SP21 MATH 2024 - Parametric Functions.pdf - MATH 2024...

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MATH 2024 Parametric Curves Adams Plane Curves and Parametric Equations Until now, we have represented graphs using a single equation involving two variables. Today we begin examining situations in which three variables are used to represent a curve in the plane. Let’s begin by considering the path followed by an object that is propelled into the air at an angle of 45 . If the initial velocity of the object is 48 feet per second, the object travels the parabolic path given by the rectangular equation y = - 1 72 x 2 + x as shown in the figure below: This equation, however, does not tell the whole story. Although it tells you where in the plane the object travels, it does not tell you when the object was at any given point ( x, y ). To determine this time, we will introduce a third variable t , called a parameter . By writing both x and y as functions of the parameter t , we obtain the set of parametric equations x ( t ) = 24 2 t and y ( t ) = - 16 t 2 + 24 2 t From this set of equations, we can determine that at time t = 0, the object is at the point (0 , 0). Similarly, at time t = 1, the object is at the point ( 24 2 , 24 2 - 16 ) , and so on. For this particular motion problem, x and y are continuous functions of t , and the resulting path is called a plane curve . If f and g are continuous functions of t on an interval I , then the equations x = f ( t ) and y = g ( t ) Page 1 of 8
MATH 2024 Parametric Curves Adams are called parametric equations and t is called the parameter . The set of points ( x, y ) obtained as t varies over the interval I is called the graph of the parametric equations. Taken together, the parametric equations and the graph are called a plane curve , denoted by C . Sketching a Curve When sketching a curve represented by a set of parametric equations, you can plot points in the xy - plane. Each set of coordinates ( x, y ) is determined from a value chosen for the parameter t . By plotting the resulting points in order of increasing values of t , the curve is traced out in a specific direction. This is called the orientation of the curve.