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Writing Parametric Equations definitions

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  • Parametric Equations
    A pair of equations expressing x and y as functions of a third variable, often t, to describe a curve.
  • Rectangular Equation
    An equation involving only x and y, representing a curve in the Cartesian plane without a parameter.
  • Parameter
    An independent variable, typically denoted t, used to express both x and y in parametric equations.
  • Domain Restriction
    A limitation on the set of allowable values for a variable, often caused by square roots or even powers.
  • Pythagorean Identity
    The trigonometric relationship cos²(t) + sin²(t) = 1, used to parameterize circles and ellipses.
  • Parameterization
    The process of expressing a rectangular equation as a set of parametric equations using a chosen parameter.
  • Ellipse
    A curve described by an equation of the form ax² + by² = c, often parameterized with sine and cosine.
  • Circle
    A special case of an ellipse where the coefficients of x² and y² are equal, parameterized using trigonometric functions.
  • Cosine Function
    A trigonometric function commonly used in parameterizations to represent the x-component of circular or elliptical motion.
  • Sine Function
    A trigonometric function often used in parameterizations to represent the y-component of circular or elliptical motion.
  • Elimination of Parameter
    The process of removing the parameter from parametric equations to recover the original rectangular equation.
  • Odd Power
    An exponent such as 1 or 3, preferred in parameter choices to avoid domain issues with negative values.
  • Even Power
    An exponent such as 2 or 4, which can introduce domain restrictions when used in parameter choices.
  • Function of t
    An expression where a variable, such as x or y, is written in terms of the parameter t.
  • Substitution
    The act of replacing a variable in an equation with an equivalent expression, often used to derive y(t) from x(t).