Parametric Equations and Eliminating the Parameter Section 10.2 Calculus BC AP/Dual, Revised ©2018 viet.dang@humbleisd.net 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

§10.2: Parametric Equations and Eliminating the Parameter Application The parametric equations 𝒙=𝟒 𝐜𝐨𝐬 𝒕 −𝟗 𝐬𝐢𝐧 𝒕 , 𝒚=𝟒 𝐜𝐨𝐬 𝒕 +𝟗 𝐬𝐢𝐧 𝒕 define an ellipse in the 𝒙𝒚-plane. If 𝒙 and 𝒚 are measured in millions of miles and t is measured in years, you might think of the equations as giving the position of an object revolving on an elliptical orbit in two dimensional space. In this context, you might be interested in questions like the following: What is the length of the orbit of this object? [Answer: 59.914 million miles] What is maximum and minimum speed of this object? [Answer: 12.728 and 5.657 million miles per year, respectively] 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

§10.2: Parametric Equations and Eliminating the Parameter Definitions Rectangular Equation involves 𝒙 (horizontal distance) and 𝒚 (vertical distance) Third variable, 𝒕, is written as time or known as the parameter. A PLANE CURVE is whereas 𝒇 and 𝒈 are continuous functions on t on an interval and the set of ordered pairs. 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

§10.2: Parametric Equations and Eliminating the Parameter Parametric Curves Basic graphing with direction to them Describing curves in a plane that are not necessarily functions Objects that move all around in 2 dimensions When eliminating a parameter it will look like, at least, a portion of a rectangular equation just have to take into account the domain Parametric curves have a direction of motion.  The direction of motion is given by increasing 𝒕. 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

§10.2: Parametric Equations and Eliminating the Parameter Visual Example Example of 𝒚=− 𝒙 𝟐 𝟕𝟐 +𝒙 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

§10.2: Parametric Equations and Eliminating the Parameter Steps Make a table of values, setting 𝒕, 𝒙, and 𝒚 Identify the Parametric equations and/or inequality Plot points by creating a 𝑻-CHART Plug in 𝒕 to get the 𝒙 and 𝒚-coordinates Draw arrows and follow the direction where time follows. 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

§10.2: Parametric Equations and Eliminating the Parameter Example 1 Graph the curve given by 𝒙=𝒕−𝟐 𝒚= 𝒕 𝟐 +𝟑𝒕 from −𝟑≤𝒕≤𝟐 𝒕 𝒙 𝒚 −𝟑 −𝟐 −𝟏 𝟎 𝟏 𝟐 −𝟓 −𝟒 −𝟑 −𝟐 −𝟏 𝟎 𝟎 −𝟐 𝟒 𝟏𝟎 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

§10.2: Parametric Equations and Eliminating the Parameter Example 1 Graph the curve given by 𝒙=𝒕−𝟐 𝒚= 𝒕 𝟐 +𝟑𝒕 from −𝟑≤𝒕≤𝟐 𝒕 𝒙 𝒚 −𝟑 −𝟐 −𝟏 𝟎 𝟏 𝟐 −𝟓 −𝟒 −𝟑 −𝟐 −𝟏 𝟎 𝟎 −𝟐 𝟒 𝟏𝟎 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

§10.2: Parametric Equations and Eliminating the Parameter Your Turn Graph the curve given by 𝒙=𝟐𝒕+𝟐 𝒚= 𝒕 𝟐 −𝟑 from −𝟐, 𝟑 𝒕 𝒙 𝒚 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟑 −𝟐 𝟏 𝟎 𝟐 −𝟑 𝟒 𝟔 𝟖 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

Eliminate the Parameter Identify the Parametric Equation Solve for 𝒕 in one equation Substitute in other equation Convert to Rectangular equation or 𝒚 = Choose smart points Involving trig functions include two equations 𝐬𝐢𝐧 𝟐 𝜽 + 𝐜𝐨𝐬 𝟐 𝜽 =𝟏 𝐬𝐞𝐜 𝟐 𝜽 − 𝐭𝐚𝐧 𝟐 𝜽 =𝟏 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

§10.2: Parametric Equations and Eliminating the Parameter Example 2 Determine the rectangular equation and graph the curve given by 𝒙= 𝟏 𝒕+𝟏 𝒚= 𝒕 𝒕+𝟏 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

§10.2: Parametric Equations and Eliminating the Parameter Example 2 Determine the rectangular equation and graph the curve given by 𝒙= 𝟏 𝒕+𝟏 𝒚= 𝒕 𝒕+𝟏 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

§10.2: Parametric Equations and Eliminating the Parameter Example 2 Determine the rectangular equation and graph the curve given by 𝒙= 𝟏 𝒕+𝟏 𝒚= 𝒕 𝒕+𝟏 𝒕 𝒙 𝒚 –1 3 −𝟏 𝟎 𝟑 Und. 𝟏 𝟏/𝟐 Und. 𝟎 𝟑/𝟒 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

§10.2: Parametric Equations and Eliminating the Parameter Your Turn Determine the rectangular equation and graph the curve given by 𝒙= 𝟏 𝒕 𝒚=𝟐 𝒕 𝟐 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

§10.2: Parametric Equations and Eliminating the Parameter Example 3 Determine the rectangular equation and graph the curve given by 𝒙=𝟑 𝐜𝐨𝐬 𝜽 𝒚=𝟒 𝐬𝐢𝐧 𝜽 at 𝟎, 𝟐𝝅 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

§10.2: Parametric Equations and Eliminating the Parameter Example 3 Determine the rectangular equation and graph the curve given by 𝒙=𝟑 𝐜𝐨𝐬 𝜽 𝒚=𝟒 𝐬𝐢𝐧 𝜽 at 𝟎, 𝟐𝝅 𝜽 𝒙 𝒚 𝟎 /𝟐  𝟑/𝟐 𝟐 𝟑 𝟎 −𝟑 𝟎 𝟒 −𝟒 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

§10.2: Parametric Equations and Eliminating the Parameter Your Turn Determine the rectangular equation and graph the curve given by 𝒙=𝟑+𝟐𝐜𝐨𝐬𝒕 𝒚=−𝟏+𝟑𝐬𝐢𝐧𝒕 at 𝟎,𝟐𝝅 𝒕 𝒙 𝒚 𝟎 /𝟐  𝟑/𝟐 𝟐 𝟓 𝟑 𝟏 −𝟏 𝟐 −𝟒 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

AP Multiple Choice Practice Question 1 (non-calculator) The asymptotes of the graph of the parametric equations, 𝒙= 𝟏 𝒕 and 𝒚= 𝒕 𝒕+𝟏 are (A) 𝒙=𝟎 and 𝒚=𝟎 (B) 𝒙=−𝟏 and 𝒚=𝟎 (C) 𝒙=𝟎 and 𝒚=−𝟏 (D) 𝒙=−𝟏 and 𝒚=−𝟏 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

AP Multiple Choice Practice Question 1 (non-calculator) The asymptotes of the graph of the parametric equations, 𝒙= 𝟏 𝒕 and 𝒚= 𝒕 𝒕+𝟏 are Vocabulary Connections and Process Answer 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter

§10.2: Parametric Equations and Eliminating the Parameter Assignment Page 703 1-19 odd, 53-55 all 6/2/2019 9:22 PM §10.2: Parametric Equations and Eliminating the Parameter