Copyright © 2007 Pearson Education, Inc. Slide 10-1 Parametric Equations Here are some examples of trigonometric functions used in parametric equations.

Copyright © 2007 Pearson Education, Inc. Slide 10-2 INTRODUCTION Imagine that a particle moves along the curve C shown here. –It is impossible to describe C by an equation of the form y = f(x). –This is because C fails the Vertical Line Test.

Copyright © 2007 Pearson Education, Inc. Slide 10-3 However, the x- and y-coordinates of the particle are functions of time. So, we can write x = f(t) and y = g(t). INTRODUCTION

Copyright © 2007 Pearson Education, Inc. Slide 10-4 Such a pair of equations is often a convenient way of describing a curve and gives rise to the following definition. INTRODUCTION

Copyright © 2007 Pearson Education, Inc. Slide 10-5 Suppose x and y are both given as functions of a third variable t (called a parameter) by the equations x = f(t) and y = g(t) –These are called parametric equations. PARAMETRIC EQUATIONS

Copyright © 2007 Pearson Education, Inc. Slide 10-6 Each value of t determines a point (x, y), which we can plot in a coordinate plane. As t varies, the point (x, y) = (f(t), g(t)) varies and traces out a curve C. –This is called a parametric curve. PARAMETRIC CURVE

Copyright © 2007 Pearson Education, Inc. Slide 10-7 The parameter t does not necessarily represent time. PARAMETER t

Copyright © 2007 Pearson Education, Inc. Slide 10-8 PARAMETER t However, in many applications of parametric curves, t does denote time. –Thus, we can interpret (x, y) = (f(t), g(t)) as the position of a particle at time t.

Copyright © 2007 Pearson Education, Inc. Slide Graphing a Circle with Parametric Equations ExampleGraph x = 2 cos t and y = 2 sin t for 0    2 . Find an equivalent equation using rectangular coordinates. SolutionLet X 1T = 2 cos (T) and Y 1T = 2 sin (T), and graph these parametric equations as shown. Technology NoteBe sure the calculator is set in parametric mode. A square window is necessary for the curve to appear circular.

Copyright © 2007 Pearson Education, Inc. Slide Graphing a Circle with Parametric Equations To verify that this is a circle, consider the following. The parametric equations are equivalent to x 2 + y 2 = 4, which is a circle with center (0, 0) and radius 2. x = 2 cos t, y = 2sin t cos 2 t + sin 2 t = 1

Copyright © 2007 Pearson Education, Inc. Slide Graphing an Ellipse with Parametric Equations Example Graph the plane curve defined by x = 2 sin t and y = 3 cos t for t in [0, 2  ]. Solution Now add both sides of the equation.

Copyright © 2007 Pearson Education, Inc. Slide Graphing a Cycloid The path traced by a fixed point on the circumference of a circle rolling along a line is called a cycloid. A cycloid is defined by x = at – a sin t, y = a – a cos t, for t in (– ,  ) where a is the diameter. ExampleGraph the cycloid with a = 1 for t in [0, 2  ]. Analytic Solution There is no simple way to find a rectangular equation for the cycloid from its parametric equation.

Copyright © 2007 Pearson Education, Inc. Slide Graphing a Cycloid Find a table of values and plot the ordered pairs. t0  22 x  5.7 22 y

Copyright © 2007 Pearson Education, Inc. Slide Graphing a Cycloid Graphing Calculator Solution Interesting Physical Property of the Cycloid If a flexible wire goes through points P and Q, and a bead slides due to gravity without friction along this path, the path that requires the shortest time takes the shape of an inverted cycloid.

Copyright © 2007 Pearson Education, Inc. Slide Applications of Parametric Equations Parametric equations are used frequently in computer graphics to design a variety of figures and letters. ExampleGraph a “smiley” face using parametric equations. Solution HeadUse the circle centered at the origin. If the radius is 2, then let x = 2 cos t and y = 2 sin t for 0  t  2 .

Copyright © 2007 Pearson Education, Inc. Slide Applications of Parametric Equations EyesUse two small circles. The eye in the first quadrant can be modeled by x = cos t and y = sin t. This represents a circle centered at (1, 1) with radius.3. The eye in quadrant II can be modeled by x = –1 +.3 cos t and y = sin t for 0  t  2 , which is a circle centered at (–1, 1) with radius 0.3. MouthUse the lower half of a circle. Try x =.5 cos ½t and y = –.5 –.5 sin ½t. This is a semicircle centered at (0, –.5) with radius.5. Since t is in [0, 2  ], the term ½t ensures that only half the circle will be drawn.

Copyright © 2007 Pearson Education, Inc. Slide Simulating Motion with Parametric Equations If a ball is thrown with a velocity v feet per second at an angle  with the horizontal, its flight can be modeled by the parametric equations where t is in seconds and h is the ball’s initial height above the ground. The term –16t 2 occurs because gravity pulls the ball downward. Figure 80 pg

Copyright © 2007 Pearson Education, Inc. Slide Simulating Motion with Parametric Equations ExampleThree golf balls are hit simultaneously into the air at 132 feet per second making angles of 30 º, 50 º, and 70 º with the horizontal. (a)Assuming the ground is level, determine graphically which ball travels the farthest. Estimate this distance. (b)Which ball reaches the greatest height? Estimate this height. Solution (a) The three sets of parametric equations with h = 0 are as follows. X 1T = 132 cos (30 º ) T, Y 1T = 132 sin (30 º ) T – 16T 2 X 2T = 132 cos (50 º ) T, Y 2T = 132 sin (50 º ) T – 16T 2 X 3T = 132 cos (70 º ) T, Y 3T = 132 sin (70 º ) T – 16T 2

Copyright © 2007 Pearson Education, Inc. Slide Simulating Motion with Parametric Equations With 0  t  9, a graphing calculator in simultaneous mode shows all three balls in flight at the same time. The ball hit at 50 º goes the farthest at an approximate distance of 540 feet. (b)The ball hit at 70 º reaches the greatest height of about 240 feet.

Copyright © 2007 Pearson Education, Inc. Slide Examining Parametric Equations of Flight ExampleA small rocket is launched from a table that is 3.36 feet above the ground. Its initial velocity is 64 feet per second, and it is launched at an angle of 30 º with respect to the ground. Find the rectangular equation that models this path. What type of path does the rocket follow? SolutionIts path is defined by the parametric equations x = (64 cos 30 º )tandy = (64 sin 30 º )t – 16t or, equivalently, From we get

Copyright © 2007 Pearson Education, Inc. Slide Examining Parametric Equations of Flight Substituting into the other parametric equation yields The rocket follows a parabolic path.

Copyright © 2007 Pearson Education, Inc. Slide Analyzing the Path of a Projectile ExampleDetermine the total flight time and the horizontal distance traveled by the rocket in the previous example. SolutionThe equation y = –16t t tells the vertical position of the rocket at time t. Find t for which y = 0 since this corresponds to the rocket at ground level. Since t represents time, t = –.1 is an unacceptable answer. Therefore, the flight time is 2.1 seconds. Use t = 2.1 to find the horizontal distance x as follows.

Copyright © 2007 Pearson Education, Inc. Slide Homework Pages 584 – – 5, 8 -11