write a function that relates the two variables by using the substitution method.

13. 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.

14. The equation for the airplane’s altitude in terms of
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

15. The equation for the airplane’s altitude in terms of
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
Solve for t in the first equation.
y = 20t
Second equation
y = = 4x
y = 4x
Substitute and simply.
The equation for the airplane’s altitude in terms of
ground distance is y = 4x.