Holt Algebra 2

3-Ext Parametric Equations 3-Ext Parametric Equations

Holt Algebra 2

Lesson PresentationLesson Presentation

Holt Algebra 2

3-Ext Parametric Equations

Graph parametric equations, and use them to model real-world applications.

Write the function represented by a pair of parametric equations.

Objectives

Holt Algebra 2

3-Ext Parametric Equations

parameterParametric equations

Vocabulary

Holt Algebra 2

3-Ext 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

3-Ext 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.

Example 1A: Writing and Graphing Parametric Equations

Write parametric equations to model thelocation of the cargo plane described above. Then graph the equations on a coordinate grid.

Holt Algebra 2

3-Ext Parametric Equations

Example 1A Continued

Use the distance formula d = rt.

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

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

3-Ext Parametric Equations

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).

Example 1A Continued

Holt Algebra 2

3-Ext 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

y = 40t = 40(20) = 800

Substitute t = 20.

At t = 20, the airplane has a ground distance of 4800 feet from the airport and an altitude of 800 feet.

Holt Algebra 2

3-Ext Parametric Equations

Check It Out! Example 1a

Write equations for and draw a graph of the motion of the helicopter.

A helicopter takes off with a horizontal speed of 5 ft/s and a vertical speed of 20 ft/s.

Using the horizontal and vertical speeds given above, write equations for the ground distance x and altitude y in terms of t.

Use the distance formula d = rt.x = 5t

y = 20t

Holt Algebra 2

3-Ext Parametric Equations

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.

Check It Out! Example 1a Continued

t 0 2 4 6 8

x 0 10 20 30 40

y 0 40 80 120 160

Holt Algebra 2

3-Ext Parametric Equations

Check It Out! Example 1b

Describe the location of the helicopter at t = 10 seconds.

Substitute t = 10.x = 5t =5(10) = 50

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

3-Ext Parametric Equations

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

Holt Algebra 2

3-Ext 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.

Example 2: Writing Functions Based on Parametric Equations

Solve one of the two parametric equations for t. Then substitute to get one equation whose variables are x and y.

Holt Algebra 2

3-Ext Parametric Equations

Example 2 Continued

Solve for t in the first equation.

Second equation

Substitute and simply.

y = 40t

The equation for the airplane’s altitude in terms of

ground distance is .

Holt Algebra 2

3-Ext Parametric Equations

Check It Out! Example 2

Write an equation for the helicopter's motion in terms of only x and y.

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.

y = 4x

y = 20t

Solve for t in the first equation.

Second equation

Substitute and simply.

x = 5t, so

y = 20 = 4x

The equation for the airplane’s altitude in terms of ground distance is y = 4x.