3 Ext Parametric Equations Objectives Graph parametric equations
3 -Ext Parametric Equations Objectives Graph parametric equations, and use them to model real-world applications. Vocabulary parameter Parametric equations 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 Example 1 A: 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. 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. x = 240 t y = 40 t Holt Algebra 2 Use the distance formula d = rt.
3 -Ext Parametric Equations Example 1 A 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 = 240 t y = 40 t 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). Holt Algebra 2
3 -Ext Parametric Equations Example 1 B: Writing and Graphing Parametric Equations Find the location of the cargo plane 20 seconds after takeoff. x = 240 t = 240(20) = 4800 Substitute t = 20. y = 40 t = 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
3 -Ext Parametric Equations Check It Out! Example 1 a 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 = 5 t y = 20 t Holt Algebra 2 Use the distance formula d = rt.
3 -Ext Parametric Equations Check It Out! Example 1 a 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 Holt Algebra 2
3 -Ext Parametric Equations Check It Out! Example 1 b Describe the location of the helicopter at t = 10 seconds. x = 5 t =5(10) = 50 Substitute t = 10. y = 20 t =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 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. x = 240 t y = 40 t 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 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 = 5 t, so Solve for t in the first equation. y = 20 t y = 20 Second equation = 4 x Substitute and simply. y = 4 x The equation for the airplane’s altitude in terms of ground distance is y = 4 x. Holt Algebra 2
3 -Ext Parametric Equations Guided Practice: Express y in terms of x 1) x =20 t y= 100 t 2) x=4 t y= 9 t 3) x=8 t y= 80 t 4) x=4 t y= 2 t 5) x=9 t y= 36 t Holt Algebra 2
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