1. Parametric Equations and Eliminating the Parameter
Section 10.2
Calculus BC AP/Dual, Revised ©2018
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§10.2: Parametric Equations and Eliminating the Parameter

2. §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: million miles] What is maximum and minimum speed of this object? [Answer: and million miles per year, respectively]
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§10.2: Parametric Equations and Eliminating the Parameter

3. §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.
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§10.2: Parametric Equations and Eliminating the Parameter

4. §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.
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§10.2: Parametric Equations and Eliminating the Parameter

5. §10.2: Parametric Equations and Eliminating the Parameter
Visual Example
Example of 𝒚=− 𝒙 𝟐 𝟕𝟐 +𝒙
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§10.2: Parametric Equations and Eliminating the Parameter

6. §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.
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§10.2: Parametric Equations and Eliminating the Parameter

7. §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
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§10.2: Parametric Equations and Eliminating the Parameter

8. §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
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§10.2: Parametric Equations and Eliminating the Parameter

9. §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
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§10.2: Parametric Equations and Eliminating the Parameter

10. 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
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§10.2: Parametric Equations and Eliminating the Parameter

11. §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
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§10.2: Parametric Equations and Eliminating the Parameter

12. §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
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§10.2: Parametric Equations and Eliminating the Parameter

13. §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
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§10.2: Parametric Equations and Eliminating the Parameter

14. §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
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§10.2: Parametric Equations and Eliminating the Parameter

15. §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
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§10.2: Parametric Equations and Eliminating the Parameter

16. §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
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§10.2: Parametric Equations and Eliminating the Parameter

17. §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
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§10.2: Parametric Equations and Eliminating the Parameter

18. 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
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§10.2: Parametric Equations and Eliminating the Parameter

19. 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
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§10.2: Parametric Equations and Eliminating the Parameter

20. §10.2: Parametric Equations and Eliminating the Parameter
Assignment
Page odd, all
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§10.2: Parametric Equations and Eliminating the Parameter