Precalculus PreAP/Dual, Revised ©2017 §10.6A: Parametric Functions Section 10.6A Precalculus PreAP/Dual, Revised ©2017 viet.dang@humbleisd.net 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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. 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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. 2/18/2019 6:44 PM §10.6A: Parametric Functions

§10.6A: Parametric Functions Visual Example Example of 𝒚=− 𝒙 𝟐 𝟕𝟐 +𝒙 2/18/2019 6:44 PM §10.6A: Parametric Functions

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? 2/18/2019 6:44 PM §10.6A: Parametric Functions

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? 2/18/2019 6:44 PM §10.6A: Parametric Functions

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) 2/18/2019 6:44 PM §10.6A: Parametric Functions

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? 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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. 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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 2/18/2019 6:44 PM §10.6A: Parametric Functions

§10.6A: Parametric Functions Example 1 (Calc.) Graph the curve given by 𝒙=𝟏−𝟐𝒕 𝒚=𝟐−𝒕 from −𝟑≤𝒕≤𝟐 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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 2/18/2019 6:44 PM §10.6A: Parametric Functions

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 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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 2/18/2019 6:44 PM §10.6A: Parametric Functions

§10.6A: Parametric Functions Example 3 Determine the rectangular equation and graph the curve given by 𝒙= 𝟏 𝒕+𝟏 𝒚= 𝒕 𝒕+𝟏 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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) 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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. 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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. 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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. 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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. 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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. 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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? 2/18/2019 6:44 PM §10.6A: Parametric Functions

§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. 2/18/2019 6:44 PM §10.6A: Parametric Functions

§10.6A: Parametric Functions Assignment Worksheet 2/18/2019 6:44 PM §10.6A: Parametric Functions