1. Holt Algebra 2 3-Ext Parametric Equations Graph parametric equations, and use them to model real-world applications. Objectives parameter Parametric equations Vocabulary

2. 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.

3. 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. Write parametric equations to model the location of the cargo plane described above. Then graph the equations on a coordinate grid. Example 1A: Writing and Graphing Parametric Equations x = 240t y = 40t Use the distance formula d = rt. Solution: 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.

4. Holt Algebra 2 3-Ext Parametric Equations t01234 x0240480720960 y04080120160 Plot and connect (0, 0), (240, 40), (480, 80), (720, 120), and (960, 160).     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. x = 240t y = 40t

5. 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.

6. 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

7. 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 t02468 x010203040 y0 80120160    

8. 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.

9. 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.

10. 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 x = 240t y = 40t The equation for the airplane’s altitude in terms of ground distance is.

11. 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.

12. Holt Algebra 2 3-Ext Parametric Equations Guided Practice: Express y in terms of x 1)x =20t y= 100t 2)x=4t y= 9t 3)x=8t y= 80t 4)x=4t y= 2t 5)x=9t y= 36t