ch.14 Parametric Equations and Conic Sections

ch.14 Parametric Equations and Conic Sections - Chapter 14...

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Chapter 14 Parametric Equations and Conic Sections 14.1 Parametric Equations
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Parametric Equations of a Plane Curve A plane curve is a set of points ( x , y ) such that x = f ( t ), y = g ( t ), and f and g are both defined on an interval I . The equations x = f ( t ) and y = g ( t ) are parametric equations with parameter t .
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Graphing a Plane Curve Defined Parametrically Example : Let x = t 2 and y = 3 t - 1, for t in [ - 3,3]. Graph the set of ordered pairs ( x , y ). Solution : Make a table of corresponding values of t , x , and y over the domain of t .
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Finding an Equivalent Rectangular Equation Example : Find a rectangular equation for the plane curve of the previous example defined as follows. x = t 2 , y = 3 t - 1
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Graphing a Plane Curve Defined Parametrically Example : Graph the plane curve defined by x = 2 sin t , y = 3cos t , for t in [0,2 ]. Solution : Use the fact that sin 2 t + cos 2 t = 1. Square both sides of each equation; solve one for sin 2 t , the other for cos 2 t . π
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14
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Chapter 14 Parametric Equations and Conic Sections 14.2 IMPLICITLY DEFINED CURVES AND CIRCLES
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Key points Parametric and implicit equations of circles
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The curves known as conic sections include two we are already familiar with, circles and parabolas . They also include ellipses and hyperbolas . Conic sections are so-called because, as was demonstrated by the Greeks, they can be constructed by slicing, or sectioning, a cone. Conic sections arise naturally in physics, since the path of a body orbiting the sun is a conic section. We will study them in terms of parametric and implicit equations. As we have already studied parabolas, we now focus on circles, ellipses and hyperbolas. Conic Sections Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
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Example 2 Graph the parametric equations x = 5 + 2 cos t and y = 3 + 2 sin t . Solution We see that x varies between 3 and 7 with a midline of 5 while y varies between 1 and 5 with a midline of 3. The figure gives a graph of this function. It appears to be a circle of radius 2 centered at the point (5, 3).
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