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Parametric Equations in Calculus: Exploring Curves and Graphs

Learn about parametric equations, curves, and graphs using calculators to visualize and analyze mathematical relationships. Discover how to transform Cartesian equations into parametric forms and explore various examples such as parabolas and lines. Explore circles, ellipses, and historical curves like the "Witch of Agnesi." Practice drawing, parameterizing, and confirming algebraic properties of parametric curves. Dive into the world of parametric calculus with this informative guide.

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Parametric Equations in Calculus: Exploring Curves and Graphs

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  1. 1.4 Parametric Equations Date Calculus AB Ms. Brewer

  2. Vocab/Formulas • Relation: set of ordered pairs (x,y) of real numbers. • If x and y are functions of t (the parameter), you can use parametric mode in a calculator.

  3. Example 1: Graph the half parabola using the parametric equations. t x y Initial Point: Terminal Point: Direction in which curve is traced: Your calculator can only draw a parametric curve in a closed interval even when the curve has no endpoints!

  4. Example 1: Find a Cartesian equation for the curve.

  5. Vocab/Formulas • Parametric Curve and Equations: If x and y are given as functions over an interval of t-values, then the set of points (x,y)=(f(t),g(t)) defined by these equations is a parametric curve. The equations are parametric equations. • Parameter: t when a ≤ t ≤ b Parameter Interval

  6. Parameterizing Circles and Ellipses • Circles: • Ellipses:

  7. Example 2: Graph the parametric curve

  8. Example 3: Draw and identify the graph of the parametric curve determined by:

  9. Example 3: Confirm algebraically that this is a line.

  10. Example 4: Graph the “Witch of Agnesi”. See the history of Maria Gaetana Agnesi on pg. 33

  11. Example 5: Find a parameterization for the line segment with endpoints (-2,1) and (3,5)

  12. Assignment Pg. 34 1-4, 9, 13, 17, 19, 21, 23, 25

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