Parametric Equations 3-Ext Lesson Presentation Holt Algebra 2

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

Vocabulary parameter 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.

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.

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.

Example 1A Continued t 1 2 3 4 x 240 480 720 960 y 40 80 120 160 1 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).    

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.

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.

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    

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.

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

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.

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 .

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 = 20 = 4x y = 4x Substitute and simply. The equation for the airplane’s altitude in terms of ground distance is y = 4x.