1. Precalculus PreAP/Dual, Revised ©2017 §10.6A: Parametric Functions
Section 10.6A
Precalculus PreAP/Dual, Revised ©2017
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§10.6A: Parametric Functions

2. §10.6A: Parametric Functions
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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§10.6A: Parametric Functions

3. §10.6A: Parametric Functions
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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§10.6A: Parametric Functions

4. §10.6A: Parametric Functions
Visual Example
Example of 𝒚=− 𝒙 𝟐 𝟕𝟐 +𝒙
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§10.6A: Parametric Functions

5. Parametric Applications of Projectile Simulation
Suppose two bugs are crawling along linear paths. Bug 1 begins a trek towards a point of 70 inches from where he begins, traveling at a speed of 12 inches per hour. Bug 2 travels at a speed of 18 inches per hour but leaves 1 hour after the other bug from a similar starting position on a parallel path. Note: distance = rate • time (𝑫=𝑹•𝑻) Given that 𝑻 represents Bug 1’s travel time, what formulas represent the distance for each bug travels over time? Distance Bug 1 = ___________ (Rate) • ___________ (Time) Distance Bug 2 = ___________ (Rate) • ___________ (Time) Question: Which bug do you think will win the race? Why?
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§10.6A: Parametric Functions

6. Graphing Calculator Simulation
Suppose two bugs are crawling along linear paths. Bug 1 begins a trek towards a point of 70 inches from where he begins, traveling at a speed of 12 inches per hour. Bug 2 travels at a speed of 18 inches per hour but leaves 1 hour after the other bug from a similar starting position on a parallel path. Note: distance = rate • time (𝑫=𝑹•𝑻) Question: At what time are the buys the same distance from the starting points along from their paths? In other words, when are the bugs alongside each other?
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§10.6A: Parametric Functions

7. Graphing Calculator Simulation
Suppose two bugs are crawling along linear paths. Bug 1 begins a trek towards a point of 70 inches from where he begins, traveling at a speed of 12 inches per hour. Bug 2 travels at a speed of 18 inches per hour but leaves 1 hour after the other bug from a similar starting position on a parallel path. Note: distance = rate • time (𝑫=𝑹•𝑻)
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§10.6A: Parametric Functions

8. Graphing Calculator Simulation
Suppose two bugs are crawling along linear paths. Bug 1 begins a trek towards a point of 70 inches from where he begins, traveling at a speed of 12 inches per hour. Bug 2 travels at a speed of 18 inches per hour but leaves 1 hour after the other buy from a similar starting position on a parallel path. Note: distance = rate • time (𝑫=𝑹•𝑻) Let’s watch the race. Which bug wins the race?
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§10.6A: Parametric Functions

9. §10.6A: Parametric Functions
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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§10.6A: Parametric Functions

10. §10.6A: Parametric Functions
Example 1
Graph the curve given by 𝒙=𝟏−𝟐𝒕 𝒚=𝟐−𝒕 from −𝟑≤𝒕≤𝟐 𝒕 𝒙 𝒚 −𝟑 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟕 𝟓 𝟑 𝟏 −𝟏 −𝟑 𝟓 𝟒 𝟑 𝟐 𝟏 𝟎
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§10.6A: Parametric Functions

11. §10.6A: Parametric Functions
Example 1 (Calc.)
Graph the curve given by 𝒙=𝟏−𝟐𝒕 𝒚=𝟐−𝒕 from −𝟑≤𝒕≤𝟐
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§10.6A: Parametric Functions

12. §10.6A: Parametric Functions
Example 2
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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§10.6A: Parametric Functions

13. §10.6A: Parametric Functions
Example 2
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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§10.6A: Parametric Functions

14. §10.6A: Parametric Functions
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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§10.6A: Parametric Functions

15. 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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§10.6A: Parametric Functions

16. §10.6A: Parametric Functions
In General…
If only 𝒕 is involved, it is a line
If 𝒕 𝟐 is involved, it is a parabola
If 𝐜𝐨𝐬 𝜽 or 𝐬𝐢𝐧 𝜽 in involved, it is a circle or an ellipse
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§10.6A: Parametric Functions

17. §10.6A: Parametric Functions
Example 3
Determine the rectangular equation and graph the curve given by 𝒙= 𝟏 𝒕+𝟏 𝒚= 𝒕 𝒕+𝟏
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§10.6A: Parametric Functions

18. §10.6A: Parametric Functions
Example 3
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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§10.6A: Parametric Functions

19. §10.6A: Parametric Functions
Your Turn
Determine the rectangular equation and graph the curve given by 𝒙= 𝒕 𝟑 +𝟐 𝒚= 𝒕 𝟐 −𝟏 from [−𝟐,𝟐] 𝒕 𝒙 𝒚 −𝟐 −𝟏 𝟎 𝟏 𝟐 −𝟔 𝟑 𝟏 𝟎 𝟐 −𝟏 𝟏𝟎
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§10.6A: Parametric Functions

20. §10.6A: Parametric Functions
Example 4
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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§10.6A: Parametric Functions

21. §10.6A: Parametric Functions
Example 4
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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§10.6A: Parametric Functions

22. §10.6A: Parametric Functions
Example 4
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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§10.6A: Parametric Functions

23. §10.6A: Parametric Functions
Example 5
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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§10.6A: Parametric Functions

24. §10.6A: Parametric Functions
Your Turn
Determine the rectangular equation and graph the curve given by 𝒙=𝟖 𝐜𝐨𝐬 𝜽 𝒚=𝟒 𝐬𝐢𝐧 𝜽 at 𝟎, 𝟐𝝅
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§10.6A: Parametric Functions

25. §10.6A: Parametric Functions
Projectile Motion
Newton’s laws and advanced mathematics can be used to determine the path of a projectile.
𝑽 𝟎 is the initial speed of the projectile at an angle 𝜽 with the horizontal and 𝑨 𝟎 is the initial altitude of the projectile
Equations
𝒙= 𝑽 𝟎 𝐜𝐨𝐬 𝜽 𝒕
𝒚=−𝟒.𝟗 𝒕 𝟐 + 𝑽 𝟎 𝐬𝐢𝐧 𝜽 𝒕+ 𝑨 𝟎 (meters/sec)
𝒚=−𝟏𝟔 𝒕 𝟐 + 𝑽 𝟎 𝐬𝐢𝐧 𝜽 𝒕+ 𝑨 𝟎 (feet/sec)
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§10.6A: Parametric Functions

26. §10.6A: Parametric Functions
Example 6
In a pumpkin tossing contest in Morton, Illinois, a contestant won the catapult competition by using two telephone poles, huge rubber bands, and a power winch. Suppose the pumpkin was launched with an initial speed of 125 feet per second, at an angle of 𝟒𝟓°, and from an initial height of 25 feet.
Write a set of parametric equations for the motion of the pumpkin.
Use the equations to find how far the pumpkin traveled.
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§10.6A: Parametric Functions

27. §10.6A: Parametric Functions
Example 6a
In a pumpkin tossing contest in Morton, Illinois, a contestant won the catapult competition by using two telephone poles, huge rubber bands, and a power winch. Suppose the pumpkin was launched with an initial speed of 125 feet per second, at an angle of 𝟒𝟓°, and from an initial height of 25 feet. (a) Write a set of parametric equations for the motion of the pumpkin.
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§10.6A: Parametric Functions

28. §10.6A: Parametric Functions
Example 6b
In a pumpkin tossing contest in Morton, Illinois, a contestant won the catapult competition by using two telephone poles, huge rubber bands, and a power winch. Suppose the pumpkin was launched with an initial speed of 125 feet per second, at an angle of 𝟒𝟓°, and from an initial height of 25 feet. (b) Use the equations to find how far the pumpkin traveled.
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§10.6A: Parametric Functions

29. §10.6A: Parametric Functions
Your Turn
Suppose that Joe hit a golf ball with an initial velocity of 150 feet per second at an angle of 30 degrees to the horizontal. (a) Find parametric equations that describe the position of the ball as a function of time. (b) How long is the golf ball in the air? (c) Determine the distance that the ball traveled.
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§10.6A: Parametric Functions

30. §10.6A: Parametric Functions
Your Turn
Suppose that Joe hit a golf ball with an initial velocity of 150 feet per second at an angle of 30 degrees to the horizontal. (a) Find parametric equations that describe the position of the ball as a function of time.
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§10.6A: Parametric Functions

31. §10.6A: Parametric Functions
Your Turn
Suppose that Joe hit a golf ball with an initial velocity of 150 feet per second at an angle of 30 degrees to the horizontal. (b) How long is the golf ball in the air?
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§10.6A: Parametric Functions

32. §10.6A: Parametric Functions
Your Turn
Suppose that Joe hit a golf ball with an initial velocity of 150 feet per second at an angle of 30 degrees to the horizontal. (c) Determine the distance that the ball traveled.
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§10.6A: Parametric Functions

33. §10.6A: Parametric Functions
Assignment
Worksheet
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§10.6A: Parametric Functions